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geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </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></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> <h4 id="formal_geometry">Formal geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/local-global+principle">local-global principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/infinitesimal+ring+extension">infinitesimal ring extension</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Artin+algebra">Artin algebra</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a>, <a class="existingWikiWord" href="/nlab/show/formal+spectrum">formal spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/completion+of+a+ring">completion of a ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adic+topology">adic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> <p><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></p> </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='#idea'>Idea</a></li> <ul> <li><a href='#FormalGroupLaw'>Formal group laws</a></li> <li><a href='#formal_group_schemes'>Formal group schemes</a></li> <li><a href='#formal_groups_over_an_operad'>Formal groups over an operad</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#in_characteristic_0'>In characteristic 0</a></li> <li><a href='#Universal1dFormalGroupLaw'>Universal 1d commutative formal group law</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> <li><a href='#general'>General</a></li> <li><a href='#1dimensional_formal_groups'>1-Dimensional formal groups</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>formal group</em> is a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>s. More general than <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>s, which are group objects in <em>first order infinitesimal</em> spaces, formal groups may be of arbitrary infinitesimal order. They sit between <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> and finite <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> or <a class="existingWikiWord" href="/nlab/show/algebraic+groups">algebraic groups</a>.</p> <p>Since <a class="existingWikiWord" href="/nlab/show/infinitesimal+spaces">infinitesimal spaces</a> are typically modeled as <a class="existingWikiWord" href="/nlab/show/Isbell+duality">formal duals</a> to <a class="existingWikiWord" href="/nlab/show/algebras">algebras</a>, formal groups are typically conceived as group objects in formal duals to <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> algebras.</p> <p>Specifically, fixing a formal <a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a> chart, then the product operation of a formal group is entirely expressed as a <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in two variables, satisfying conditions. This is called a <em>formal group law</em>, a concept that goes back to Bochner and <a class="existingWikiWord" href="/nlab/show/Daniel+Lazard">Lazard</a>.</p> <p>Commutative formal group laws of dimension 1 notably appear in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> (originating in work by <a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Novikov</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Buchstaber">Buchstaber</a> and <a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Quillen</a>, see <a href="#Adams74">Adams 74, part II</a>), where they express the behaviour of <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theories">complex oriented cohomology theories</a> evaluated on infinite <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a> (i.e. on the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <a class="existingWikiWord" href="/nlab/show/BU%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B U(1)</annotation> </semantics> </math></a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">\simeq \mathbb{C}P^\infty</annotation></semantics></math>). In particular <a class="existingWikiWord" href="/nlab/show/complex+cobordism">complex cobordism</a> cohomology theory in this context gives the universal formal group law represented by the <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a>. The <a class="existingWikiWord" href="/nlab/show/height+of+a+formal+group">height of formal groups</a> induces a filtering on <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theories">complex oriented cohomology theories</a> called the <em><a class="existingWikiWord" href="/nlab/show/chromatic+homotopy+theory">chromatic filtration</a></em>.</p> <p>More recently <a class="existingWikiWord" href="/nlab/show/Fabien+Morel">Morel</a> and <a class="existingWikiWord" href="/nlab/show/Marc+Levine">Marc Levine</a> consider the <a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a> of smooth schemes in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>. Formal groups are also useful in local <a class="existingWikiWord" href="/nlab/show/class+field+theory">class field theory</a>; they can be used to explicitly construct the local Artin map according to Lubin and Tate.</p> <h3 id="FormalGroupLaw">Formal group laws</h3> <div class="num_defn" id="AdicRing"> <h6 id="definition">Definition</h6> <p>An (commutative) <a class="existingWikiWord" href="/nlab/show/adic+ring">adic ring</a> is a (<a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative</a>) <a class="existingWikiWord" href="/nlab/show/topological+ring">topological ring</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">A</annotation></semantics></math> and an ideal <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">I \subset A</annotation></semantics></math> such that</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> on <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 the <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">I</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adic+topology">adic topology</a>;</p> </li> <li> <p>the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⟶</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msup><mi>I</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \longrightarrow \underset{\longleftarrow}{\lim}_n (A/I^n) </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> over <a class="existingWikiWord" href="/nlab/show/quotient+rings">quotient rings</a> by powers of the ideal is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </li> </ol> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of adic rings is a ring <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> that is also a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> (hence a function that preserves the filtering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊃</mo><mi>⋯</mi><mo>⊃</mo><mi>A</mi><mo stretchy="false">/</mo><msup><mi>I</mi> <mn>2</mn></msup><mo>⊃</mo><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">A \supset \cdots \supset A/I^2 \supset A/I </annotation></semantics></math>). This gives a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AdicRing</mi></mrow><annotation encoding="application/x-tex">AdicRing</annotation></semantics></math> and a subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AdicCRing</mi></mrow><annotation encoding="application/x-tex">AdicCRing</annotation></semantics></math> of commutative adic rings.</p> <p>The <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AdicRing</mi></mrow><annotation encoding="application/x-tex">AdicRing</annotation></semantics></math> (on <a class="existingWikiWord" href="/nlab/show/Noetherian+rings">Noetherian rings</a>) is that of affine <a class="existingWikiWord" href="/nlab/show/formal+schemes">formal schemes</a>.</p> <p>Similarly, 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> any fixed <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, then adic rings under <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> are <em>adic <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>-algebras</em>. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Adic</mi><mi>R</mi><mi>Alg</mi></mrow><annotation encoding="application/x-tex">Adic R Alg</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Adic</mi><mi>R</mi><mi>CAlg</mi></mrow><annotation encoding="application/x-tex">Adic R CAlg</annotation></semantics></math> for the corresponding categories.</p> </div> <div class="num_example" id="PowerSeriesAlgebra"> <h6 id="example">Example</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/commutative+ring">commutative ring</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> then the <a class="existingWikiWord" href="/nlab/show/formal+power+series+ring">formal power series ring</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> R[ [ x_1, x_2, \cdots, x_n ] ] </annotation></semantics></math></div> <p>in <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> <a class="existingWikiWord" href="/nlab/show/variables">variables</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> 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> and equipped with the ideal</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I = (x_1, \cdots , x_n) </annotation></semantics></math></div> <p>is an adic ring (def. <a class="maruku-ref" href="#AdicRing"></a>).</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>AdicRing</mi><mo>↪</mo><mi>ProRing</mi></mrow><annotation encoding="application/x-tex"> AdicRing \hookrightarrow ProRing </annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/adic+rings">adic rings</a> (def. <a class="maruku-ref" href="#AdicRing"></a>) to <a class="existingWikiWord" href="/nlab/show/pro-rings">pro-rings</a>, given by</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>A</mi><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msup><mi>I</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (A,I) \mapsto ( (A/I^{\bullet})) \,, </annotation></semantics></math></div> <p>i.e. for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>AdicRing</mi></mrow><annotation encoding="application/x-tex">A,B \in AdicRing</annotation></semantics></math> two adic rings, then there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>AdicRing</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><msub><mi>Hom</mi> <mi>Ring</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msup><mi>I</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>,</mo><mi>B</mi><mo stretchy="false">/</mo><msup><mi>I</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{AdicRing}(A,B) \simeq \underset{\longleftarrow}{\lim}_{n_2} \underset{\longrightarrow}{\lim}_{n_1} Hom_{Ring}(A/I^{n_1},B/I^{n_2}) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="GroupObjectFormalGroupLaw"> <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><mo>∈</mo><mi>CRing</mi></mrow><annotation encoding="application/x-tex">R \in CRing</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, a <strong>formal group law</strong> of dimension <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> 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> is the structure of a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Adic</mi><mi>R</mi><msup><mi>CAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Adic R CAlg^{op}</annotation></semantics></math> from def. <a class="maruku-ref" href="#AdicRing"></a> on the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R [ [x_1, \cdots ,x_n] ]</annotation></semantics></math> from example <a class="maruku-ref" href="#PowerSeriesAlgebra"></a>.</p> <p>Hence this is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>⟶</mo><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>y</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mu \;\colon\; R[ [ x_1, \cdots, x_n ] ] \longrightarrow R [ [ x_1, \cdots, x_n, \, y_1, \cdots, y_n ] ] </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Adic</mi><mi>R</mi><mi>CAlg</mi></mrow><annotation encoding="application/x-tex">Adic R CAlg</annotation></semantics></math> satisfying unitality, associativity.</p> <p>This is a <strong>commutative formal group law</strong> if it is an abelian group object, hence if it in addition satisfies the corresponding commutativity condition.</p> </div> <p>This is equivalently a set of <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> power series <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">F_i</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2n</annotation></semantics></math> variables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>y</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1,\ldots,x_n,y_1,\ldots,y_n</annotation></semantics></math> such that (in notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x=(x_1,\ldots,x_n)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>y</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=(y_1,\ldots,y_n)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>F</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x,y) = (F_1(x,y),\ldots,F_n(x,y))</annotation></semantics></math>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F(x,F(y,z))=F(F(x,y),z) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>+</mo><msub><mi>y</mi> <mi>i</mi></msub><mo>+</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>higher</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>order</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>terms</mi></mrow><annotation encoding="application/x-tex"> F_i(x,y) = x_i+y_i+\,\,higher\,\,order\,\,terms </annotation></semantics></math></div> <div class="num_example" id="Commutative1DimFormalGroupLaw"> <h6 id="example_2">Example</h6> <p>A 1-dimensional commutative formal group law according to def. <a class="maruku-ref" href="#GroupObjectFormalGroupLaw"></a> is equivalently a <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>≥</mo><mn>0</mn></mrow></munder><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex"> \mu(x,y) = \underset{i,j \geq 0}{\sum} a_{i,j} x^i y^j </annotation></semantics></math></div> <p>(the <a class="existingWikiWord" href="/nlab/show/image">image</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[ [ x,y ] ]</annotation></semantics></math> of the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">t \in R [ [ t ] ]</annotation></semantics></math>) such that</p> <ol> <li> <p>(unitality)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> \mu(x,0) = x</annotation></semantics></math></div></li> <li> <p>(associativity)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> \mu(x,\mu(y,z)) = \mu(\mu(x,y),z) \,; </annotation></semantics></math></div></li> <li> <p>(commutativity)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu(x,y) = \mu(y,x) \,. </annotation></semantics></math></div></li> </ol> <p>The first condition means equivalently that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>a</mi> <mrow><mn>0</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a_{i,0} = \left\{ \array{ 1 & if \; i = 1 \\ 0 & otherwise } \right. \;\;\;\;\,, \;\;\;\;\; a_{0,i} = \left\{ \array{ 1 & if \; i = 1 \\ 0 & otherwise } \right. \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> is necessarily of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></munder><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu(x,y) \;=\; x + y + \underset{i,j \geq 1}{\sum} a_{i,j} x^i y^j \,. </annotation></semantics></math></div> <p>The existence of inverses is no extra condition: by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> on the index <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">i</annotation></semantics></math> one finds that there exists a unique</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>≥</mo><mn>1</mn></mrow></munder><mi>ι</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex"> \iota(x) = \underset{i \geq 1}{\sum} \iota(x)_i x^i </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>ι</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>μ</mi><mo stretchy="false">(</mo><mi>ι</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu(x,\iota(x)) = 0 \;\;\;\,, \;\;\; \mu(\iota(x),x) = 0 \,. </annotation></semantics></math></div> <p>Hence 1-dimensional formal group laws 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> are equivalently <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Adic</mi><mi>R</mi><msup><mi>CAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Adic R CAlg^{op}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[ [ x ] ]</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/power+series">power series</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>3</mn></msub><msup><mi>x</mi> <mn>3</mn></msup><mo>+</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">f(x) = x + a_2 x^2 + a_3 x^3 + \ldots</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[ [x] ]</annotation></semantics></math> has a functional or compositional inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(x)</annotation></semantics></math> in the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x R[ [x] ]</annotation></semantics></math> under composition. Thus we may define a 1-dimensional formal group law by the formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu(x, y) = f^{-1}(f(x) + f(y))</annotation></semantics></math>. That this is in some sense the typical way that 1-dimensional formal group laws arise is the content of <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard's theorem</a>.</p> </div> <h3 id="formal_group_schemes">Formal group schemes</h3> <p>Much more general are <strong>formal group schemes</strong> from (<a href="#Grothendieck">Grothendieck</a>)</p> <p><strong>Formal group schemes</strong> are simply the <a class="existingWikiWord" href="/nlab/show/group+object">group object</a>s in a category of <a class="existingWikiWord" href="/nlab/show/formal+schemes">formal schemes</a>; however usually only the case of the formal spectra of complete <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>-algebras is considered; this category is equivalent to the category of complete cocommutative <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>-<a class="existingWikiWord" href="/nlab/show/Hopf+algebras">Hopf algebras</a>.</p> <h3 id="formal_groups_over_an_operad">Formal groups over an operad</h3> <p>For a generalization over <a class="existingWikiWord" href="/nlab/show/operads">operads</a> see (<a href="#Fresse">Fresse</a>).</p> <h2 id="properties">Properties</h2> <h3 id="in_characteristic_0">In characteristic 0</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The quotient <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℳ</mi> <mi>FG</mi></msub><mo>×</mo><mi>Spec</mi><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}_{FG} \times Spec \mathbb{Q}</annotation></semantics></math> of formal group over the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}_m</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a> (over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">Spec \mathbb{Q}</annotation></semantics></math>). This means that in <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0 every formal group is determined, up to unique <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, by its <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>.</p> </div> <p>For instance (<a href="#Lurie10">Lurie 10, lecture 12, corollary 3</a>).</p> <h3 id="Universal1dFormalGroupLaw">Universal 1d commutative formal group law</h3> <p>It is immediate that there exists a ring carrying a universal formal group law. For observe that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><msubsup><mi>x</mi> <mn>1</mn> <mi>i</mi></msubsup><msubsup><mi>x</mi> <mn>1</mn> <mi>j</mi></msubsup></mrow><annotation encoding="application/x-tex">\underset{i,j}{\sum} a_{i,j} x_1^i x_1^j</annotation></semantics></math> an element in a <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> algebra, then the condition that it defines a <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> is equivalently a sequence of polynomial <a class="existingWikiWord" href="/nlab/show/equations">equations</a> on the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</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></mrow><annotation encoding="application/x-tex">a_k</annotation></semantics></math>. For instance the commutativity condition means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>a</mi> <mrow><mi>j</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> a_{i,j} = a_{j,i} </annotation></semantics></math></div> <p>and the unitality constraint means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mn>0</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a_{i 0} = \left\{ \array{ 1 & if \; i = 1 \\ 0 & otherwise } \right. \,. </annotation></semantics></math></div> <p>Similarly associativity is equivalently a condition on combinations of triple products of the coefficients. It is not necessary to even write this out, the important point is only that it is some polynomial equation.</p> <p>This allows to make the following definition</p> <div class="num_defn" id="LazardRing"> <h6 id="definition_3">Definition</h6> <p>The <strong><a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a></strong> is the <a class="existingWikiWord" href="/nlab/show/graded+commutative+ring">graded commutative ring</a> generated by elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i j}</annotation></semantics></math> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2(i+j-1)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">i,j \in \mathbb{N}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>relations</mi><mspace width="thickmathspace"></mspace><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mspace width="thickmathspace"></mspace><mi>below</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below) </annotation></semantics></math></div> <p>quotiented by the relations</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>a</mi> <mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i j} = a_{j i}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>10</mn></msub><mo>=</mo><msub><mi>a</mi> <mn>01</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a_{10} = a_{01} = 1</annotation></semantics></math>; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>i</mi><mo>≠</mo><mn>1</mn><mo>:</mo><msub><mi>a</mi> <mrow><mi>i</mi><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\forall i \neq 1: a_{i 0} = 0</annotation></semantics></math></p> </li> <li> <p>the obvious associativity relation</p> </li> </ol> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">i,j,k</annotation></semantics></math>.</p> <p>The <strong>universal 1-dimensional commutative <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a></strong> is the formal power series with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the Lazard ring given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup><mo>∈</mo><mi>L</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \ell(x,y) \coloneqq \sum_{i,j} a_{i j} x^i y^j \in L[ [ x , y ] ] \,. </annotation></semantics></math></div></div> <div class="num_remark" id="LazardRing"> <h6 id="remark">Remark</h6> <p>The grading is chosen with regards to the formal group laws arising from <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theories">complex oriented cohomology theories</a> (<a href="complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw">prop.</a>) where the <a class="existingWikiWord" href="/nlab/show/variable">variable</a> <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> naturally has degree -2. This way</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>deg</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>j</mi><mi>deg</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg(a_{i j} x^i y^j) = deg(a_{i j}) + i deg(x) + j deg(y) = -2 \,. </annotation></semantics></math></div></div> <p>The following is immediate from the definition:</p> <div class="num_prop" id="LazardRingIsUniversal"> <h6 id="proposition_3">Proposition</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/ring">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> and 1-dimensional commutative <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> 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> (example <a class="maruku-ref" href="#Commutative1DimFormalGroupLaw"></a>), there exists a unique ring <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>L</mi><mo>⟶</mo><mi>R</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; L \longrightarrow R </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a> (def. <a class="maruku-ref" href="#LazardRing"></a>) to <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>, such that it takes the universal formal group law <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</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>f</mi> <mo>*</mo></msub><mi>ℓ</mi><mo>=</mo><mi>μ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f_\ast \ell = \mu \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>If the formal group law <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> has coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>c</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{c_{i,j}\}</annotation></semantics></math>, then in order that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mi>ℓ</mi><mo>=</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">f_\ast \ell = \mu</annotation></semantics></math>, i.e. that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>c</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex"> \underset{i,j}{\sum} f(a_{i,j}) x^i y^j = \underset{i,j}{\sum} c_{i,j} x^i y^j </annotation></semantics></math></div> <p>it must be that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>c</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> f(a_{i,j}) = c_{i,j} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i,j}</annotation></semantics></math> are the generators of the Lazard ring. Hence it only remains to see that this indeed constitutes a ring homomorphism. But this is guaranteed by the vary choice of relations imposed in the definition of the Lazard ring.</p> </div> <p>What is however highly nontrivial is this statement:</p> <div class="num_theorem" id="LazardTheorem"> <h6 id="theorem">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Lazard%27s+theorem">Lazard's theorem</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> (def. <a class="maruku-ref" href="#LazardRing"></a>) is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> L \simeq \mathbb{Z}[ t_1, t_2, \cdots ] </annotation></semantics></math></div> <p>in countably many generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t_i</annotation></semantics></math> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>i</mi></mrow><annotation encoding="application/x-tex">2 i</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Lazard+theorem">Lazard theorem</a> <a class="maruku-ref" href="#LazardTheorem"></a> first of all implies, via prop. <a class="maruku-ref" href="#LazardRingIsUniversal"></a>, that there exists an abundance of 1-dimensional formal group laws: given any 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> then every choice of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>t</mi> <mi>i</mi></msub><mo>∈</mo><mi>R</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t_i \in R\}</annotation></semantics></math> defines a formal group law. (On the other hand, it is nontrivial to say which formal group law that is.)</p> <p>Deeper is the fact expressed by the <a class="existingWikiWord" href="/nlab/show/Milnor-Quillen+theorem+on+MU">Milnor-Quillen theorem on MU</a>: the Lazard ring in its polynomial incarnation of prop. <a class="maruku-ref" href="#LazardTheorem"></a> is canonically identified with the <a class="existingWikiWord" href="/nlab/show/graded+commutative+ring">graded commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(M U)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> of the universal complex <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> <a class="existingWikiWord" href="/nlab/show/MU">MU</a>. Moreover:</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a> carries a <a class="existingWikiWord" href="/nlab/show/universal+complex+orientation+on+MU">universal complex orientation</a> in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/homotopy+commutative+ring+spectrum">homotopy commutative ring spectrum</a> then homotopy classes of homotopy ring homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">M U \to E</annotation></semantics></math> are in bijection to <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex orientations</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology">complex orientation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> induced a 1-dimensional commutative <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> (<a href="complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw">prop.</a>)</p> </li> <li> <p>under forming stable homotopy groups every ring spectrum homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">M U \to E</annotation></semantics></math> induces a ring homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>≃</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>U</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L \simeq \pi_\bullet(M U) \longrightarrow \pi_\bullet(E) </annotation></semantics></math></div> <p>and hence, by the universality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>, a formal group law over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(E)</annotation></semantics></math>.</p> </li> </ol> <p>This is the formal group law given by the above complex orientation.</p> <p>Hence the universal group law over the Lazard ring is a kind of <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> of the <a class="existingWikiWord" href="/nlab/show/universal+complex+orientation+on+MU">universal complex orientation on MU</a>.</p> </div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+multiplicative+group">formal multiplicative group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a> of an <a class="existingWikiWord" href="/nlab/show/algebraic+variety">algebraic variety</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+Picard+group">formal Picard group</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lubin-Tate+formal+group">Lubin-Tate formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Honda+formal+group">Honda formal group</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/height+of+a+formal+group+law">height of a formal group law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-typical+formal+group+law">p-typical formal group law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morava+stabilizer+group">Morava stabilizer group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Landweber+exact+functor+theorem">Landweber exact functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chromatic+homotopy+theory">chromatic homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></p> </li> </ul> <p>Formal geometry is closely related also to the <a class="existingWikiWord" href="/nlab/show/rigid+analytic+geometry">rigid analytic geometry</a>.</p> <p>(nlab remark: we should explain connections to the Witt rings, Cartier/Dieudonné modules).</p> <div> <p><strong>Examples of sequences of local structures</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>point</th><th>first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a> = arbitrary order infinitesimal</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>local = <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>finite</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <strong><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/integration">integration</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a> (path)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tangent+vector">tangent vector</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/jet">jet</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a> of <a class="existingWikiWord" href="/nlab/show/curve">curve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ+of+a+space">germ of a space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/square-0+ring+extension">square-0 ring extension</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nilpotent+ring+extension">nilpotent ring extension</a>/<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring+extension">ring extension</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></td><td style="text-align: left;"><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/finite+field">finite field</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><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> <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{(p)}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization at (p)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><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> <a class="existingWikiWord" href="/nlab/show/integers">integers</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></td><td style="text-align: left;"></td><td style="text-align: left;">local strict deformation quantization</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li> <p>Shigkaki Tôgô, <em>Note of formal Lie groups</em> , American Journal of Mathematics, Vol. 81, No. 3, Jul., 1959 (<a href="http://www.jstor.org/stable/2372919">JSTOR</a>)</p> </li> <li> <p>A. Fröhlich, <em>Formal group</em>, Lecture Notes in Mathematics Volume 74, Springer (1968)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+M.+Buchstaber">Victor M. Buchstaber</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+P.+Novikov">Sergei P. Novikov</a>, <em>Formal groups, power systems and Adams operators</em>, Math. USSR-Sb.13 (1971) 80-116 [<a href="https://iopscience.iop.org/article/10.1070/SM1971v013n01ABEH001030">doi:10.1070/SM1971v013n01ABEH001030</a>]</p> </li> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, Part II.1 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</p> </li> <li id="Grothendieck"> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a> et al. <a class="existingWikiWord" href="/nlab/show/SGA">SGA</a> III, vol. 1, Expose VIIB (P. Gabriel) ETUDE INFINITESIMALE DES SCHEMAS EN GROUPES (part B) 474-560</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, section 4.4 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li id="Fresse"> <p><a class="existingWikiWord" href="/nlab/show/Benoit+Fresse">Benoit Fresse</a>, <em>Lie theory of formal groups over an operad</em>, J. Alg. <strong>202</strong>, 455–511, 1998, <a href="http://dx.doi.org/10.1006/jabr.1997.7280">doi</a></p> </li> <li> <p>Michiel Hazewinkel, Formal Groups and Applications, <a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183548600">projecteuclid</a></p> </li> <li> <p>Michel <a class="existingWikiWord" href="/nlab/show/Demazure%2C+lectures+on+p-divisible+groups">Demazure, lectures on p-divisible groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean+Dieudonn%C3%A9">Jean Dieudonné</a>, <em>Introduction to the theory of formal groups</em>, Marcel Dekker, New York 1973.</p> </li> </ul> <h3 id="1dimensional_formal_groups">1-Dimensional formal groups</h3> <p>A basic introduction is in</p> <ul> <li>Carl Erickson, <em>One-dimensional formal groups</em> (<a href="http://people.brandeis.edu/~cwe/pdfs/formal_groups.pdf">pdf</a>)</li> </ul> <p>Specifically formal group laws of <a class="existingWikiWord" href="/nlab/show/elliptic+curves">elliptic curves</a>:</p> <ul> <li> <p>Antonia W. Bluher, <em>Formal groups, elliptic curves, and some theorems of Couveignes</em>, in: J.P. Buhler (eds.) <em>Algorithmic Number Theory</em> ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 (<a href="https://arxiv.org/abs/math/9708215">arXiv:math/9708215</a>, <a href="https://doi.org/10.1007/BFb0054887">doi:10.1007/BFb0054887</a>)</p> </li> <li id="Friedl17"> <p>Stefan Friedl, <em>An elementary proof of the group law for elliptic curves</em> (<a href="https://arxiv.org/abs/1710.00214">arXiv:1710.00214</a>)</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Quillen%27s+theorem+on+MU">Quillen's theorem on MU</a> is due to</p> <ul> <li id="Quillen69"><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>On the formal group laws of unoriented and complex cobordism theory</em>, 1969, <a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183530915">projecteuclid</a></li> </ul> <p>See also</p> <ul> <li id="Lurie10"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Chromatic+Homotopy+Theory">Chromatic Homotopy Theory</a></em>, Lecture series (<a href="http://www.math.harvard.edu/~lurie/252x.html">lecture notes</a>) Lecture 11 <em>Formal groups</em> (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Takeshi+Torii">Takeshi Torii</a>, <em>One dimensional formal group laws of height <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N-1</annotation></semantics></math></em>, PhD thesis 2001 (<a href="http://mathnt.mat.jhu.edu/mathnew/Thesis/Torii.thesis.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Takeshi+Torii">Takeshi Torii</a>, <em>On Degeneration of One-Dimensional Formal Group Laws and Applications to Stable Homotopy Theory</em>, American Journal of Mathematics Vol. 125, No. 5 (Oct., 2003), pp. 1037-1077 (<a href="http://www.jstor.org/stable/25099207">JSTOR</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em>Formal groups and stable homotopy of commutative rings</em>, Geom. Topol. 8 (2004) 335-412 (<a href="http://arxiv.org/abs/math/0402372">arXiv:math/0402372</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of formal groups and its incarnation as a <a class="existingWikiWord" href="/nlab/show/Hopf+algebroid">Hopf algebroid</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Niko+Naumann">Niko Naumann</a>, <em>The stack of formal groups in stable homotopy theory</em>, Advances in Mathematics Volume 215, Issue 2, 10 November 2007, Pages 569–600 doi:<a href="http://dx.doi.org/10.1016/j.aim.2007.04.007">10.1016/j.aim.2007.04.007</a>, arXiv:<a href="http://front.math.ucdavis.edu/math.AT/0503308">math.AT/0503308</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 4, 2024 at 23:55:29. 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