Tate cohomology group - Revision history

Vipul at 23:01, 3 October 2011

2011-10-03T23:01:38Z

← Older revision		Revision as of 23:01, 3 October 2011
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	! Value of <math>n</math> !! Definition of <math>\hat H^n(G,A)</math>		! 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).		\| <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).

Vipul: /* Definition */

2011-10-03T23:01:18Z

Definition

← Older revision		Revision as of 23:01, 3 October 2011
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	Suppose <math>G</math> is a [[finite group]], and <math>A</math> is an [[abelian group]], and <math>\varphi:G \to \operatorname{Aut}(A)</math> is a homomorphism of groups, making <math>A</math> into a <math>G</math>-module (i.e., <math>G</math> acts on <math>A</math>).		Suppose <math>G</math> is a [[finite group]], and <math>A</math> is an [[abelian group]], and <math>\varphi:G \to \operatorname{Aut}(A)</math> is a homomorphism of groups, making <math>A</math> into a <math>G</math>-module (i.e., <math>G</math> acts on <math>A</math>).

	The '''Tate cohomology groups''' <math>\hat{H}^n(G,A)</math> are a collection of groups for <math>n</math> varying over all integers (both positive and negative) that combine information about the [[cohomology group ~~for a group action\|cohomology groups~~]] and [[homology group ~~for a group action\|homology groups~~]].		The '''Tate cohomology groups''' <math>\hat{H}^n(G,A)</math> are a collection of groups for <math>n</math> varying over all integers (both positive and negative) that combine information about the [[defining ingredient::cohomology group]]s and [[defining ingredient::homology group]]s.

	They are defined as follows:		They are defined as follows:

Vipul: Created page with "==Definition== Suppose $G$ is a finite group, and $A$ is an abelian group, and $\varphi:G \to \operatorname{Aut}(A)$ is a homomorphism o..."

2011-10-03T23:00:36Z

Created page with "==Definition== Suppose <math>G</math> is a finite group, and <math>A</math> is an abelian group, and <math>\varphi:G \to \operatorname{Aut}(A)</math> is a homomorphism o..."

New page

==Definition==

Suppose <math>G</math> is a [[finite group]], and <math>A</math> is an [[abelian group]], and <math>\varphi:G \to \operatorname{Aut}(A)</math> is a homomorphism of groups, making <math>A</math> into a <math>G</math>-module (i.e., <math>G</math> acts on <math>A</math>).

The '''Tate cohomology groups''' <math>\hat{H}^n(G,A)</math> are a collection of groups for <math>n</math> varying over all integers (both positive and negative) that combine information about the [[cohomology group for a group action|cohomology groups]] and [[homology group for a group action|homology groups]].

They are defined as follows:

{| class="sortable" border="1"
! 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 \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 = 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).
|-
| <math>n = -1</math> || Kernel of the map <math>N</math> defined for the <math>n = 0</math> case.
|}

Tate cohomology group - Revision history

Vipul at 23:01, 3 October 2011

Vipul: /* Definition */

Vipul: Created page with "==Definition== Suppose G is a finite group, and A is an abelian group, and \varphi:G \to \operatorname{Aut}(A) is a homomorphism o..."

Vipul: Created page with "==Definition== Suppose $G$ is a finite group, and $A$ is an abelian group, and $\varphi:G \to \operatorname{Aut}(A)$ is a homomorphism o..."