First cohomology group

This article gives a basic definition in the following area: group cohomology
View other basic definitions in group cohomology |View terms related to group cohomology |View facts related to group cohomology

Definition

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