First cohomology group

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This article gives a basic definition in the following area: group cohomology
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Let G be a group acting on an abelian group A, via an action \varphi:G \to \operatorname{Aut}(A). Equivalently, A is a module over the (possibly non-commutative) unital group ring \mathbb{Z}G of G over the ring of integers.

Definition in cohomology terms

The first cohomology group H^1_\varphi(G,A) is an abelian group defined in the following equivalent ways.

When \varphi is understood from context, the subscript {}_\varphi may be omitted in the notation for the cohomology group, as well as the notation for the groups of 1-cocycles and 1-coboundaries.

No. Shorthand Detailed description of H^1_\varphi(G,A), the second cohomology group
1 Explicit, using the bar resolution H^1_\varphi(G,A), is defined as the quotient Z^1_\varphi(G,A)/B^1_\varphi(G,A) where Z^1_\varphi(G,A) is the group of 1-cocycles for the action and B^1_\varphi(G,A) is the group of 1-coboundaries.
1' Explicit, using the normalized bar resolution Same as definition (1), but we use normalized cocycles and normalized coboundaries instead of arbitrary cocycles and coboundaries.
2 Complex based on arbitrary resolution Let \mathcal{F} be a projective resolution for \mathbb{Z} as a \mathbb{Z}G-module with the trivial action. Let \mathcal{C} be the complex \operatorname{Hom}_{\mathbb{Z}G}(\mathcal{F},A). The cohomology group H^1_\varphi(G,A) is defined as the first cohomology group for this complex.
3 As an \operatorname{Ext}^1 functor \operatorname{Ext}^1_{\mathbb{Z}G}(\mathbb{Z},A) where \mathbb{Z} is a trivial \mathbb{Z}G-module and A has the module structure specified by \varphi.
4 As a right derived functor H^1_\varphi(G,A) = R^1(-^G)(A), i.e., it is the first right derived functor of the invariants functor for G (denoted -^G) evaluated at A. The invariants functor sends a \mathbb{Z}G-module to its submodule of elements fixed by all elements of G.

All these definitions have natural analogues for the n^{th} cohomology group H^n_\varphi(G,A) for all n \ge 0. For more, see cohomology group.

Definition in terms of stability automorphisms of extensions

Suppose E is a group that has an abelian normal subgroup identified with A and such that the quotient group E/A is identified with G (we abuse notation here and treat A and G as the actual subgroup and quotient group respectively). Further, assume that the induced action of the quotient on the subgroup is the same as the group action \varphi.

Then, H^1_\varphi(G,A) is a group quotient:

(Group of stability automorphisms of the chain 1 \underline{\triangleleft} A \underline{\triangleleft} E)/(Subgroup comprising the stability automorphisms of the chain that are induced from conjugation by elements of A)

More precisely, the group of all stability automorphisms can be naturally identified with the 1-cocycle group Z^1_\varphi(G,A) and the group of stability automorphisms arising via conjugation by an element of A can be naturally identified with the 1-coboundary group B^1_\varphi(G,A).

Particular cases

If the action of G on A is the trivial group action (i.e., every element of G fixes every element of A), then the first cohomology group H^1(G,A) can be naturally identified with the set \operatorname{Hom}(G,A) endowed with a group structure under pointwise addition. Specifically, the group of 1-cocycles is identified with the group of homomorphisms under pointwise addition and the group of 1-coboundaries is the trivial group.Further information: First cohomology group for trivial group action is naturally isomorphic to group of homomorphisms

Related notions