Kunneth formula for group cohomology - Revision history

Vipul at 05:30, 22 September 2015

2015-09-22T05:30:21Z

← Older revision		Revision as of 05:30, 22 September 2015
Line 1:		Line 1:
	{{quotation\|'''~~POTENTIAL INACCURACIES HAVE BEEN POINTED OUT IN THE CONTENT OF THIS PAGE~~. ~~PLEASE READ WITH CAUTION WHILE THIS MESSAGE IS STILL PRESENT~~.'''}}		{{quotation\|'''The details of the page have been blanked out due to a notification about an error in the statement; see [http://mathoverflow.net/questions/75472/kuenneth-formula-for-group-cohomology-with-nontrivial-action-on-the-coefficient Math Overflow for more].'''}}

	==Statement==		==Statement==
Line 5:		Line 5:
	===For trivial group action===		===For trivial group action===

	Suppose <math>G_1,G_2</math> are [[group]]s and <math>M</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1\| ]][[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>M</math> in terms of the [[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>M</math>:		{{fillin}}

	~~<math>H^p(G_1 \times G_2; M) \cong \left(\sum_~~{~~i+j = p} H^i(G_1;M) \otimes H^j(G_2;M) \right) \oplus \left(\sum_{u + v = p + 1} \operatorname{Tor}^1_{\mathbb~~{Z}}~~(H^u(G_1;M),H^v(G_2;M))\right)</math>~~

	===For nontrivial group action===		===For nontrivial group action===

Vipul at 05:28, 22 September 2015

2015-09-22T05:28:48Z

← Older revision		Revision as of 05:28, 22 September 2015
Line 1:		Line 1:
			{{quotation\|'''POTENTIAL INACCURACIES HAVE BEEN POINTED OUT IN THE CONTENT OF THIS PAGE. PLEASE READ WITH CAUTION WHILE THIS MESSAGE IS STILL PRESENT.'''}}

	==Statement==		==Statement==

Vipul at 22:52, 12 January 2013

2013-01-12T22:52:52Z

← Older revision		Revision as of 22:52, 12 January 2013
Line 5:		Line 5:
	Suppose <math>G_1,G_2</math> are [[group]]s and <math>M</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1\| ]][[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>M</math> in terms of the [[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>M</math>:		Suppose <math>G_1,G_2</math> are [[group]]s and <math>M</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1\| ]][[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>M</math> in terms of the [[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>M</math>:

	<math>H^p(G_1 \times G_2; M) \cong \left(\sum_{i+j = p} H^i(G_1;M) \otimes H^j(G_2;M) \right) \oplus \left(\sum_{u + v = p + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^u(G_1;M),H^v(G_2;M)\right)</math>		<math>H^p(G_1 \times G_2; M) \cong \left(\sum_{i+j = p} H^i(G_1;M) \otimes H^j(G_2;M) \right) \oplus \left(\sum_{u + v = p + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^u(G_1;M),H^v(G_2;M))\right)</math>

	===For nontrivial group action===		===For nontrivial group action===

Vipul at 21:58, 12 October 2011

2011-10-12T21:58:45Z

← Older revision		Revision as of 21:58, 12 October 2011
Line 3:		Line 3:
	===For trivial group action===		===For trivial group action===

	Suppose <math>G_1,G_2</math> are [[group]]s and <math>A</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1\| ]][[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>A</math> in terms of the [[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>A</math>:		Suppose <math>G_1,G_2</math> are [[group]]s and <math>M</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1\| ]][[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>M</math> in terms of the [[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>M</math>:

	<math>H^p(G_1 \times G_2; A) \cong \left(\sum_{i+j = p} H^i(G_1;A) \otimes H^j(G_2;A) \right) \oplus \left(\sum_{u + v = p + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^u(G_1;A),H^v(G_2;A)\right)</math>		<math>H^p(G_1 \times G_2; M) \cong \left(\sum_{i+j = p} H^i(G_1;M) \otimes H^j(G_2;M) \right) \oplus \left(\sum_{u + v = p + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^u(G_1;M),H^v(G_2;M)\right)</math>

	===For nontrivial group action===		===For nontrivial group action===

Vipul at 21:57, 12 October 2011

2011-10-12T21:57:29Z

← Older revision		Revision as of 21:57, 12 October 2011
Line 5:		Line 5:
	Suppose <math>G_1,G_2</math> are [[group]]s and <math>A</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1\| ]][[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>A</math> in terms of the [[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>A</math>:		Suppose <math>G_1,G_2</math> are [[group]]s and <math>A</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1\| ]][[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>A</math> in terms of the [[cohomology group for trivial group action\|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>A</math>:

	<math>H^n(G_1 \times G_2; A) \cong \left(\sum_{i+j = n} H^i(G_1;A) \otimes H^j(G_2;A) \right) \oplus \left(\sum_{p + q = n + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^p(G_1;A),H^q(G_2;A)\right)</math>		<math>H^p(G_1 \times G_2; A) \cong \left(\sum_{i+j = p} H^i(G_1;A) \otimes H^j(G_2;A) \right) \oplus \left(\sum_{u + v = p + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^u(G_1;A),H^v(G_2;A)\right)</math>

	===For nontrivial group action===		===For nontrivial group action===

Vipul: /* Related facts */

2011-10-12T21:01:17Z

Related facts

← Older revision		Revision as of 21:01, 12 October 2011
Line 13:		Line 13:
	==Related facts==		==Related facts==

			* [[Kunneth formula for group homology]]
	* [[Cohomology group for trivial group action commutes with direct product in second coordinate]]		* [[Cohomology group for trivial group action commutes with direct product in second coordinate]]

Vipul: /* Related facts */

2011-10-03T20:07:38Z

Related facts

← Older revision		Revision as of 20:07, 3 October 2011
Line 13:		Line 13:
	==Related facts==		==Related facts==

	* [[~~Second cohomology~~ group for trivial group action commutes with direct product in second coordinate]]		* [[Cohomology group for trivial group action commutes with direct product in second coordinate]]

Vipul at 19:53, 3 October 2011

2011-10-03T19:53:53Z

← Older revision		Revision as of 19:53, 3 October 2011
Line 10:		Line 10:

	{{fillin}}		{{fillin}}

			==Related facts==

			* [[Second cohomology group for trivial group action commutes with direct product in second coordinate]]

Vipul: Created page with "==Statement== ===For trivial group action=== Suppose $G_1,G_2$ are groups and $A$ is an abelian group. We have the following formula for the [[fac..."

2011-10-03T19:51:35Z

Created page with "==Statement== ===For trivial group action=== Suppose <math>G_1,G_2</math> are groups and <math>A</math> is an abelian group. We have the following formula for the [[fac..."

New page

==Statement==

===For trivial group action===

Suppose <math>G_1,G_2</math> are [[group]]s and <math>A</math> is an [[abelian group]]. We have the following formula for the [[fact about::cohomology group for trivial group action;1| ]][[cohomology group for trivial group action|cohomology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>A</math> in terms of the [[cohomology group for trivial group action|cohomology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <matH>A</math>:

<math>H^n(G_1 \times G_2; A) \cong \left(\sum_{i+j = n} H^i(G_1;A) \otimes H^j(G_2;A) \right) \oplus \left(\sum_{p + q = n + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^p(G_1;A),H^q(G_2;A)\right)</math>

===For nontrivial group action===

{{fillin}}

Kunneth formula for group cohomology - Revision history

Vipul at 05:30, 22 September 2015

Vipul at 05:28, 22 September 2015

Vipul at 22:52, 12 January 2013

Vipul at 21:58, 12 October 2011

Vipul at 21:57, 12 October 2011

Vipul: /* Related facts */

Vipul: /* Related facts */

Vipul at 19:53, 3 October 2011

Vipul: Created page with "==Statement== ===For trivial group action=== Suppose G_1,G_2 are groups and A is an abelian group. We have the following formula for the [[fac..."

Vipul: Created page with "==Statement== ===For trivial group action=== Suppose $G_1,G_2$ are groups and $A$ is an abelian group. We have the following formula for the [[fac..."