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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="arithmetic_geometry">Arithmetic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+arithmetic+geometry">higher arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/number">number</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/integer+number">integer number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, <a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a>, <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, <a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a>, <a class="existingWikiWord" href="/nlab/show/octonion">octonion</a>, <a class="existingWikiWord" href="/nlab/show/adic+number">adic number</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+number">cardinal number</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal number</a>, <a class="existingWikiWord" href="/nlab/show/surreal+number">surreal number</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/second-order+arithmetic">second-order arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+arithmetic">transfinite arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+arithmetic">cardinal arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+arithmetic">ordinal arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+integer">p-adic integer</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+rational+number">p-adic rational number</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></strong>, <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+scheme">arithmetic scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chow+group">arithmetic Chow group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil-%C3%A9tale+topology+for+arithmetic+schemes">Weil-étale topology for arithmetic schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+cohomology">absolute cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Iwasawa-Tate+theory">Iwasawa-Tate theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adelic+integration">adelic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shtuka">shtuka</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenioid">Frobenioid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Riemann-Roch+theorem">arithmetic Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Motivation'>Motivation</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>One proposal for a precise realization of the idea of “absolute” <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a> over <a class="existingWikiWord" href="/nlab/show/F1">Spec(F1)</a> is <em>Borger’s absolute geometry</em> (<a href="#Borger09">Borger 09</a>). Here the structure of a <a class="existingWikiWord" href="/nlab/show/Lambda-ring">Lambda-ring</a> on a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, hence on its <a class="existingWikiWord" href="/nlab/show/spectrum+of+a+commutative+ring">spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R) \to Spec(\mathbb{Z})</annotation></semantics></math>, is interpreted as a collection of lifts of all <a class="existingWikiWord" href="/nlab/show/Frobenius+morphisms">Frobenius morphisms</a> and hence as <a class="existingWikiWord" href="/nlab/show/descent">descent</a> data for descent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{F}_1)</annotation></semantics></math> (which is defined thereby). This definition yields an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a> of <a class="existingWikiWord" href="/nlab/show/gros+topos">gros</a> <a class="existingWikiWord" href="/nlab/show/etale+toposes">etale toposes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Et</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mover><munder><mo>⟶</mo><mrow></mrow></munder><mover><mo>⟵</mo><mrow></mrow></mover></mover><mover><mo>⟶</mo><mrow></mrow></mover></mover><mi>Et</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Et(Spec(\mathbb{Z})) \stackrel{\overset{}{\longrightarrow}}{\stackrel{\overset{}{\longleftarrow}}{\underset{}{\longrightarrow}}} Et(Spec(\mathbb{F}_1)) \,, </annotation></semantics></math></div> <p>where on the right the notation is just suggestive, the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> is a suitable one over <a class="existingWikiWord" href="/nlab/show/Lambda-rings">Lambda-rings</a>. Here the middle <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> is the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> which forgets the Lambda structure, and its <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> is given by the <a class="existingWikiWord" href="/nlab/show/ring+of+Witt+vectors">ring of Witt vectors</a> construction and may be thought of as producing <a class="existingWikiWord" href="/nlab/show/arithmetic+jet+spaces">arithmetic jet spaces</a>. In this sense the <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> here would be directly analogous to the <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of an <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> whose induced <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> is the <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a>.</p> <p>This proposal seems to subsume many aspects of other existing proposals (see e.g. <a href="#LeBruyn13">Le Bruyn 13</a>) and stands out as yielding an “absolute <a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Et</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Et(Spec(\mathbb{F}_1))</annotation></semantics></math> which is rich and genuinely interesting in its own right.</p> <h2 id="Motivation">Motivation</h2> <p>The following is an attempt to motivate or make intuitively clear why lifts of <a class="existingWikiWord" href="/nlab/show/Frobenius+morphisms">Frobenius morphisms</a> may be related to “absolute geometry” over <a class="existingWikiWord" href="/nlab/show/F1">F1</a>.</p> <p>First of all, the <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a> says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is analogous to the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[z]</annotation></semantics></math> over a finite field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, as well as to the ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{\mathbb{C}}</annotation></semantics></math> of (<a class="existingWikiWord" href="/nlab/show/entire+function">entire</a>) <a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a>.</p> <p>To make this analogy more concrete, notice that one characteristic property of the rings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[z]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{\mathbb{C}}</annotation></semantics></math>, witnessing their affine-ness in one variable, is that they carry a canonical <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial z}</annotation></semantics></math>, and that the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is recovered as the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> by the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>. More to the point, for each <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z-x)</annotation></semantics></math> there is the first order translation operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">id</mi><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathrm{id} + (z-x)\frac{\partial}{\partial (z-x)}</annotation></semantics></math> and the quotient by its difference from the identity is the <a class="existingWikiWord" href="/nlab/show/residue+field">residue field</a> of functions at the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> <p>Therefore if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is analogous to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[z]</annotation></semantics></math> and to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{\mathbb{C}}</annotation></semantics></math>, then it ought to admit analogous operators, one for each of its <a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p)</annotation></semantics></math> given by a <a class="existingWikiWord" href="/nlab/show/prime+number">prime number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">p \in \mathbb{Z}</annotation></semantics></math>. Remarkably, such a collection of operations indeed exists on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>: the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-power operations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>:</mo><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">(-)^p : \mathbb{Z} \to \mathbb{Z}</annotation></semantics></math> (acting on the underlying set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>) which by <a class="existingWikiWord" href="/nlab/show/Fermat%27s+little+theorem">Fermat's little theorem</a> is indeed of the above form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>:</mo><mi>n</mi><mo>↦</mo><msup><mi>n</mi> <mi>p</mi></msup><mo>=</mo><mi>n</mi><mo>+</mo><mi>p</mi><mo>⋅</mo><msub><mo>∂</mo> <mi>p</mi></msub><mi>n</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-)^p : n \mapsto n^p = n + p \cdot\partial_p n \,. </annotation></semantics></math></div> <p>Here the expression <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>p</mi></msub><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\partial_p n \in \mathbb{Z}</annotation></semantics></math> is uniquely defined by this equation, it is given by the <a class="existingWikiWord" href="/nlab/show/Fermat+quotient">Fermat quotient</a> operation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>p</mi></msub><mo>:</mo><mi>n</mi><mo>↦</mo><msub><mo>∂</mo> <mi>p</mi></msub><mi>n</mi><mo>≔</mo><mfrac><mrow><msup><mi>n</mi> <mi>p</mi></msup><mo>−</mo><mi>n</mi></mrow><mi>p</mi></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_p : n \mapsto \partial_p n \coloneqq \frac{n^p - n}{p} \,. </annotation></semantics></math></div> <p>Hence by <a class="existingWikiWord" href="/nlab/show/analogy">analogy</a> it makes sense to think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>p</mi></msub><mo>:</mo><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\partial_p : \mathbb{Z} \to \mathbb{Z}</annotation></semantics></math> as being like a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(\mathbb{Z})</annotation></semantics></math> – it is called a <em><a class="existingWikiWord" href="/nlab/show/p-derivation">p-derivation</a></em>. This is the beginning of the theory of <em><a class="existingWikiWord" href="/nlab/show/arithmetic+differential+equations">arithmetic differential equations</a></em> (<a href="#Buium05">Buium 05</a>).</p> <p>More generally, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[z]</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is to be thought of as a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(A)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(k[z])</annotation></semantics></math>. The canonical <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> on the latter canonically lifts to the former, and is given by the same formula: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">id</mi><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathrm{id} + (z-x)\frac{\partial}{\partial (z-x)}</annotation></semantics></math>. This exhibits the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(A) \to \mathrm{Spec}(k[z])</annotation></semantics></math> is simply a <a class="existingWikiWord" href="/nlab/show/product">product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(A/(z)) \to \mathrm{Spec}(k)</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/affine+line">affine line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔸</mi> <mi>k</mi></msub><mo>=</mo><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{A}_k = \mathrm{Spec}(k[z])</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. Analogously, any <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is to be thought of as a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(R)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(\mathbb{Z})</annotation></semantics></math> and the above arithmetic translation operator canonically lifts to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(R)</annotation></semantics></math>, by the same formula: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>↦</mo><msup><mi>a</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">a \mapsto a^p</annotation></semantics></math>. However, inspecting this one finds that not only is the derivation-like part lifted non-trivially, but also the identity-part is lifted in general to some <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>: in particular if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(R)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/support">supported</a> over the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(\mathbb{F}_p)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(\mathbb{Z})</annotation></semantics></math>, then the standard fact that in <a class="existingWikiWord" href="/nlab/show/positive+characteristic">characteristic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-power operation is a ring homomorphism now means that the derivation-like action vanishes here, as expected from the analogy, but that a pure homomorphism part covering the identity remains – the <a class="existingWikiWord" href="/nlab/show/Frobenius+homomorphism">Frobenius homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">Frob</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{Frob}_p</annotation></semantics></math>. If by some abuse of notation we allow ourselves to write <a class="existingWikiWord" href="/nlab/show/Isbell+duality">formally dual</a> morphisms for maps between rings that are not necessarily homomorphisms, then this situation is, possibly, usefully visualized as follows.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi mathvariant="normal">Frob</mi> <mi>p</mi></msub><mo>+</mo><mn>0</mn></mrow></mover></mtd> <mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi mathvariant="normal">id</mi><mo>+</mo><mn>0</mn></mrow></mover></mtd> <mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi mathvariant="normal">id</mi><mo>+</mo><mi>p</mi><mo>⋅</mo><msub><mo>∂</mo> <mi>p</mi></msub></mrow></mover></mtd> <mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathrm{Spec}(R) &\stackrel{\mathrm{Frob}_p + 0}{\longrightarrow} & \mathrm{Spec}(R) \\ \downarrow && \downarrow \\ \mathrm{Spec}(\mathbb{F}_p) &\stackrel{\mathrm{id} + 0}{\longrightarrow} & \mathrm{Spec}(\mathbb{F}_p) \\ \downarrow && \downarrow \\ \mathrm{Spec}(\mathbb{Z}) &\stackrel{\mathrm{id} + p \cdot \partial_p}{\longrightarrow} & \mathrm{Spec}(\mathbb{Z}) } </annotation></semantics></math></div> <p>This suggests that we are to think of the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>=</mo><mi mathvariant="normal">id</mi><mo>+</mo><mi>p</mi><mo>⋅</mo><msub><mo>∂</mo> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">(-)^p = \mathrm{id} + p \cdot \partial_p</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(\mathbb{Z})</annotation></semantics></math> not just as analogous to the identity transformation plus a derivation, but as analogous to the sum of a general finite transformation plus a derivation. In other words, a lift of this operation to some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(R)</annotation></semantics></math> is to be a choice of ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\Phi : R \to R</annotation></semantics></math> (a “finite translation”) in addition to a derivation-like operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>p</mi></msub><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\partial_p : R \to R</annotation></semantics></math> (the “infinitesimal translation”), such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>=</mo><mi>Φ</mi><mo>+</mo><mi>p</mi><mo>⋅</mo><msub><mo>∂</mo> <mi>p</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-)^p = \Phi + p \cdot\partial_p \,. </annotation></semantics></math></div> <p>As before, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is invertible in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, hence away from the fiber over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(\mathbb{F}_p)</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\partial_p</annotation></semantics></math> is uniquely fixed by this equation once <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> is chosen, and hence it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> alone which is to be chosen on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. A pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>Φ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(R,\Phi)</annotation></semantics></math> satisfying the above equation is equivalently a “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\Lambda_p</annotation></semantics></math>-ring” (see at <em><a class="existingWikiWord" href="/nlab/show/Lambda+ring">Lambda ring</a></em>). Or rather, by the analogy we are to lift the whole collection of operators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">id</mi><mo>+</mo><mi>p</mi><mo>⋅</mo><msub><mo>∂</mo> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{id} + p \cdot\partial_p</annotation></semantics></math> for all primes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and hence are to ask that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> satisfies the above equation for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and hence defines for all primes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> the derivation-like operator (function on the underlying set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>p</mi> <mi>Φ</mi></msubsup><mo>≔</mo><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>−</mo><mi>Φ</mi></mrow><mi>p</mi></mfrac><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial^\Phi_p \coloneqq \frac{(-)^p - \Phi}{p} : R \to R \,. </annotation></semantics></math></div> <p>This is the general form of an “<a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic differential operator</a>”, see for instance the first pages of (<a href="#Buium13">Buium 13</a>) for review.</p> <p>In terms of the abuse of notation already employed previously, this situation is usefully visualized as follows.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Φ</mi><mo>+</mo><mi>p</mi><mo>⋅</mo><msubsup><mo>∂</mo> <mi>p</mi> <mi>Φ</mi></msubsup></mrow></mover></mtd> <mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi mathvariant="normal">id</mi><mo>+</mo><mi>p</mi><mo>⋅</mo><msub><mo>∂</mo> <mi>p</mi></msub></mrow></mover></mtd> <mtd><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathrm{Spec}(R) &\stackrel{\Phi + p\cdot \partial^{\Phi}_p}{\longrightarrow} & \mathrm{Spec}(R) \\ \downarrow && \downarrow \\ \mathrm{Spec}(\mathbb{Z}) &\stackrel{\mathrm{id} + p\cdot \partial_p}{\longrightarrow} & \mathrm{Spec}(\mathbb{Z}) } </annotation></semantics></math></div> <p>A ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> equipped with such an endomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/Lambda+ring">Lambda ring</a> (an insight highlighted by <a class="existingWikiWord" href="/nlab/show/James+Borger">James Borger</a>). In this way a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>-ring structure on a commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>-analog of exhibiting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(R) \to \mathrm{Spec}(\mathbb{Z})</annotation></semantics></math> as being like a product by the <a class="existingWikiWord" href="/nlab/show/affine+line">affine line</a> over the non-existent field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math> of a map to “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Spec}(\mathbb{F}_1)</annotation></semantics></math>”. This should be one way to think of how <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>-rings embody “geometry over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math>” as proposed in (<a href="#Borger09">Borger 09</a>).</p> <p>Finally notice that the refinement of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">(-)^p</annotation></semantics></math>-operation from <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a> to <a class="existingWikiWord" href="/nlab/show/E-infinity+arithmetic+geometry">E-infinity arithmetic geometry</a> is given by the <a class="existingWikiWord" href="/nlab/show/power+operations">power operations</a> in <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a> (<a href="#Lurie">Lurie, remark 2.2.9</a>)</p> <h2 id="definition">Definition</h2> <blockquote> <p>under construction</p> </blockquote> <p>Write <a class="existingWikiWord" href="/nlab/show/CRing">CRing</a> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of (finitely generated) <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mi>Ring</mi></mrow><annotation encoding="application/x-tex">\Lambda Ring</annotation></semantics></math> for that of <a class="existingWikiWord" href="/nlab/show/Lambda-rings">Lambda-rings</a>.</p> <p>By the discussion at <em><a href="Lambda-ring#FreeAndCofreeLambdaRings">here</a></em> the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Λ</mi><mi>Ring</mi><mo>⟶</mo><mi>CRing</mi></mrow><annotation encoding="application/x-tex">U \;\colon\; \Lambda Ring \longrightarrow CRing</annotation></semantics></math> from <a class="existingWikiWord" href="/nlab/show/Lambda-rings">Lambda-rings</a> to <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a> has</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, given by forming the ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Symm</mi></mrow><annotation encoding="application/x-tex">Symm</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/symmetric+functions">symmetric functions</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> given by forming the <a class="existingWikiWord" href="/nlab/show/ring+of+Witt+vectors">ring of Witt vectors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Symm</mi><mo>⊣</mo><mi>U</mi><mo>⊣</mo><mi>Witt</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Λ</mi><mi>Ring</mi><mover><mover><mover><mo>←</mo><mi>Witt</mi></mover><mover><mo>⟶</mo><mi>U</mi></mover></mover><mover><mo>←</mo><mi>Symm</mi></mover></mover><mi>CRing</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Symm \dashv U \dashv Witt) \;\colon\; \Lambda Ring \stackrel{\overset{Symm}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\overset{Witt}{\leftarrow}}} CRing \,. </annotation></semantics></math></div> <p>Hence</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/rings+of+Witt+vectors">rings of Witt vectors</a> are the <em><a class="existingWikiWord" href="/nlab/show/co-free+functors">co-free</a> Lambda-rings;</em></p> </li> <li> <p>rings of <a class="existingWikiWord" href="/nlab/show/symmetric+functions">symmetric functions</a> are the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> Lambda-rings.</p> </li> </ul> <p>Regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ring</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Ring^{op}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/site">site</a> for <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, the order of the adjoints is reversed by forming <a class="existingWikiWord" href="/nlab/show/opposite+categories">opposite categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Witt</mi><mo>⊣</mo><mi>U</mi><mo>⊣</mo><mi>Symm</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>CRing</mi> <mi>op</mi></msup><mover><mover><mover><mo>⟶</mo><mi>Symm</mi></mover><mover><mo>⟵</mo><mi>U</mi></mover></mover><mover><mo>⟶</mo><mi>Witt</mi></mover></mover><mi>Λ</mi><msup><mi>Ring</mi> <mi>op</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Witt \dashv U \dashv Symm) \;\colon\; CRing^{op} \stackrel{\overset{Witt}{\longrightarrow}}{ \stackrel{\overset{U}{\longleftarrow}}{ \overset{Symm}{\longrightarrow} } } \Lambda Ring^{op} \,. </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mi>et</mi></msub></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z})_{et}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math> equipped with the <a class="existingWikiWord" href="/nlab/show/etale+topology">etale topology</a>. Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Et</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≔</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Et(Spec(\mathbb{Z})) \coloneqq Sh(Spec(\mathbb{Z})_{et}) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/gros+topos">gros</a> <a class="existingWikiWord" href="/nlab/show/etale+topos">etale topos</a> of <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>.</p> <p>Put a compatible <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><msup><mi>Ring</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Lambda Ring^{op}</annotation></semantics></math>(…) and write the resulting <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>et</mi></msub></mrow><annotation encoding="application/x-tex">Spec(\mathbb{F}_1)_{et}</annotation></semantics></math>.</p> <p>In analogy we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Et</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≔</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Et(Spec(\mathbb{F}_1)) \coloneqq Sh(Spec(\mathbb{F}_1)_{et}) </annotation></semantics></math></div> <p>and speak of the “<a class="existingWikiWord" href="/nlab/show/etale+topos">etale topos</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{F}_1)</annotation></semantics></math>”, or the “absolute <a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>” or something like this.</p> <p>The above <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> on sites then induces a sequence of adjoint functors on the <a class="existingWikiWord" href="/nlab/show/categories+of+presheaves">categories of presheaves</a> by left and right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mi>et</mi></msub><mo stretchy="false">)</mo><mover><mover><mover><mover><mover><mo>⟶</mo><mrow><msub><mi>Symm</mi> <mo>*</mo></msub></mrow></mover><mover><mo>⟵</mo><mrow><msup><mi>Symm</mi> <mo>*</mo></msup><mo>≃</mo><msub><mi>U</mi> <mo>*</mo></msub></mrow></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>Symm</mi> <mo>!</mo></msub><mo>≃</mo><msup><mi>U</mi> <mo>*</mo></msup><mo>≃</mo><msub><mi>Witt</mi> <mo>*</mo></msub></mrow></mover></mover><mover><mo>⟵</mo><mrow><msub><mi>U</mi> <mo>!</mo></msub><mo>≃</mo><msup><mi>Witt</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>Witt</mi> <mo>!</mo></msub></mrow></mover></mover><mi>PSh</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> PSh(Spec(\mathbb{Z})_{et}) \stackrel{\stackrel{Witt_!}{\longrightarrow}}{ \stackrel{\stackrel{U_! \simeq Witt^\ast}{\longleftarrow}}{ \stackrel{\stackrel{Symm_! \simeq U^\ast \simeq Witt_\ast}{\longrightarrow}}{ \stackrel{\stackrel{Symm^\ast \simeq U_\ast}{\longleftarrow}}{ \stackrel{Symm_\ast}{\longrightarrow} } } } } PSh(Spec(\mathbb{F}_1)_{et}) </annotation></semantics></math></div> <p>The top three restrict to sheaves</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mi>et</mi></msub><mo stretchy="false">)</mo><mover><mover><mover><mo>⟶</mo><mrow><msub><mi>Symm</mi> <mo>!</mo></msub><mo>≃</mo><msup><mi>U</mi> <mo>*</mo></msup><mo>≃</mo><msub><mi>Witt</mi> <mo>*</mo></msub></mrow></mover><mover><mo>⟵</mo><mrow><msub><mi>U</mi> <mo>!</mo></msub><mo>≃</mo><msup><mi>Witt</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>⟶</mo><mrow><msub><mi>Witt</mi> <mo>!</mo></msub></mrow></mover></mover><mi>Sh</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sh(Spec(\mathbb{Z})_{et}) \stackrel{\stackrel{Witt_!}{\longrightarrow}}{ \stackrel{\stackrel{U_! \simeq Witt^\ast}{\longleftarrow}}{ \stackrel{Symm_! \simeq U^\ast \simeq Witt_\ast}{\longrightarrow} } } Sh(Spec(\mathbb{F}_1)_{et}) </annotation></semantics></math></div> <p>The induced <a class="existingWikiWord" href="/nlab/show/adjoint+pair">adjoint pair</a> of <a class="existingWikiWord" href="/nlab/show/monad">monad</a>/<a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>W</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>W</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>U</mi> <mo>!</mo></msub><mo>∘</mo><msub><mi>Witt</mi> <mo>!</mo></msub><mo>⊣</mo><msub><mi>U</mi> <mo>!</mo></msub><mo>∘</mo><msub><mi>Witt</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (W^\ast \dashv W_\ast) \coloneqq ( U_! \circ Witt_! \dashv U_! \circ Witt_\ast) \,. </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a></p> <h2 id="references">References</h2> <p>The original article which proposes the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> over <a class="existingWikiWord" href="/nlab/show/Lambda-rings">Lambda-rings</a> as a realization of <a class="existingWikiWord" href="/nlab/show/F1">F1</a>-geometry is</p> <ul> <li id="Borger09"><a class="existingWikiWord" href="/nlab/show/James+Borger">James Borger</a>, <em>Lambda-rings and the field with one element</em> (<a href="http://arxiv.org/abs/0906.3146">arXiv/0906.3146</a>)</li> </ul> <p>This is based on technical details laid out in</p> <ul> <li id="Borger08"> <p><a class="existingWikiWord" href="/nlab/show/James+Borger">James Borger</a>, <em>The basic geometry of Witt vectors, I: The affine case</em> (<a href="http://arxiv.org/abs/0801.1691">arXiv:0801.1691</a>)</p> </li> <li id="Borger10"> <p><a class="existingWikiWord" href="/nlab/show/James+Borger">James Borger</a>, <em>The basic geometry of Witt vectors, II: Spaces</em> (<a href="http://arxiv.org/abs/1006.0092">arXiv:1006.0092</a>)</p> </li> </ul> <p>More discussion relating to this includes</p> <ul> <li id="LeBruyn13"><a class="existingWikiWord" href="/nlab/show/Lieven+Le+Bruyn">Lieven Le Bruyn</a>, <em>Absolute geometry and the Habiro topology</em> (<a href="http://arxiv.org/abs/1304.6532">arXiv:1304.6532</a>)</li> </ul> <p>Related discussion of <a class="existingWikiWord" href="/nlab/show/arithmetic+jet+spaces">arithmetic jet spaces</a> is in</p> <ul> <li id="Buium05"> <p><a class="existingWikiWord" href="/nlab/show/Alexandru+Buium">Alexandru Buium</a>, <em>Arithmetic differential equations</em>, volume 118 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005.</p> </li> <li id="Buium13"> <p><a class="existingWikiWord" href="/nlab/show/Alexandru+Buium">Alexandru Buium</a>, <em>Differential calculus with integers</em> (<a href="http://arxiv.org/abs/1308.5194">arXiv:1308.5194</a>, <a href="http://www.math.unm.edu/~buium/statupdated.pdf">slighly differing pdf</a>)</p> </li> </ul> <p>Related discussion of <a class="existingWikiWord" href="/nlab/show/power+operations">power operations</a> in <a class="existingWikiWord" href="/nlab/show/E-infinity+arithmetic+geometry">E-infinity arithmetic geometry</a> is around remark 2.2.9 of</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Rational+and+p-adic+Homotopy+Theory">Rational and p-adic Homotopy Theory</a></em></li> </ul> <p>Further discussion of and speculation on an analogy between <a class="existingWikiWord" href="/nlab/show/power+operations">power operations</a> and Borger’s absolute geometry is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Guillot">Pierre Guillot</a>, <em>Adams operations in cohomotopy</em> (<a href="http://arxiv.org/abs/math/0612327">arXiv:0612327</a>)</p> </li> <li id="MoravaSanthanam"> <p><a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, Rakha Santhanam, <em>Power operations and Absolute geometry</em> (<a href="http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 9, 2020 at 17:27:13. 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