Tate cohomology group: Difference between revisions

Latest revision as of 23:01, 3 October 2011

Definition

Suppose $G$ is a finite group, and $A$ is an abelian group, and $φ : G \to A u t (A)$ is a homomorphism of groups, making $A$ into a $G$ -module (i.e., $G$ acts on $A$ ).

The Tate cohomology groups ${\hat{H}}^{n} (G, A)$ are a collection of groups for $n$ varying over all integers (both positive and negative) that combine information about the cohomology groups and homology groups.

They are defined as follows:

Value of $n$	Definition of ${\hat{H}}^{n} (G, A)$
$n \geq 1$	Same as the cohomology group $H^{n} (G, A)$
$n \leq - 2$	Same as the homology group $H_{m} (G, A)$ where $m + n = - 1$ .
$n = 0$	Cokernel of the map $N : H_{0} (G, A) \to H^{0} (G, A)$ that is obtained by passing to homology/cohomology classes the following map at the level of cycles and cocycles: an element $a \in A$ (a 0-cycle) is sent to $\sum_{g \in G} φ (g) (a)$ (a 0-cocycle).
$n = - 1$	Kernel of the map $N$ defined for the $n = 0$ case.

@@ Line 10: / Line 10: @@
 ! Value of <math>n</math> !! Definition of <math>\hat H^n(G,A)</math>
 |-
-| <math>n \ge 1</math> || Same as the [[cohomology group for a group action|cohomology group]] <math>H^n(G,A)</math>
+| <math>n \ge 1</math> || Same as the [[cohomology group]] <math>H^n(G,A)</math>
 |-
-| <math>n \le -2</math> || Same as the [[homology group for a group action|homology group]] <math>H_m(G,A)</math> where <math>m + n = - 1</math>.
+| <math>n \le -2</math> || Same as the [[homology group]] <math>H_m(G,A)</math> where <math>m + n = - 1</math>.
 |-
 | <math>n = 0</math> || Cokernel of the map <math>N:H_0(G,A) \to H^0(G,A)</math> that is obtained by passing to homology/cohomology classes the following map at the level of cycles and cocycles: an element <math>a \in A</math> (a 0-cycle) is sent to <math>\sum_{g \in G}\varphi(g)(a)</math> (a 0-cocycle).