From 4587d765854390c7104e71eb30c7cfdef94720e2 Mon Sep 17 00:00:00 2001 From: zeramorphic <50671761+zeramorphic@users.noreply.github.com> Date: Thu, 25 Jan 2024 11:55:42 +0000 Subject: [PATCH] Lectures 6 --- iii/gc/01_definitions_and_resolutions.tex | 146 ++++++++++++++++++++++ 1 file changed, 146 insertions(+) diff --git a/iii/gc/01_definitions_and_resolutions.tex b/iii/gc/01_definitions_and_resolutions.tex index 3907dff..18cdb49 100644 --- a/iii/gc/01_definitions_and_resolutions.tex +++ b/iii/gc/01_definitions_and_resolutions.tex @@ -318,3 +318,149 @@ \subsection{???} \begin{proof} The bar resolution gives a suitable resolution. \end{proof} + +\subsection{Cohomology} +\begin{definition} + Consider a projective resolution + % https://q.uiver.app/#q=WzAsOCxbMCwwLCJcXGNkb3RzIl0sWzEsMCwiUF97bisxfSJdLFsyLDAsIlBfbiJdLFszLDAsIlxcY2RvdHMiXSxbNCwwLCJQXzEiXSxbNSwwLCJQXzAiXSxbNiwwLCJcXG1hdGhiYiBaIl0sWzcsMCwiMCJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdLFs0LDVdLFs1LDZdLFs2LDddXQ== +\[\begin{tikzcd} + \cdots & {P_{n+1}} & {P_n} & \cdots & {P_1} & {P_0} & {\mathbb Z} & 0 + \arrow[from=1-1, to=1-2] + \arrow[from=1-2, to=1-3] + \arrow[from=1-3, to=1-4] + \arrow[from=1-4, to=1-5] + \arrow[from=1-5, to=1-6] + \arrow[from=1-6, to=1-7] + \arrow[from=1-7, to=1-8] +\end{tikzcd}\] + of \( \mathbb Z \) by \( \mathbb Z G \)-modules. + Let \( M \) be a (left) \( \mathbb Z G \)-module. + Applying \( \Hom_G(-,M) \), we obtain a sequence + % https://q.uiver.app/#q=WzAsNixbMCwwLCJcXGNkb3RzIl0sWzEsMCwiXFxIb21fRyhQX3tuKzF9LE0pIl0sWzIsMCwiXFxIb21fRyhQX24sTSkiXSxbMywwLCJcXGNkb3RzIl0sWzQsMCwiXFxIb21fRyhQXzEsTSkiXSxbNSwwLCJcXEhvbV9HKFBfMCxNKSJdLFsxLDBdLFsyLDFdLFszLDJdLFs0LDNdLFs1LDQsImReMSIsMl1d +\[\begin{tikzcd} + \cdots & {\Hom_G(P_{n+1},M)} & {\Hom_G(P_n,M)} & \cdots & {\Hom_G(P_1,M)} & {\Hom_G(P_0,M)} + \arrow[from=1-2, to=1-1] + \arrow[from=1-3, to=1-2] + \arrow[from=1-4, to=1-3] + \arrow[from=1-5, to=1-4] + \arrow["{d^1}"', from=1-6, to=1-5] +\end{tikzcd}\] + where \( d^n = d_n^\star \). + Then the \emph{\( n \)th cohomology group} \( H^n(G, M) \) with coefficients in \( M \) is + \[ H^n(G, M) = {\ker d^{n+1}}{\im d^n};\quad H^0(G, M) = \ker d^1 \] +\end{definition} +\begin{remark} + We have removed the \( \mathbb Z \) term in the \( \Hom_G(-, M) \) sequence. + These cohomology groups are the homology groups of a chain complex \( C_n = \Hom_G(P_{-n}, M) \) for \( n \leq 0 \). + We will show that these cohomology groups are independent of the choice of projective resolution. +\end{remark} +\begin{example} + Let \( G = \langle t \rangle \) be an infinite cyclic group. + We have a projective resolution + % https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiXFxtYXRoYmIgWiBHIl0sWzIsMCwiXFxtYXRoYmIgWiBHIl0sWzMsMCwiXFxtYXRoYmIgWiJdLFs0LDAsIjAiXSxbMCwxXSxbMSwyLCJcXGNkb3RcXCwodC0xKSJdLFsyLDNdLFszLDRdXQ== +\[\begin{tikzcd} + 0 & {\mathbb Z G} & {\mathbb Z G} & {\mathbb Z} & 0 + \arrow[from=1-1, to=1-2] + \arrow["{\cdot\,(t-1)}", from=1-2, to=1-3] + \arrow[from=1-3, to=1-4] + \arrow[from=1-4, to=1-5] +\end{tikzcd}\] + For \( \varphi \in \Hom_G(\mathbb Z G, M) \) and \( x \in \mathbb Z G \), + \[ d^1(\varphi)(x) = \varphi(d_1(x)) = \varphi(x(t-1)) \] + Recall that we have an isomorphism \( i : \Hom_G(\mathbb Z G, M) \cong M \) by \( \theta \mapsto \theta(1) \). + In particular, + \[ d^1(\varphi) \mapsto d^1(\varphi)(1) = \varphi(t-1) = (t - 1)\varphi(1) = (t - 1) i(\varphi) \] + We thus obtain + % https://q.uiver.app/#q=WzAsMyxbMiwwLCJNIl0sWzEsMCwiTSJdLFswLDAsIjAiXSxbMCwxLCJcXGFscGhhIiwyXSxbMSwyXV0= +\[\begin{tikzcd} + 0 & M & M + \arrow["\alpha"', from=1-3, to=1-2] + \arrow[from=1-2, to=1-1] +\end{tikzcd}\] + where \( \alpha \) is multiplication on the left by \( t - 1 \). + Therefore, the cohomology groups are + \[ H^0(G, M) = \qty{m \in M \mid tm = m} = M^G;\quad H^1(G, M) = \faktor{M}{(t-1)M} = M_G;\quad H^n(G, M) = 0 \text{ for } n \neq 0, 1 \] + Note that the group of invariants \( M^G \) is the largest submodule with trivial \( G \)-action, and the group of coinvariants \( M_G \) is the largest quotient module with trivial \( G \)-action. +\end{example} +\begin{remark} + It is generally true that \( H^0(G, M) = M^G \), but in general \( H^1(G, M) = M_G \) does not hold. + In general, \( M_G \) is the \( 0 \)th homology group, which will be discussed later. + Note that for any group of type \( FP \), the cohomology groups vanish for all but finitely many indices \( n \). +\end{remark} +\begin{definition} + \( G \) is of \emph{cohomological dimension \( m \)} over \( \mathbb Z \) if there exists some \( \mathbb Z G \)-module \( M \) with \( H^m(G, M) \neq 0 \) but \( H^n(G, M_1) = 0 \) for all \( n > m \) and all \( \mathbb Z G \)-modules \( M_1 \). +\end{definition} +\begin{remark} + For all \( G \), we have \( H^0(G, \mathbb Z) = \mathbb Z \neq 0 \) so all groups have dimension at least zero. +\end{remark} +\begin{example} + Infinite cyclic groups have cohomological dimension 1 over \( \mathbb Z \). + One can show that if \( G \) is a free group of finite rank, then it is also of cohomological dimension 1 over \( \mathbb Z \). + Stallings showed in 1968 that the converse is true: a finitely generated group of cohomological dimension 1 is free. + Swan strengthened this in 1969 by removing the assumption of finite generation. +\end{example} +We now consider the bar resolution in our definition of cohomology. +Note that +\[ \Hom_G(\mathbb Z G \qty{G^{(n)}}, M) \cong C^n(G, M) \] +where \( C^n(G, M) \) is the set of functions \( G^{(n)} \to M \), since a \( \mathbb Z G \)-map is determined by its action on a basis. +Moreover, \( C^n(G, M) \) corresponds to the set of functions \( G^n \to M \). +For \( n = 0 \), note that \( C^0(G, M) \) is the set of functions \( G^0 \to M \) which bijects with \( M \). +\begin{definition} + The abelian group of \emph{\( n \)-cochains} of \( G \) with coefficients in \( M \) is \( C^n(G, M) \). + The \emph{\( n \)th coboundary map} \( d^n : C^{n-1}(G, M) \to C^n(G, M) \) is dual to the \( d_n \) from the bar resolution: + \begin{align*} + d^n(\varphi)(g_1, \dots, g_n) &= g_1 \varphi(g_2, \dots, g_n) \\ + &- \varphi(g_1 g_2, g_3, \dots, g_n) \\ + &+ \varphi(g_1, g_2 g_3, \dots, g_n) - \cdots \\ + &+ (-1)^{n-1} \varphi(g_1, g_2, \dots, g_{n-1} g_n) \\ + &+ (-1)^n \varphi(g_1, g_2, \dots, g_{n-1}) + \end{align*} + The group of \emph{\( n \)-cocycles} is \( Z^n(G, M) = \ker d^{n+1} \leq C^n(G, M) \). + The group of \emph{\( n \)-coboundaries} is \( B^n(G, M) = \im d^n \leq C^n(G, M) \). + Thus the \( n \)th cohomology group is + \[ H^n(G, M) = \faktor{Z^n(G, M)}{B^n(G, M)} \] +\end{definition} +\begin{corollary} + \( H^0(G, M) = M^G \) for all \( G \). +\end{corollary} +\begin{definition} + A \emph{derivation} of \( G \) with coefficients in \( M \) is a function \( \varphi : G \to M \) such that \( \varphi(gh) = g \varphi(h) + \varphi(g) \). +\end{definition} +Note that \( Z^1(G, M) \) is exactly the set of derivations of \( G \) with coefficients in \( M \), so a derivation is precisely a 1-cocycle. +\begin{definition} + An \emph{inner derivation} of \( G \) with coefficients in \( M \) is a function \( \varphi : G \to M \) of the form \( \varphi(g) = gm - m \) for a fixed \( m \in M \). +\end{definition} +Such maps are derivations. +\begin{corollary} + \( H^1(G, M) \) is the group of derivations modulo the inner derivations. + In particular, if \( M \) is a trivial \( \mathbb Z G \)-module, then + \[ H^1(G, M) = \qty{\text{group homomorphisms } G \to M} \] + treating \( M \) as an abelian group under addition. +\end{corollary} + +\subsection{Independence of cohomology groups} +We now prove that cohomology groups are independent of the choice of resolution. +\begin{definition} + Let \( (A_n, \alpha_n), (B_n, \beta_n) \) be chain complexes of \( \mathbb Z G \)-modules. + A \emph{chain map} \( (f_n) \) is a sequence of \( \mathbb Z G \)-maps \( f_n : A_n \to B_n \) such that the following diagram commutes. + % https://q.uiver.app/#q=WzAsMTAsWzAsMCwiXFxjZG90cyJdLFsxLDAsIkFfbiJdLFsyLDAsIkFfe24tMX0iXSxbMCwxLCJcXGNkb3RzIl0sWzEsMSwiQl9uIl0sWzIsMSwiQl97bi0xfSJdLFszLDEsIkJfe24tMn0iXSxbNCwxLCJcXGNkb3RzIl0sWzMsMCwiQV97bi0yfSJdLFs0LDAsIlxcY2RvdHMiXSxbMCwxXSxbMSwyLCJcXGFscGhhX24iXSxbMyw0XSxbNCw1LCJcXGJldGFfbiIsMl0sWzEsNCwiZl9uIl0sWzIsNSwiZl97bi0xfSJdLFs1LDYsIlxcYmV0YV97bi0xfSIsMl0sWzYsN10sWzIsOCwiXFxhbHBoYV97bi0xfSJdLFs4LDldLFs4LDYsImZfe24tMn0iXV0= +\[\begin{tikzcd} + \cdots & {A_n} & {A_{n-1}} & {A_{n-2}} & \cdots \\ + \cdots & {B_n} & {B_{n-1}} & {B_{n-2}} & \cdots + \arrow[from=1-1, to=1-2] + \arrow["{\alpha_n}", from=1-2, to=1-3] + \arrow[from=2-1, to=2-2] + \arrow["{\beta_n}"', from=2-2, to=2-3] + \arrow["{f_n}", from=1-2, to=2-2] + \arrow["{f_{n-1}}", from=1-3, to=2-3] + \arrow["{\beta_{n-1}}"', from=2-3, to=2-4] + \arrow[from=2-4, to=2-5] + \arrow["{\alpha_{n-1}}", from=1-3, to=1-4] + \arrow[from=1-4, to=1-5] + \arrow["{f_{n-2}}", from=1-4, to=2-4] +\end{tikzcd}\] +\end{definition} +\begin{lemma} + A chain map \( (f_n) \) as above induces a map on homology groups + \[ f_\star : H_n(A_\bullet) \to H_n(B_\bullet) \] +\end{lemma}