群上同調

在同調代數中，群上同調是一套研究群及其表示的代數工具。群上同調源於代數拓撲，在代數數論上也有重要應用；它是現代類域論的基本構件之一。

起源

群論中的指導思想之一，是研究群 $G$ 及其表示的關係。群 $G$ 的表示是 $G$-模的特例：一個 $G$-模是一個阿貝爾群 $M$ 配上 $G$ 在 $M$ 上的群作用 $G\to \mathrm {End} (M)$。等價的說法是：$M$ 是群環 $\mathbb {Z} [G]$ 上的模。通常將 $G$ 的作用寫成乘法 $m\mapsto gm$。全體 $G$-模自然地構成一個阿貝爾範疇。

對給定的 $G$-模 $G$，最重要的子群之一是其 $G$-不變子群

$M^{G}=\lbrace x\in M:\forall g\in G\ gx=x\rbrace .$

若 $N\subset M$ 是一個 $G$-子模（即：是 $M$ 的子群，且在 $G$ 的作用下不變），則 $M/N$ 上賦有自然的 $G$-模結構，$N^{G}\subset M^{G}$，但是未必有 $(M/N)^{G}=M^{G}/N^{G}$。第一個群上同調群 $H^{1}(G,N)$ 可以設想為兩者間差異的某種量度。一般而言，可以定義一族函子 $H^{n}(G,-)$，其間關係可以由長正合序列表示。

形式建構

以下假設 $G$ 為有限群，全體 $G$-模構成阿貝爾範疇，其間的態射 $\mathrm {Hom} _{G}(M,N)$ 定義為滿足 $f(gx)=gf(x)$ 的群同態 $f:M\to N$。由於此範疇等價於 $\mathbb {Z} [G]$-模範疇，故有充足的內射對象。

函子 $M\to M^{G}$ 是從 $G$-模範疇映至阿貝爾群範疇的左正合函子。定義 $H^{n}(G,M)$ 為其導函子。根據導函子的一般理論，可知：

• $H^{0}(G,M)=M^{G}$
• 長正合序列：若 $0\to M'\to M\to M''\to 0$ 為 $G$-模的短正合序列，則導出相應的長正合序列

$\cdots \to H^{i-1}(G,M'')\to H^{i}(G,M')\to H^{i}(G,M)\to H^{i}(G,M'')\to H^{i+1}(M')\to H^{i+1}(M)\to \cdots$

在上述定義中，若固定一個域 $k$，並以 $k[G]$ 代替 $\mathbb {Z} [G]$，得到的上同調群依然同構。

標準分解

導出函子的定義來自內射分解，不便於具體計算。然而注意到 $M^{G}=\mathrm {Hom} _{G}(\mathbb {Z} ,M)$，其中 $\mathbb {Z}$ 被賦予平凡的 $G$ 作用：$gx=x$，故

$H^{i}(G,M)=\mathrm {Ext} ^{i}(\mathbb {Z} ,M)$

另一方面，$G$-模範疇中也有充足的射影對象，若取一 $\mathbb {Z}$ 的射影分解 $0\leftarrow \mathbb {Z} \leftarrow P_{\bullet }$，則有自然的同構 $\mathrm {Ext} ^{i}(\mathbb {Z} ,M)\simeq H^{i}(\mathrm {Hom} (P_{\bullet },M))$。最自然的分解是標準分解

$L_{i}:=\sum _{(g_{0},\ldots ,g_{i})\in G}\mathbb {Z} (g_{0},\ldots ,g_{i})$

$g(g_{0},\ldots ,g_{i})=(gg_{0},\ldots ,gg_{i})$

$d(g_{0},\ldots ,g_{i})=\sum _{j=0}^{i}(g_{0},\ldots ,{\hat {g}}_{j},\ldots ,g_{i})$

而 $L_{0}\to \mathbb {Z}$ 由 $g_{0}\mapsto 1$ 給出。

定義 $K^{i}:=\mathrm {Hom} _{G}(L_{i},M)$，其元素為形如 $f:G^{i+1}\mapsto M$ 的函數，並滿足 $f(gg_{0},\ldots ,gg_{i})=gf(g_{0},\ldots ,g_{i})$，稱之為齊次上鏈。根據 $G$ 在 $L_{i}$ 上的作用，這種 $f$ 由它在形如 $(e,g_{1},g_{1}g_{2},\ldots ,g_{1}\ldots ,g_{i})$ 的元素上的取值確定。藉此，可將上鏈複形 $K^{i}$ 描述為

• $K^{i}$ 的元素為 $G^{i}\to M$ 之函數。
• $(df)(g_{1},\ldots ,g_{i+1})=g_{1}f(g_{2},\ldots ,g_{i+1})+\sum _{j=1}^{i}(-1)^{j}f(g_{1},\ldots ,g_{j}g_{j+1},\ldots ,g_{i+1})+(-1)^{i+1}f(g_{1},\ldots ,g_{i})$

其中的元素稱為非齊次上鏈。

綜上所述，得到 $H^{i}(K^{\bullet })=H^{i}(G,M)$。

例子

較常用的上同調是 $H^{1}$ 與 $H^{2}$。從標準分解可導出以下的描述：

$H^{1}(G,M)={\dfrac {\{f:G\to M|\forall g,g',\;f(gg')=gf(g')+f(g)\}}{\{f:G\to M:\exists m\,\forall g,\;f(g)=gm-m\}}}$

準此要領，亦得到

$H^{2}(G,M)={\dfrac {\{f:G^{2}\to M|gf(g',g'')-f(gg',g'')+f(g,g'g'')-f(g,g')=0\}}{\{f:G^{2}\to M:\exists h:G\to M,f(g,g')=gh(g')-h(gg')+h(g)\}}}$

群同調

Dually to the construction of group cohomology there is the following definition of group homology: given a G-module M, set DM to be the submodule generated by elements of the form g·m-m, g∈G, m∈M. Assigning to M its so-called co-invariants, the quotient

M_G := M/DM,

is a right exact functor. Its left derived functors are by definition the group homology

H_n(G, M).

The functor which assigns M_G to M is isomorphic to the functor which sends M to $\mathbb {Z} \otimes _{\mathbb {Z} [G]}M$, where $\mathbb {Z}$ is endowed with the trivial G-action. Hence one also gets an expression for group homology in terms of the Tor functors,

$H_{n}(G,M)=\operatorname {Tor} _{n}^{\mathbb {Z} [G]}(\mathbb {Z} ,M)$

Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups.

Non-abelian group cohomology

Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G.

The zeroth cohomology of G with coefficients in A is

$H^{0}(G,A)=A^{G},\,$

which is a subgroup of A.

The first cohomology of G with coefficents in A is defined as above using 1-cocycles and 1-coboundaries. However, it is generally not a group when A is non-abelian. It instead has the structure of a pointed set.

Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let

$0\to A\to B\to C\to 0\,$

be a short exact sequence of G-groups, then there is an exact sequence of pointed sets

$0\to A^{G}\to B^{G}\to C^{G}\to H^{1}(G,A)\to H^{1}(G,B)\to H^{1}(G,C).\,$

Properties

Group cohomology depends contravariantly on the group G, in the following sense: if f : G → H is a group homomorphism and M is an H-module, then we have a naturally induced morphism Hⁿ(H,M) → Hⁿ(G,M) (where in the latter case, M is treated as a G-module via f).

If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H²(G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H²(G,M) classifies the isomorphism classes of all extensions of G by M in which the induced action of G on M by inner automorphisms agrees with the given action.

The Hochschild-Serre spectral sequence relates the cohomology of a normal subgroup N of G and the quotient G/N to the cohomology of the group G (for (pro-)finite groups G).

參考文獻

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• Milne, James, Class Field Theory, 2007 , Chapter II
• Rotman, Joseph, An Introduction to the Theory of Groups, Springer-Verlag, 1995, ISBN 978-0-387-94285-8, MR1307623 
• , Local fields, Graduate Texts in Mathematics 67, Berlin, New York: , 1979, ISBN 978-0-387-90424-5, MR554237 , Chapter VII
• Serre, Jean-Pierre, Cohomologie galoisienne, Lecture Notes in Mathematics 5 Fifth, Berlin, New York: , 1994, ISBN 978-3-540-58002-7, MR1324577 
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