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href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_the_multiplicative_group'>Relation to the multiplicative group</a></li> <li><a href='#relation_to_the_group_ring'>Relation to the group ring</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>For <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 <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, its <em>group of units</em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">R^\times</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(R)</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/group">group</a> whose <a class="existingWikiWord" href="/nlab/show/elements">elements</a> are the elements 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> that are <a class="existingWikiWord" href="/nlab/show/invertible+element">invertible</a> under the product (<em><a class="existingWikiWord" href="/nlab/show/units">units</a></em> of the ring), and whose group operation is the multiplication 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>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(R)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/affine+variety">affine variety</a> (in fact an affine <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>) over <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>, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>R</mi> <mn>2</mn></msup><mo>:</mo><mi>x</mi><mi>y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(x, y) \in R^2: x y = 1\}</annotation></semantics></math>.</p> </div> <p>This leads us to the following alternative perspective:</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>, with <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 <a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a> therein, the <em>group of units</em> 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> is the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of the two maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">m, c_1: R \times R \to R</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> is the ring multiplication and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">c_1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/constant+map">constant map</a> with value the multiplicative identity.</p> </div> <p>Cf. Example <a class="maruku-ref" href="#idele"></a> below.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_the_multiplicative_group">Relation to the multiplicative group</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The group of units 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> is equivalently the collection of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">Spec R</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{G}_m</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>R</mi> <mo>×</mo></msup><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>Spec</mi><mi>R</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> GL_1(R) = R^\times \simeq Hom(Spec R, \mathbb{G}_m) \,. </annotation></semantics></math></div></div> <h3 id="relation_to_the_group_ring">Relation to the group ring</h3> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</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>R</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>⊣</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>×</mo></msup><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>Alg</mi> <mi>R</mi></msub><mover><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>×</mo></msup></mrow></munder><mover><mo>←</mo><mrow><mi>R</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow></mover></mover><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> (R[-]\dashv (-)^\times) \colon Alg_R \stackrel{\overset{R[-]}{\leftarrow}}{\underset{(-)^\times}{\to}} Grp </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> over <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> and that of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[-]</annotation></semantics></math> forms the <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> over <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> and where <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> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">(-)^\times</annotation></semantics></math> assigns to an <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>-algebra its group of units.</p> </div> <h2 id="examples">Examples</h2> <div class="num_example"> <h6 id="example">Example</h6> <p>The multiplicative group of the <a class="existingWikiWord" href="/nlab/show/ring">ring</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/multiplicative+group+of+integers+modulo+n">multiplicative group of integers modulo n</a></em>.</p> </div> <div class="num_example" id="idele"> <h6 id="example_2">Example</h6> <p>The group of units of the <a class="existingWikiWord" href="/nlab/show/ring+of+adeles">ring of adeles</a> <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">\mathbb{A}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/group+of+ideles">group of ideles</a>. The topology on the idele group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕀</mi></mrow><annotation encoding="application/x-tex">\mathbb{I}</annotation></semantics></math> arises by considering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕀</mi></mrow><annotation encoding="application/x-tex">\mathbb{I}</annotation></semantics></math> as an affine variety in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^2</annotation></semantics></math> as above, and giving it the subspace topology. This is <em>not</em> the subspace topology induced by the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕀</mi><mo>↪</mo><mi>𝔸</mi></mrow><annotation encoding="application/x-tex">\mathbb{I} \hookrightarrow\mathbb{A}</annotation></semantics></math> into the ring of adeles.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>The group of units of 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>-adic integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> fits in an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mn>1</mn><mo>+</mo><mi>p</mi><msub><mi>ℤ</mi> <mi>p</mi></msub><mo>↪</mo><msubsup><mi>ℤ</mi> <mi>p</mi> <mo>×</mo></msubsup><mo>→</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>×</mo></msup><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \to 1 + p \mathbb{Z}_p \hookrightarrow \mathbb{Z}_p^\times \to (\mathbb{Z}/(p))^\times \to 1</annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> is isomorphic to the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}/(p-1)</annotation></semantics></math> (see <a class="existingWikiWord" href="/nlab/show/root+of+unity">root of unity</a>) and the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> is, at least when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p \gt 2</annotation></semantics></math>, isomorphic to the <em>additive group</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math>. Explicitly, for such <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 formal <a class="existingWikiWord" href="/nlab/show/exponential+map">exponential map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><mfrac><mrow><msup><mi>x</mi> <mi>n</mi></msup></mrow><mrow><mi>n</mi><mo>!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}</annotation></semantics></math> converges when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>p</mi><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x \in p \mathbb{Z}_p</annotation></semantics></math> and maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">p \mathbb{Z}_p</annotation></semantics></math> isomorphically onto the multiplicative group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mi>p</mi><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">1 + p \mathbb{Z}_p</annotation></semantics></math>. The formal logarithm <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>log</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow><mi>n</mi></mfrac></mrow><annotation encoding="application/x-tex">\log(x) = \sum_{n \geq 1} \frac{(-1)^{n-1} (x - 1)^n}{n}</annotation></semantics></math> is also <a class="existingWikiWord" href="/nlab/show/convergence">convergent</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mn>1</mn><mo>+</mo><mi>p</mi><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x \in 1 + p \mathbb{Z}_p</annotation></semantics></math> and provides the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a>.</p> <p>By <a class="existingWikiWord" href="/nlab/show/Henselian+ring">Hensel's lemma</a>, the group of units <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℤ</mi> <mi>p</mi> <mo>×</mo></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p^\times</annotation></semantics></math> has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">(p-1)^{th}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/roots+of+unity">roots of unity</a> and therefore the exact sequence above <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">splits</a>. This splitting descends to the quotient ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>p</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}/(p^n)</annotation></semantics></math> and its group of units, giving an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>p</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>p</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(\mathbb{Z}/(p^n)) \cong \mathbb{Z}/(p^{n-1}) \oplus \mathbb{Z}/(p-1)</annotation></semantics></math>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>group of units</strong>/<a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+submonoid+of+cancellative+elements">multiplicative submonoid of cancellative elements</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+units">∞-group of units</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exponential+sequence">exponential sequence</a>, <a class="existingWikiWord" href="/nlab/show/Kummer+sequence">Kummer sequence</a></p> </li> </ul> <h2 id="references">References</h2> <p>For instance:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anthony+Knapp">Anthony Knapp</a>, p. 143 in: <em>Basic Algebra</em>, Springer (2006) [<a href="https://doi.org/10.1007/978-0-8176-4529-8">doi:10.1007/978-0-8176-4529-8</a>, <a href="https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/knapp.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 25, 2024 at 04:08:00. 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