Cocycle for a group action: Difference between revisions

Latest revision as of 03:09, 20 October 2010

Definition

Suppose $G$ is a group and $A$ is an abelian group, with an action $\varphi$ of $G$ on $A$ . In other words, $\varphi$ is a homomorphism of groups from $G$ to $\operatorname {Aut} (A)$ , the automorphism group of $A$ .

Definition in terms of bar resolution

A $n$ -cocycle is an element in the $n^{th}$ cocycle group for the Hom complex from the bar resolution of $G$ to $A$ , in the sense of $\mathbb {Z} G$ -modules.

Explicit definition

For $n$ a nonnegative integer, a $n$ -cocycle for the action $\varphi$ of $G$ on $A$ is a function $f:G^{n}\to A$ such that, for all $g_{1},g_{2},\dots ,g_{n+1}\in G$ :

$\!\varphi (g_{1})(f(g_{2},g_{3},\dots ,g_{n+1}))+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]+(-1)^{n+1}f(g_{1},g_{2},\dots ,g_{n})=0$

If we suppress the symbol $\varphi$ and denote the action by $\cdot$ , this becomes:

$\!g_{1}\cdot f(g_{2},g_{3},\dots ,g_{n+1}))+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]+(-1)^{n+1}f(g_{1},g_{2},\dots ,g_{n})=0$

In particular, when the action is trivial, this is equivalent to saying that:

$\!f(g_{2},g_{3},\dots ,g_{n+1})+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]+(-1)^{n+1}f(g_{1},g_{2},\dots ,g_{n})=0$

Particular cases

$n$	Condition for being a $n$ -cocycle	Further information
1	For all $g_{1},g_{2}\in G$ , we have $\!g_{1}\cdot f(g_{2})-f(g_{1}g_{2})+f(g_{1})=0$ , equivalently $\!f(g_{1}g_{2})=f(g_{1})+g_{1}\cdot f(g_{2})$	1-cocycle for a group action
2	For all $g_{1},g_{2},g_{3}\in G$ , we have $\!g_{1}\cdot f(g_{2},g_{3})-f(g_{1}g_{2},g_{3})+f(g_{1},g_{2}g_{3})-f(g_{1},g_{2})=0$ , equivalently $g_{1}\cdot f(g_{2},g_{3})+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{2},g_{3})$	2-cocycle for a group action
3	For all $g_{1},g_{2},g_{3},g_{4}\in G$ , we have $\!g_{1}\cdot f(g_{2},g_{3},g_{4})-f(g_{1}g_{2},g_{3},g_{4})+f(g_{1},g_{2}g_{3},g_{4})-f(g_{1},g_{2},g_{3}g_{4})+f(g_{1},g_{2},g_{3})=0$ , or equivalently, $\!g_{1}\cdot f(g_{2},g_{3},g_{4})+f(g_{1},g_{2}g_{3},g_{4})+f(g_{1},g_{2},g_{3})=f(g_{1}g_{2},g_{3},g_{4})+f(g_{1},g_{2},g_{3}g_{4})$	3-cocycle for a group action

@@ Line 1: / Line 1: @@
 ==Definition==
-Suppose <math>G</math> is a [[group]] and <math>A</math> is an [[abelian group]], with an action of <math>G</math> on <math>A</math>.
+Suppose <math>G</math> is a [[group]] and <math>A</math> is an [[abelian group]], with an action <math>\varphi</math> of <math>G</math> on <math>A</math>. In other words, <math>\varphi</math> is a [[homomorphism of groups]] from <math>G</math> to <math>\operatorname{Aut}(A)</math>, the [[automorphism group]] of <math>A</math>.
-For <math>n</math> a nonnegative integer, a <math>n</math>-cocycle for the action <math>\varphi</math> of <math>G</math> on <math>A</math> is a function <math>f:G^n \to A</math> such that, for all <math>g_1,g_2, \dots, g_{n+1} \in G</math>:
+===Definition in terms of bar resolution===
-<math>\! g_1 \cdot f(g_2,g_3,\dots,g_{n+1}) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] - f(g_1,g_2,\dots,g_n) = 0</math>
+A <math>n</math>-cocycle is an element in the <math>n^{th}</math> cocycle group for the Hom complex from the [[defining ingredient::bar resolution]] of <math>G</math> to <math>A</math>, in the sense of <math>\mathbb{Z}G</math>-modules.
-In particular, when the action is trivial, this is equivalent to saying that:
+===Explicit definition===
+For <math>n</math> a nonnegative integer, a <math>n</math>-cocycle for the action <math>\varphi</math> of <math>G</math> on <math>A</math> is a function <math>f:G^n \to A</math> such that, for all <math>g_1,g_2, \dots, g_{n+1} \in G</math>:
-<math>\! f(g_2,g_3,\dots,g_{n+1}) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] - f(g_1,g_2,\dots,g_n) = 0</math>
+<math>\! \varphi(g_1)(f(g_2,g_3,\dots,g_{n+1})) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n + 1}f(g_1,g_2,\dots,g_n) = 0</math>
-==Particular cases==
+If we suppress the symbol <math>\varphi</math> and denote the action by <math>\cdot</math>, this becomes:
-===A 1-cocycle===
+<math>\! g_1 \cdot f(g_2,g_3,\dots,g_{n+1})) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} f(g_1,g_2,\dots,g_n) = 0</math>
-{{further|[[1-cocycle for a group action]]}}
+In particular, when the action is trivial, this is equivalent to saying that:
-A 1-cocycle is a function <math>f:G \to A</math> such that:
+<math>\! f(g_2,g_3,\dots,g_{n+1}) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} f(g_1,g_2,\dots,g_n) = 0</math>
-<math>\! g_1 \cdot f(g_2) - f(g_1g_2) + f(g_1) = 0</math>
+==Particular cases==
-In particular,a 1-cocycle for the trivial group action is a [[homomorphism of groups]] from <math>G</math> to <math>A</math>.
-===A 2-cocycle===
-{{further|[[2-cocycle for a group action]], [[2-cocycle for trivial group action]]}}
-A 2-cocycle is a function <math>f:G \times G \to A</math> such that:
-<math>\! g_1 \cdot f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0</math>
-In particular, a 2-cocycle for the trivial group action is a function <math>f:G \times G \to A</math> such that:
-<math>\! f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0</math>
+{| class="sortable" border="1"
+! <math>n</math> !! Condition for being a <math>n</math>-cocycle !! Further information
+|-
+| 1 || For all <math>g_1,g_2\in G</math>, we have <math>\! g_1 \cdot f(g_2) - f(g_1g_2) + f(g_1) = 0</math>, equivalently <math>\! f(g_1g_2) = f(g_1) + g_1 \cdot f(g_2)</math> || [[1-cocycle for a group action]]
+|-
+| 2 ||  For all <math>g_1,g_2,g_3\in G</math>, we have <math>\! g_1 \cdot f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0</math>, equivalently <math>g_1 \cdot f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_2,g_3)</math> || [[2-cocycle for a group action]]
+|-
+| 3 || For all <math>g_1,g_2,g_3,g_4 \in G</math>, we have <math>\!g_1 \cdot f(g_2,g_3,g_4) - f(g_1g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) - f(g_1,g_2,g_3g_4) + f(g_1,g_2,g_3) = 0</math>, or equivalently, <math>\! g_1 \cdot f(g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) + f(g_1,g_2,g_3) = f(g_1g_2,g_3,g_4) + f(g_1,g_2,g_3g_4)</math> || [[3-cocycle for a group action]]
+|}