Frobenius endomorphism

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</ul> </li> <li id="toc-As_a_generator_of_Galois_groups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#As_a_generator_of_Galois_groups"> <div class="vector-toc-text"> 3 As a generator of Galois groups </div> </a> <ul id="toc-As_a_generator_of_Galois_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frobenius_for_schemes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Frobenius_for_schemes"> <div class="vector-toc-text"> 4 Frobenius for schemes </div> </a> <button aria-controls="toc-Frobenius_for_schemes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Frobenius for schemes subsection </button> <ul id="toc-Frobenius_for_schemes-sublist" class="vector-toc-list"> <li id="toc-The_absolute_Frobenius_morphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_absolute_Frobenius_morphism"> <div class="vector-toc-text"> 4.1 The absolute Frobenius morphism </div> </a> <ul id="toc-The_absolute_Frobenius_morphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Restriction_and_extension_of_scalars_by_Frobenius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Restriction_and_extension_of_scalars_by_Frobenius"> <div class="vector-toc-text"> 4.2 Restriction and extension of scalars by Frobenius </div> </a> <ul id="toc-Restriction_and_extension_of_scalars_by_Frobenius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relative_Frobenius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relative_Frobenius"> <div class="vector-toc-text"> 4.3 Relative Frobenius </div> </a> <ul id="toc-Relative_Frobenius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_Frobenius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic_Frobenius"> <div class="vector-toc-text"> 4.4 Arithmetic Frobenius </div> </a> <ul id="toc-Arithmetic_Frobenius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_Frobenius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_Frobenius"> <div class="vector-toc-text"> 4.5 Geometric Frobenius </div> </a> <ul id="toc-Geometric_Frobenius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_and_geometric_Frobenius_as_Galois_actions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic_and_geometric_Frobenius_as_Galois_actions"> <div class="vector-toc-text"> 4.6 Arithmetic and geometric Frobenius as Galois actions </div> </a> <ul id="toc-Arithmetic_and_geometric_Frobenius_as_Galois_actions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Frobenius_for_local_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Frobenius_for_local_fields"> <div class="vector-toc-text"> 5 Frobenius for local fields </div> </a> <ul id="toc-Frobenius_for_local_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frobenius_for_global_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Frobenius_for_global_fields"> <div class="vector-toc-text"> 6 Frobenius for global fields </div> </a> <ul id="toc-Frobenius_for_global_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 7 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label 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data-title="Endomorphisme de Frobenius" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%84%EB%A1%9C%EB%B2%A0%EB%8B%88%EC%9A%B0%EC%8A%A4_%EC%82%AC%EC%83%81" title="프로베니우스 사상 – Korean" lang="ko" hreflang="ko" data-title="프로베니우스 사상" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Endomorfismo_di_Frobenius" title="Endomorfismo di Frobenius – Italian" lang="it" hreflang="it" data-title="Endomorfismo di Frobenius" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a 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Find sources: <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Frobenius+endomorphism%22">"Frobenius endomorphism"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Frobenius+endomorphism%22+-wikipedia&tbs=ar:1">news</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Frobenius+endomorphism%22&tbs=bkt:s&tbm=bks">newspapers</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Frobenius+endomorphism%22+-wikipedia">books</a> · <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Frobenius+endomorphism%22">scholar</a> · <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Frobenius+endomorphism%22&acc=on&wc=on">JSTOR</a> (November 2013) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> In <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a> and <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, the Frobenius endomorphism (after <a href="/wiki/Ferdinand_Georg_Frobenius" title="Ferdinand Georg Frobenius">Ferdinand Georg Frobenius</a>) is a special <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> of <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> with <a href="/wiki/Prime_number" title="Prime number">prime</a> <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> p, an important class that includes <a href="/wiki/Finite_fields" class="mw-redirect" title="Finite fields">finite fields</a>. The endomorphism maps every element to its p-th power. In certain contexts it is an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>, but this is not true in general. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> Let R be a commutative ring with prime characteristic p (an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism F is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(r)=r^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(r)=r^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5188b35673c10ffd81bbd1abda0ae414336cd1bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.805ex; height:2.843ex;" alt="{\displaystyle F(r)=r^{p}}"></dd></dl> for all r in R. It respects the multiplication of R: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(rs)=(rs)^{p}=r^{p}s^{p}=F(r)F(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(rs)=(rs)^{p}=r^{p}s^{p}=F(r)F(s),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9153e28d363e8e373dd1e063a9f45355e384682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.136ex; height:2.843ex;" alt="{\displaystyle F(rs)=(rs)^{p}=r^{p}s^{p}=F(r)F(s),}"></dd></dl> and F(1) is 1 as well. Moreover, it also respects the addition of R. The expression (r + s)p can be expanded using the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>. Because p is prime, it divides p! but not any q! for q < p; it therefore will divide the <a href="/wiki/Numerator" class="mw-redirect" title="Numerator">numerator</a>, but not the <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a>, of the explicit formula of the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p!}{k!(p-k)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p!}{k!(p-k)!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c6b5c1ff638e50e2a82e3b98fcd7644cb1864c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.018ex; height:6.176ex;" alt="{\displaystyle {\frac {p!}{k!(p-k)!}},}"></dd></dl> if 1 ≤ k ≤ p − 1. Therefore, the coefficients of all the terms except rp and sp are divisible by p, and hence they vanish.<a href="#cite_note-1">[1]</a> Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(r+s)=(r+s)^{p}=r^{p}+s^{p}=F(r)+F(s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(r+s)=(r+s)^{p}=r^{p}+s^{p}=F(r)+F(s).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641ed196e2f783f1597e197be658faed7ccdb64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.497ex; height:2.843ex;" alt="{\displaystyle F(r+s)=(r+s)^{p}=r^{p}+s^{p}=F(r)+F(s).}"></dd></dl> This shows that F is a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a>. If φ : R → S is a homomorphism of rings of characteristic p, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x^{p})=\varphi (x)^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x^{p})=\varphi (x)^{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f7c3d42d276b667d6479744d2aa24ae7fa7d0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.182ex; height:2.843ex;" alt="{\displaystyle \varphi (x^{p})=\varphi (x)^{p}.}"></dd></dl> If FR and FS are the Frobenius endomorphisms of R and S, then this can be rewritten as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ F_{R}=F_{S}\circ \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∘</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>∘</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \circ F_{R}=F_{S}\circ \varphi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251251055a7d81da813a980d708e95e6a2118930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.936ex; height:2.676ex;" alt="{\displaystyle \varphi \circ F_{R}=F_{S}\circ \varphi .}"></dd></dl> This means that the Frobenius endomorphism is a <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformation</a> from the identity <a href="/wiki/Functor" title="Functor">functor</a> on the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of characteristic p rings to itself. If the ring R is a ring with no <a href="/wiki/Nilpotent" title="Nilpotent">nilpotent</a> elements, then the Frobenius endomorphism is <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>: F(r) = 0 means rp = 0, which by definition means that r is nilpotent of order at most p. In fact, this is necessary and sufficient, because if r is any nilpotent, then one of its powers will be nilpotent of order at most p. In particular, if R is a field then the Frobenius endomorphism is injective. The Frobenius morphism is not necessarily <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>, even when R is a field. For example, let K = Fp(t) be the finite field of p elements together with a single <a href="/wiki/Transcendental_element" class="mw-redirect" title="Transcendental element">transcendental element</a>; equivalently, K is the field of <a href="/wiki/Rational_function" title="Rational function">rational functions</a> with coefficients in Fp. Then the image of F does not contain t. If it did, then there would be a rational function q(t)/r(t) whose p-th power q(t)p/r(t)p would equal t. But the degree of this p-th power (the difference between the degrees of its numerator and denominator) is p deg(q) − p deg(r), which is a multiple of p. In particular, it can't be 1, which is the degree of t. This is a contradiction; so t is not in the image of F. A field K is called <a href="/wiki/Perfect_field" title="Perfect field">perfect</a> if either it is of characteristic zero or it is of positive characteristic and its Frobenius endomorphism is an automorphism. For example, all finite fields are perfect. <div class="mw-heading mw-heading2"><h2 id="Fixed_points_of_the_Frobenius_endomorphism">Fixed points of the Frobenius endomorphism</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=2" title="Edit section: Fixed points of the Frobenius endomorphism">edit</a>]</div> Consider the finite field Fp. By <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>, every element x of Fp satisfies xp = x. Equivalently, it is a root of the polynomial Xp − X. The elements of Fp therefore determine p roots of this equation, and because this equation has degree p it has no more than p roots over any <a href="/wiki/Field_extension" title="Field extension">extension</a>. In particular, if K is an algebraic extension of Fp (such as the algebraic closure or another finite field), then Fp is the fixed field of the Frobenius automorphism of K. Let R be a ring of characteristic p > 0. If R is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field. However, if R is not a domain, then Xp − X may have more than p roots; for example, this happens if R = Fp × Fp. A similar property is enjoyed on the finite field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9c8fdcca891dcfe7fbeb429813af2eae3bc63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.707ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{p^{n}}}"> by the nth iterate of the Frobenius automorphism: Every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9c8fdcca891dcfe7fbeb429813af2eae3bc63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.707ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{p^{n}}}"> is a root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{p^{n}}-X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{p^{n}}-X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26c5444ae298745e11c69beb75e5308379f41ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.841ex; height:2.843ex;" alt="{\displaystyle X^{p^{n}}-X}">, so if K is an algebraic extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9c8fdcca891dcfe7fbeb429813af2eae3bc63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.707ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{p^{n}}}"> and F is the Frobenius automorphism of K, then the fixed field of Fn is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9c8fdcca891dcfe7fbeb429813af2eae3bc63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.707ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{p^{n}}}">. If R is a domain that is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9c8fdcca891dcfe7fbeb429813af2eae3bc63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.707ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{p^{n}}}">-algebra, then the fixed points of the nth iterate of Frobenius are the elements of the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9c8fdcca891dcfe7fbeb429813af2eae3bc63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.707ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{p^{n}}}">. Iterating the Frobenius map gives a sequence of elements in R: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,x^{p},x^{p^{2}},x^{p^{3}},\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,x^{p},x^{p^{2}},x^{p^{3}},\ldots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/018faa04be1f4e613227b328e35c6074d77cb274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.052ex; height:3.343ex;" alt="{\displaystyle x,x^{p},x^{p^{2}},x^{p^{3}},\ldots .}"></dd></dl> This sequence of iterates is used in defining the <a href="/w/index.php?title=Frobenius_closure&action=edit&redlink=1" class="new" title="Frobenius closure (page does not exist)">Frobenius closure</a> and the <a href="/wiki/Tight_closure" title="Tight closure">tight closure</a> of an ideal. <div class="mw-heading mw-heading2"><h2 id="As_a_generator_of_Galois_groups">As a generator of Galois groups</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=3" title="Edit section: As a generator of Galois groups">edit</a>]</div> The <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of an extension of finite fields is generated by an iterate of the Frobenius automorphism. First, consider the case where the ground field is the prime field Fp. Let Fq be the finite field of q elements, where q = pn. The Frobenius automorphism F of Fq fixes the prime field Fp, so it is an element of the Galois group Gal(Fq/Fp). In fact, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{q}^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{q}^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c98903cb1b9f77d79d3c62b5dd800008fe2685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.194ex; height:3.176ex;" alt="{\displaystyle \mathbf {F} _{q}^{\times }}"> is <a href="/wiki/Finite_field#Multiplicative_structure" title="Finite field">cyclic with q − 1 elements</a>, we know that the Galois group is cyclic and F is a generator. The order of F is n because Fj acts on an element x by sending it to xpj, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{p^{j}}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{p^{j}}=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19bca1b1e2f829546a1f06c0701a1bd4c3bb2612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.531ex; height:3.009ex;" alt="{\displaystyle x^{p^{j}}=x}"> can only have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e79163ba3c2248a6275212b1fdb7ce321e682c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.169ex; height:3.009ex;" alt="{\displaystyle p^{j}}"> many roots, since we are in a field. Every automorphism of Fq is a power of F, and the generators are the powers Fi with i coprime to n. Now consider the finite field Fqf as an extension of Fq, where q = pn as above. If n > 1, then the Frobenius automorphism F of Fqf does not fix the ground field Fq, but its nth iterate Fn does. The Galois group Gal(Fqf /Fq) is cyclic of order f and is generated by Fn. It is the subgroup of Gal(Fqf /Fp) generated by Fn. The generators of Gal(Fqf /Fq) are the powers Fni where i is coprime to f. The Frobenius automorphism is not a generator of the <a href="/wiki/Absolute_Galois_group" title="Absolute Galois group">absolute Galois group</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Gal} \left({\overline {\mathbf {F} _{q}}}/\mathbf {F} _{q}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Gal</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Gal} \left({\overline {\mathbf {F} _{q}}}/\mathbf {F} _{q}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b629a5638e009559e4f7ece79670599e66f153f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.676ex; height:4.843ex;" alt="{\displaystyle \operatorname {Gal} \left({\overline {\mathbf {F} _{q}}}/\mathbf {F} _{q}\right),}"></dd></dl> because this Galois group is isomorphic to the <a href="/wiki/Profinite_integer" title="Profinite integer">profinite integers</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathbf {Z} }}=\varprojlim _{n}\mathbf {Z} /n\mathbf {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>←</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbf {Z} }}=\varprojlim _{n}\mathbf {Z} /n\mathbf {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9011f66c715c7e30e8ec64850c794c601fa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.016ex; height:6.009ex;" alt="{\displaystyle {\widehat {\mathbf {Z} }}=\varprojlim _{n}\mathbf {Z} /n\mathbf {Z} ,}"></dd></dl> which are not cyclic. However, because the Frobenius automorphism is a generator of the Galois group of every finite extension of Fq, it is a generator of every finite quotient of the absolute Galois group. Consequently, it is a topological generator in the usual Krull topology on the absolute Galois group. <div class="mw-heading mw-heading2"><h2 id="Frobenius_for_schemes">Frobenius for schemes</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=4" title="Edit section: Frobenius for schemes">edit</a>]</div> There are several different ways to define the Frobenius morphism for a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>. The most fundamental is the absolute Frobenius morphism. However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme. There are several different ways of adapting the Frobenius morphism to the relative situation, each of which is useful in certain situations. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Absolute_and_relative_Frobenius.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Absolute_and_relative_Frobenius.svg/220px-Absolute_and_relative_Frobenius.svg.png" decoding="async" width="220" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Absolute_and_relative_Frobenius.svg/330px-Absolute_and_relative_Frobenius.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Absolute_and_relative_Frobenius.svg/440px-Absolute_and_relative_Frobenius.svg.png 2x" data-file-width="138" data-file-height="139" /></a><figcaption>Let φ : X → S be a morphism of schemes, and denote the absolute Frobenius morphisms of S and X by FS and FX, respectively. Define X(p) to be the base change of X by FS. Then the above diagram commutes and the square is <a href="/wiki/Cartesian_square_(category_theory)" class="mw-redirect" title="Cartesian square (category theory)">Cartesian</a>. The morphism FX/S is relative Frobenius.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="The_absolute_Frobenius_morphism">The absolute Frobenius morphism</h3>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=5" title="Edit section: The absolute Frobenius morphism">edit</a>]</div> Suppose that X is a scheme of characteristic p > 0. Choose an open affine subset U = Spec A of X. The ring A is an Fp-algebra, so it admits a Frobenius endomorphism. If V is an open affine subset of U, then by the naturality of Frobenius, the Frobenius morphism on U, when restricted to V, is the Frobenius morphism on V. Consequently, the Frobenius morphism glues to give an endomorphism of X. This endomorphism is called the absolute Frobenius morphism of X, denoted FX. By definition, it is a homeomorphism of X with itself. The absolute Frobenius morphism is a natural transformation from the identity functor on the category of Fp-schemes to itself. If X is an S-scheme and the Frobenius morphism of S is the identity, then the absolute Frobenius morphism is a morphism of S-schemes. In general, however, it is not. For example, consider the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\mathbf {F} _{p^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\mathbf {F} _{p^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26041a306d06db346f90eefd91a898a2040bfe99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.415ex; height:3.009ex;" alt="{\displaystyle A=\mathbf {F} _{p^{2}}}">. Let X and S both equal Spec A with the structure map X → S being the identity. The Frobenius morphism on A sends a to ap. It is not a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5d1b9f50001dd9cc15cd86d77ac5c76648f26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.573ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{p^{2}}}">-algebras. If it were, then multiplying by an element b in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5d1b9f50001dd9cc15cd86d77ac5c76648f26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.573ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{p^{2}}}"> would commute with applying the Frobenius endomorphism. But this is not true because: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\cdot a=ba\neq F(b)\cdot a=b^{p}a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>≠</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\cdot a=ba\neq F(b)\cdot a=b^{p}a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4f06030152d5561ed0604e6740c33851e2605a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.819ex; height:2.843ex;" alt="{\displaystyle b\cdot a=ba\neq F(b)\cdot a=b^{p}a.}"></dd></dl> The former is the action of b in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5d1b9f50001dd9cc15cd86d77ac5c76648f26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.573ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{p^{2}}}">-algebra structure that A begins with, and the latter is the action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5d1b9f50001dd9cc15cd86d77ac5c76648f26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.573ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{p^{2}}}"> induced by Frobenius. Consequently, the Frobenius morphism on Spec A is not a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{p^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{p^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5d1b9f50001dd9cc15cd86d77ac5c76648f26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.573ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{p^{2}}}">-schemes. The absolute Frobenius morphism is a purely inseparable morphism of degree p. Its differential is zero. It preserves products, meaning that for any two schemes X and Y, FX×Y = FX × FY. <div class="mw-heading mw-heading3"><h3 id="Restriction_and_extension_of_scalars_by_Frobenius">Restriction and extension of scalars by Frobenius</h3>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=6" title="Edit section: Restriction and extension of scalars by Frobenius">edit</a>]</div> Suppose that φ : X → S is the structure morphism for an S-scheme X. The base scheme S has a Frobenius morphism FS. Composing φ with FS results in an S-scheme XF called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an S-morphism X → Y induces an S-morphism XF → YF. For example, consider a ring A of characteristic p > 0 and a finitely presented algebra over A: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e00f8d177d656b5de83ecac70c3cc28c75eadd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.003ex; height:2.843ex;" alt="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}).}"></dd></dl> The action of A on R is given by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\cdot \sum a_{\alpha }X^{\alpha }=\sum ca_{\alpha }X^{\alpha },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>⋅</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mo>∑</mo> <mi>c</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\cdot \sum a_{\alpha }X^{\alpha }=\sum ca_{\alpha }X^{\alpha },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fb6ab5053f8b02953c47d0831d9e37af0bece2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.512ex; height:3.843ex;" alt="{\displaystyle c\cdot \sum a_{\alpha }X^{\alpha }=\sum ca_{\alpha }X^{\alpha },}"></dd></dl> where α is a multi-index. Let X = Spec R. Then XF is the affine scheme Spec R, but its structure morphism Spec R → Spec A, and hence the action of A on R, is different: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\cdot \sum a_{\alpha }X^{\alpha }=\sum F(c)a_{\alpha }X^{\alpha }=\sum c^{p}a_{\alpha }X^{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>⋅</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mo>∑</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mo>∑</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\cdot \sum a_{\alpha }X^{\alpha }=\sum F(c)a_{\alpha }X^{\alpha }=\sum c^{p}a_{\alpha }X^{\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a40944808365c50929d332ac2fb1f93d9e355c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:44.764ex; height:3.843ex;" alt="{\displaystyle c\cdot \sum a_{\alpha }X^{\alpha }=\sum F(c)a_{\alpha }X^{\alpha }=\sum c^{p}a_{\alpha }X^{\alpha }.}"></dd></dl> Because restriction of scalars by Frobenius is simply composition, many properties of X are inherited by XF under appropriate hypotheses on the Frobenius morphism. For example, if X and SF are both finite type, then so is XF. The extension of scalars by Frobenius is defined to be: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{(p)}=X\times _{S}S_{F}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>X</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{(p)}=X\times _{S}S_{F}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c849b18a55643255b694a795fdd8da7e7d46092d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.082ex; height:3.176ex;" alt="{\displaystyle X^{(p)}=X\times _{S}S_{F}.}"></dd></dl> The projection onto the S factor makes X(p) an S-scheme. If S is not clear from the context, then X(p) is denoted by X(p/S). Like restriction of scalars, extension of scalars is a functor: An S-morphism X → Y determines an S-morphism X(p) → Y(p). As before, consider a ring A and a finitely presented algebra R over A, and again let X = Spec R. Then: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{(p)}=\operatorname {Spec} R\otimes _{A}A_{F}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>R</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{(p)}=\operatorname {Spec} R\otimes _{A}A_{F}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdef1d2599a734889b214645073446f59dec18c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.393ex; height:3.176ex;" alt="{\displaystyle X^{(p)}=\operatorname {Spec} R\otimes _{A}A_{F}.}"></dd></dl> A global section of X(p) is of the form: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}=\sum _{i}\sum _{\alpha }X^{\alpha }\otimes a_{i\alpha }^{p}b_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>⊗</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}=\sum _{i}\sum _{\alpha }X^{\alpha }\otimes a_{i\alpha }^{p}b_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fcef253ca6af43837b1ad1df5a70150013a817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.395ex; height:7.509ex;" alt="{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}=\sum _{i}\sum _{\alpha }X^{\alpha }\otimes a_{i\alpha }^{p}b_{i},}"></dd></dl> where α is a multi-index and every aiα and bi is an element of A. The action of an element c of A on this section is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\cdot \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}=\sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>⋅</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\cdot \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}=\sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de2c9c8e6a74711206d64730df47ad9377828ce9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.769ex; height:7.509ex;" alt="{\displaystyle c\cdot \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}=\sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}c.}"></dd></dl> Consequently, X(p) is isomorphic to: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} A[X_{1},\ldots ,X_{n}]/\left(f_{1}^{(p)},\ldots ,f_{m}^{(p)}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} A[X_{1},\ldots ,X_{n}]/\left(f_{1}^{(p)},\ldots ,f_{m}^{(p)}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27273cdaea62053fdc4ba1ced3dfa23ce11eae76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.228ex; height:4.843ex;" alt="{\displaystyle \operatorname {Spec} A[X_{1},\ldots ,X_{n}]/\left(f_{1}^{(p)},\ldots ,f_{m}^{(p)}\right),}"></dd></dl> where, if: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}=\sum _{\beta }f_{j\beta }X^{\beta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>β</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{j}=\sum _{\beta }f_{j\beta }X^{\beta },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e1813345c6bbfa2ff11d6b6f6722c71b4c9f0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.698ex; height:5.843ex;" alt="{\displaystyle f_{j}=\sum _{\beta }f_{j\beta }X^{\beta },}"></dd></dl> then: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}^{(p)}=\sum _{\beta }f_{j\beta }^{p}X^{\beta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </munder> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{j}^{(p)}=\sum _{\beta }f_{j\beta }^{p}X^{\beta }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00bc38294a6e91bf6af9e89ebc3e8edfcdda8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.308ex; height:6.009ex;" alt="{\displaystyle f_{j}^{(p)}=\sum _{\beta }f_{j\beta }^{p}X^{\beta }.}"></dd></dl> A similar description holds for arbitrary A-algebras R. Because extension of scalars is base change, it preserves limits and coproducts. This implies in particular that if X has an algebraic structure defined in terms of finite limits (such as being a group scheme), then so does X(p). Furthermore, being a base change means that extension of scalars preserves properties such as being of finite type, finite presentation, separated, affine, and so on. Extension of scalars is well-behaved with respect to base change: Given a morphism S′ → S, there is a natural isomorphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{(p/S)}\times _{S}S'\cong (X\times _{S}S')^{(p/S')}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>≅</mo> <mo stretchy="false">(</mo> <mi>X</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <msup> <mi>S</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{(p/S)}\times _{S}S'\cong (X\times _{S}S')^{(p/S')}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cb9be61ca4685916cde65cdc71ff73df23005b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.198ex; height:3.509ex;" alt="{\displaystyle X^{(p/S)}\times _{S}S'\cong (X\times _{S}S')^{(p/S')}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relative_Frobenius">Relative Frobenius</h3>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=7" title="Edit section: Relative Frobenius">edit</a>]</div> Let X be an S-scheme with structure morphism φ. The relative Frobenius morphism of X is the morphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X/S}:X\to X^{(p)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X/S}:X\to X^{(p)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08ffca492309101294a48baae9f4b5cb2624b90f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.876ex; height:3.676ex;" alt="{\displaystyle F_{X/S}:X\to X^{(p)}}"></dd></dl> defined by the universal property of the <a href="/wiki/Pullback_(category_theory)" title="Pullback (category theory)">pullback</a> X(p) (see the diagram above): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X/S}=(F_{X},\varphi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X/S}=(F_{X},\varphi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eaa57d14dc7fc377dbc80dddf754f729fc16f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.245ex; height:3.176ex;" alt="{\displaystyle F_{X/S}=(F_{X},\varphi ).}"></dd></dl> Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of S-schemes. Consider, for example, the A-algebra: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e00f8d177d656b5de83ecac70c3cc28c75eadd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.003ex; height:2.843ex;" alt="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}).}"></dd></dl> We have: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/(f_{1}^{(p)},\ldots ,f_{m}^{(p)}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/(f_{1}^{(p)},\ldots ,f_{m}^{(p)}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5958a3ca8b10c768163b6715285865c6cb7246fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.652ex; height:3.676ex;" alt="{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/(f_{1}^{(p)},\ldots ,f_{m}^{(p)}).}"></dd></dl> The relative Frobenius morphism is the homomorphism R(p) → R defined by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\sum _{\alpha }X^{\alpha }\otimes a_{i\alpha }\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }X^{p\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>⊗</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>α</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\sum _{\alpha }X^{\alpha }\otimes a_{i\alpha }\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }X^{p\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5168e132de25be0cc34836e2a216df07505b714d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.621ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\sum _{\alpha }X^{\alpha }\otimes a_{i\alpha }\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }X^{p\alpha }.}"></dd></dl> Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of X(p/S) ×S S′ and (X ×S S′)(p/S′), we have: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X/S}\times 1_{S'}=F_{X\times _{S}S'/S'}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> </msub> <mo>×</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <msup> <mi>S</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>S</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X/S}\times 1_{S'}=F_{X\times _{S}S'/S'}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5fb93b9f317cfe30bfd88ddffaf62c3bf87d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.064ex; height:3.176ex;" alt="{\displaystyle F_{X/S}\times 1_{S'}=F_{X\times _{S}S'/S'}.}"></dd></dl> Relative Frobenius is a universal homeomorphism. If X → S is an open immersion, then it is the identity. If X → S is a closed immersion determined by an ideal sheaf I of OS, then X(p) is determined by the ideal sheaf Ip and relative Frobenius is the augmentation map OS/Ip → OS/I. X is unramified over S if and only if FX/S is unramified and if and only if FX/S is a monomorphism. X is étale over S if and only if FX/S is étale and if and only if FX/S is an isomorphism. <div class="mw-heading mw-heading3"><h3 id="Arithmetic_Frobenius">Arithmetic Frobenius</h3>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=8" title="Edit section: Arithmetic Frobenius">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Arithmetic_and_geometric_Frobenius" title="Arithmetic and geometric Frobenius">Arithmetic and geometric Frobenius</a></div> The arithmetic Frobenius morphism of an S-scheme X is a morphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X/S}^{a}:X^{(p)}\to X\times _{S}S\cong X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>:</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mi>X</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>S</mi> <mo>≅</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X/S}^{a}:X^{(p)}\to X\times _{S}S\cong X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73916db9d5bf2a450566d4458a560c58efeec9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:27.586ex; height:4.009ex;" alt="{\displaystyle F_{X/S}^{a}:X^{(p)}\to X\times _{S}S\cong X}"></dd></dl> defined by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X/S}^{a}=1_{X}\times F_{S}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X/S}^{a}=1_{X}\times F_{S}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e50e2c687b663b08dc8e9b6f1308ea7b12f9448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:17.176ex; height:3.343ex;" alt="{\displaystyle F_{X/S}^{a}=1_{X}\times F_{S}.}"></dd></dl> That is, it is the base change of FS by 1X. Again, if: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdf109f0aa7fb76d0e194c75ad7a297599d0147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.003ex; height:2.843ex;" alt="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}),}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m})\otimes _{A}A_{F},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m})\otimes _{A}A_{F},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657ae74319024e58a6f6444604319ec06dae2445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.853ex; height:3.343ex;" alt="{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m})\otimes _{A}A_{F},}"></dd></dl> then the arithmetic Frobenius is the homomorphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }b_{i}^{p}X^{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }b_{i}^{p}X^{\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d03a90ce09361ccc6ab839491f219653d88bb730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.33ex; height:7.509ex;" alt="{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }b_{i}^{p}X^{\alpha }.}"></dd></dl> If we rewrite R(p) as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/\left(f_{1}^{(p)},\ldots ,f_{m}^{(p)}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/\left(f_{1}^{(p)},\ldots ,f_{m}^{(p)}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ea54d6240e00724a61bece15bb9944be07a1a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.393ex; height:4.843ex;" alt="{\displaystyle R^{(p)}=A[X_{1},\ldots ,X_{n}]/\left(f_{1}^{(p)},\ldots ,f_{m}^{(p)}\right),}"></dd></dl> then this homomorphism is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum a_{\alpha }X^{\alpha }\mapsto \sum a_{\alpha }^{p}X^{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <mo>∑</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum a_{\alpha }X^{\alpha }\mapsto \sum a_{\alpha }^{p}X^{\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45b852331e52c7cf904d8605a5c0c324991b16d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.335ex; height:3.843ex;" alt="{\displaystyle \sum a_{\alpha }X^{\alpha }\mapsto \sum a_{\alpha }^{p}X^{\alpha }.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Geometric_Frobenius">Geometric Frobenius</h3>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=9" title="Edit section: Geometric Frobenius">edit</a>]</div> Assume that the absolute Frobenius morphism of S is invertible with inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{S}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{S}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8848050832ef234bfdcca3fa28e4e21b9e6da8f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.148ex; height:3.343ex;" alt="{\displaystyle F_{S}^{-1}}">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{F^{-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{F^{-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabd77166954be1cb424c70113f2c7639c2b975e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.81ex; height:2.843ex;" alt="{\displaystyle S_{F^{-1}}}"> denote the S-scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{S}^{-1}:S\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{S}^{-1}:S\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1691c1a0ca9ec1ebab75dd25755c44555c186b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.697ex; height:3.343ex;" alt="{\displaystyle F_{S}^{-1}:S\to S}">. Then there is an extension of scalars of X by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{S}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{S}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8848050832ef234bfdcca3fa28e4e21b9e6da8f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.148ex; height:3.343ex;" alt="{\displaystyle F_{S}^{-1}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{(1/p)}=X\times _{S}S_{F^{-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>X</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{(1/p)}=X\times _{S}S_{F^{-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6969feb02518936cc4863d5a3b438e1cb7f3b238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.647ex; height:3.509ex;" alt="{\displaystyle X^{(1/p)}=X\times _{S}S_{F^{-1}}.}"></dd></dl> If: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdf109f0aa7fb76d0e194c75ad7a297599d0147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.003ex; height:2.843ex;" alt="{\displaystyle R=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m}),}"></dd></dl> then extending scalars by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{S}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{S}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8848050832ef234bfdcca3fa28e4e21b9e6da8f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.148ex; height:3.343ex;" alt="{\displaystyle F_{S}^{-1}}"> gives: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(1/p)}=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m})\otimes _{A}A_{F^{-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{(1/p)}=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m})\otimes _{A}A_{F^{-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91726d2526a3215437beee3866efb8e61bec51a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.419ex; height:3.509ex;" alt="{\displaystyle R^{(1/p)}=A[X_{1},\ldots ,X_{n}]/(f_{1},\ldots ,f_{m})\otimes _{A}A_{F^{-1}}.}"></dd></dl> If: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}=\sum _{\beta }f_{j\beta }X^{\beta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>β</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{j}=\sum _{\beta }f_{j\beta }X^{\beta },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e1813345c6bbfa2ff11d6b6f6722c71b4c9f0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.698ex; height:5.843ex;" alt="{\displaystyle f_{j}=\sum _{\beta }f_{j\beta }X^{\beta },}"></dd></dl> then we write: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}^{(1/p)}=\sum _{\beta }f_{j\beta }^{1/p}X^{\beta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </munder> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msubsup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{j}^{(1/p)}=\sum _{\beta }f_{j\beta }^{1/p}X^{\beta },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee045e08696e464ab2992370611ea909312bcc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.985ex; height:6.009ex;" alt="{\displaystyle f_{j}^{(1/p)}=\sum _{\beta }f_{j\beta }^{1/p}X^{\beta },}"></dd></dl> and then there is an isomorphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(1/p)}\cong A[X_{1},\ldots ,X_{n}]/(f_{1}^{(1/p)},\ldots ,f_{m}^{(1/p)}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≅</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{(1/p)}\cong A[X_{1},\ldots ,X_{n}]/(f_{1}^{(1/p)},\ldots ,f_{m}^{(1/p)}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36dba91f86d3a422bafdceba95027fcf10d5d34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.584ex; height:3.676ex;" alt="{\displaystyle R^{(1/p)}\cong A[X_{1},\ldots ,X_{n}]/(f_{1}^{(1/p)},\ldots ,f_{m}^{(1/p)}).}"></dd></dl> The geometric Frobenius morphism of an S-scheme X is a morphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X/S}^{g}:X^{(1/p)}\to X\times _{S}S\cong X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msubsup> <mo>:</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mi>X</mi> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>S</mi> <mo>≅</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X/S}^{g}:X^{(1/p)}\to X\times _{S}S\cong X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b68874792d098ea95127e9abb61717c32feb8e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:29.23ex; height:4.009ex;" alt="{\displaystyle F_{X/S}^{g}:X^{(1/p)}\to X\times _{S}S\cong X}"></dd></dl> defined by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X/S}^{g}=1_{X}\times F_{S}^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>×</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X/S}^{g}=1_{X}\times F_{S}^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9b1fc4da28d0466951cc33586f1b1daa28270a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.537ex; height:3.843ex;" alt="{\displaystyle F_{X/S}^{g}=1_{X}\times F_{S}^{-1}.}"></dd></dl> It is the base change of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{S}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{S}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8848050832ef234bfdcca3fa28e4e21b9e6da8f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.148ex; height:3.343ex;" alt="{\displaystyle F_{S}^{-1}}"> by 1X. Continuing our example of A and R above, geometric Frobenius is defined to be: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }b_{i}^{1/p}X^{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msub> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msubsup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }b_{i}^{1/p}X^{\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b986e6a34ba362c33d1168c5e663c0136967b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.974ex; height:7.509ex;" alt="{\displaystyle \sum _{i}\left(\sum _{\alpha }a_{i\alpha }X^{\alpha }\right)\otimes b_{i}\mapsto \sum _{i}\sum _{\alpha }a_{i\alpha }b_{i}^{1/p}X^{\alpha }.}"></dd></dl> After rewriting R(1/p) in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{j}^{(1/p)}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f_{j}^{(1/p)}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465fc15df8503caed8d2220804521905c2fd96b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.628ex; height:4.009ex;" alt="{\displaystyle \{f_{j}^{(1/p)}\}}">, geometric Frobenius is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum a_{\alpha }X^{\alpha }\mapsto \sum a_{\alpha }^{1/p}X^{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <mo>∑</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msubsup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum a_{\alpha }X^{\alpha }\mapsto \sum a_{\alpha }^{1/p}X^{\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d815e85d8830253c678a5d22961404929cfe10c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.754ex; height:4.009ex;" alt="{\displaystyle \sum a_{\alpha }X^{\alpha }\mapsto \sum a_{\alpha }^{1/p}X^{\alpha }.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Arithmetic_and_geometric_Frobenius_as_Galois_actions">Arithmetic and geometric Frobenius as Galois actions</h3>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=10" title="Edit section: Arithmetic and geometric Frobenius as Galois actions">edit</a>]</div> Suppose that the Frobenius morphism of S is an isomorphism. Then it generates a subgroup of the automorphism group of S. If S = Spec k is the spectrum of a finite field, then its automorphism group is the Galois group of the field over the prime field, and the Frobenius morphism and its inverse are both generators of the automorphism group. In addition, X(p) and X(1/p) may be identified with X. The arithmetic and geometric Frobenius morphisms are then endomorphisms of X, and so they lead to an action of the Galois group of k on X. Consider the set of K-points X(K). This set comes with a Galois action: Each such point x corresponds to a homomorphism OX → K from the structure sheaf to K, which factors via k(x), the residue field at x, and the action of Frobenius on x is the application of the Frobenius morphism to the residue field. This Galois action agrees with the action of arithmetic Frobenius: The composite morphism <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}\to k(x){\xrightarrow {\overset {}{F}}}k(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mover> <mi>F</mi> <mrow /> </mover> </mpadded> </mover> </mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}\to k(x){\xrightarrow {\overset {}{F}}}k(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/715e23676f866a04b23725d6f3ed990d9e5d122b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.341ex; width:18.351ex; height:4.676ex;" alt="{\displaystyle {\mathcal {O}}_{X}\to k(x){\xrightarrow {\overset {}{F}}}k(x)}"></dd></dl> is the same as the composite morphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}{\xrightarrow {{\overset {}{F}}_{X/S}^{a}}}{\mathcal {O}}_{X}\to k(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mrow /> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mpadded> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}{\xrightarrow {{\overset {}{F}}_{X/S}^{a}}}{\mathcal {O}}_{X}\to k(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c83731f698123c4c03374ee8d2d1be8cca22f4e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.399ex; width:20.312ex; height:5.343ex;" alt="{\displaystyle {\mathcal {O}}_{X}{\xrightarrow {{\overset {}{F}}_{X/S}^{a}}}{\mathcal {O}}_{X}\to k(x)}"></dd></dl> by the definition of the arithmetic Frobenius. Consequently, arithmetic Frobenius explicitly exhibits the action of the Galois group on points as an endomorphism of X. <div class="mw-heading mw-heading2"><h2 id="Frobenius_for_local_fields">Frobenius for local fields</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=11" title="Edit section: Frobenius for local fields">edit</a>]</div> Given an <a href="/wiki/Unramified" class="mw-redirect" title="Unramified">unramified</a> <a href="/wiki/Finite_extension" class="mw-redirect" title="Finite extension">finite extension</a> L/K of <a href="/wiki/Local_field" title="Local field">local fields</a>, there is a concept of Frobenius endomorphism that induces the Frobenius endomorphism in the corresponding extension of <a href="/wiki/Residue_field" title="Residue field">residue fields</a>.<a href="#cite_note-FT144-2">[2]</a> Suppose L/K is an unramified extension of local fields, with <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> OK of K such that the residue field, the integers of K modulo their unique maximal ideal φ, is a finite field of order q, where q is a power of a prime. If Φ is a prime of L lying over φ, that L/K is unramified means by definition that the integers of L modulo Φ, the residue field of L, will be a finite field of order qf extending the residue field of K where f is the degree of L/K. We may define the Frobenius map for elements of the ring of integers OL of L as an automorphism sΦ of L such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\Phi }(x)\equiv x^{q}\mod \Phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Φ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\Phi }(x)\equiv x^{q}\mod \Phi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f933c9bbfd83a1c494c75af5d6e1009c27e691f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.652ex; height:2.843ex;" alt="{\displaystyle s_{\Phi }(x)\equiv x^{q}\mod \Phi .}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Frobenius_for_global_fields">Frobenius for global fields</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=12" title="Edit section: Frobenius for global fields">edit</a>]</div> In <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, Frobenius elements are defined for extensions L/K of <a href="/wiki/Global_field" title="Global field">global fields</a> that are finite <a href="/wiki/Galois_extension" title="Galois extension">Galois extensions</a> for <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> Φ of L that are unramified in L/K. Since the extension is unramified the <a href="/wiki/Decomposition_group" class="mw-redirect" title="Decomposition group">decomposition group</a> of Φ is the Galois group of the extension of residue fields. The Frobenius element then can be defined for elements of the ring of integers of L as in the local case, by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\Phi }(x)\equiv x^{q}\mod \Phi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Φ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\Phi }(x)\equiv x^{q}\mod \Phi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5c8061158e90bf6595716e9384a0c77d05444e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.652ex; height:2.843ex;" alt="{\displaystyle s_{\Phi }(x)\equiv x^{q}\mod \Phi ,}"></dd></dl> where q is the order of the residue field OK/(Φ ∩ OK). Lifts of the Frobenius are in correspondence with <a href="/wiki/P-derivations" class="mw-redirect" title="P-derivations">p-derivations</a>. <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=13" title="Edit section: Examples">edit</a>]</div> The polynomial <dl><dd>x5 − x − 1</dd></dl> has <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> <dl><dd>19 × 151,</dd></dl> and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root ρ of it to the field of 3-adic numbers Q3 gives an unramified extension Q3(ρ) of Q3. We may find the image of ρ under the Frobenius map by locating the root nearest to ρ3, which we may do by <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>. We obtain an element of the ring of integers Z3[ρ] in this way; this is a polynomial of degree four in ρ with coefficients in the 3-adic integers Z3. Modulo 38 this polynomial is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{3}+3(460+183\rho -354\rho ^{2}-979\rho ^{3}-575\rho ^{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mn>460</mn> <mo>+</mo> <mn>183</mn> <mi>ρ</mi> <mo>−</mo> <mn>354</mn> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>979</mn> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>575</mn> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{3}+3(460+183\rho -354\rho ^{2}-979\rho ^{3}-575\rho ^{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98df0a874e890b0d66801b0f37a47c9e642bece5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.837ex; height:3.176ex;" alt="{\displaystyle \rho ^{3}+3(460+183\rho -354\rho ^{2}-979\rho ^{3}-575\rho ^{4})}">.</dd></dl> This is algebraic over Q and is the correct global Frobenius image in terms of the embedding of Q into Q3; moreover, the coefficients are algebraic and the result can be expressed algebraically. However, they are of degree 120, the order of the Galois group, illustrating the fact that explicit computations are much more easily accomplished if p-adic results will suffice. If L/K is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime φ in the base field K. For an example, consider the extension Q(β) of Q obtained by adjoining a root β satisfying <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{5}+\beta ^{4}-4\beta ^{3}-3\beta ^{2}+3\beta +1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>β</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{5}+\beta ^{4}-4\beta ^{3}-3\beta ^{2}+3\beta +1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49853647f41e29294871512d1f2e0df4d60c9ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.01ex; height:3.009ex;" alt="{\displaystyle \beta ^{5}+\beta ^{4}-4\beta ^{3}-3\beta ^{2}+3\beta +1=0}"></dd></dl> to Q. This extension is cyclic of order five, with roots <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cos {\tfrac {2\pi n}{11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> </mrow> <mn>11</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cos {\tfrac {2\pi n}{11}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fab7e4270d2af36d10835973990888d8a01d722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.634ex; height:3.509ex;" alt="{\displaystyle 2\cos {\tfrac {2\pi n}{11}}}"></dd></dl> for integer n. It has roots that are <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a> of β: <dl><dd>β2 − 2, β3 − 3β, β5 − 5β3 + 5β</dd></dl> give the result of the Frobenius map for the primes 2, 3 and 5, and so on for larger primes not equal to 11 or of the form 22n + 1 (which split). It is immediately apparent how the Frobenius map gives a result equal mod p to the p-th power of the root β. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=14" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Perfect_field" title="Perfect field">Perfect field</a></li> <li><a href="/wiki/Frobenioid" title="Frobenioid">Frobenioid</a></li> <li><a href="/wiki/Finite_field#Frobenius_automorphism_and_Galois_theory" title="Finite field">Finite field § Frobenius automorphism and Galois theory</a></li> <li><a href="/wiki/Universal_homeomorphism" title="Universal homeomorphism">Universal homeomorphism</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Frobenius_endomorphism&action=edit&section=15" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> This is known as the <a href="/wiki/Freshman%27s_dream" title="Freshman's dream">freshman's dream</a>. </li> <li id="cite_note-FT144-2"><a href="#cite_ref-FT144_2-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFröhlichTaylor1991" class="citation book cs1"><a href="/wiki/Albrecht_Fr%C3%B6hlich" title="Albrecht Fröhlich">Fröhlich, A.</a>; <a href="/wiki/Martin_J._Taylor" title="Martin J. Taylor">Taylor, M.J.</a> (1991). Algebraic number theory. Cambridge studies in advanced mathematics. Vol. 27. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 144. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-36664-X" title="Special:BookSources/0-521-36664-X"><bdi>0-521-36664-X</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0744.11001">0744.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+number+theory&rft.series=Cambridge+studies+in+advanced+mathematics&rft.pages=144&rft.pub=Cambridge+University+Press&rft.date=1991&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0744.11001%23id-name%3DZbl&rft.isbn=0-521-36664-X&rft.aulast=Fr%C3%B6hlich&rft.aufirst=A.&rft.au=Taylor%2C+M.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrobenius+endomorphism" class="Z3988"> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Frobenius_automorphism">"Frobenius automorphism"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Frobenius+automorphism&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFrobenius_automorphism&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrobenius+endomorphism" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Frobenius_endomorphism">"Frobenius endomorphism"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Frobenius+endomorphism&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFrobenius_endomorphism&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrobenius+endomorphism" class="Z3988"></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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Frobenius endomorphism - Wikipedia