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Let $G$ be a group, $\Lambda = \mathbb Z[G]$ the group ring, and

$$\cdots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb Z \rightarrow 0 $$

the standard resolution of $\mathbb Z$, regarded as a trivial $\Lambda$-module, by free $\Lambda$-modules $P_i = \mathbb Z[G^{i+1}]$. Each $P_i \rightarrow P_{i-1}$ is defined on generators by

$$(g_0, ... , g_i) \mapsto \sum\limits_{j=0}^{i} (-1)^j (g_0, ... , \hat{g_j}, ... , g_i)$$

To define the homology group $H_i(G,A)$, we apply the functor $(-) \otimes_{\Lambda} A$ to the free resolution, getting a complex

$$ \cdots \rightarrow K_2 \rightarrow K_1 \rightarrow K_0 \rightarrow 0$$

where $K_i = P_i \otimes_{\Lambda} A$, and take homology at $i$.

I have seen the following explicit description of the "one-cycles" defining $H_1(G,A)$ (Caessels and Frohlich, ANT, Chapter 4, page 98):

A one-cycle of $G$ with values in $A$ is a function $f: G \rightarrow A$ such that $f(s) = 0$ for almost all $s \in G$ and such that $df = \sum\limits_{s \in G} (s^{-1}-1)f(s) = 0$.

How can one think of elements of $P_1 \otimes_{\Lambda} A = \mathbb Z[G^2] \otimes_{\Lambda} A$ as functions $G \rightarrow A$?

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  • $\begingroup$ Is there a confusion between 1-cycle in homology and 1-cocyle in cohomology ? Also, you might want to learn the difference between inhomogeneous and homogeneous cocycles. Inhomogeneous cocycles are maps $G^n\to A$ instead of $G^{n+1}\to A$. $\endgroup$ – Roland May 28 '18 at 17:12
  • $\begingroup$ I understand the difference between 1-cycle and 1-cocycle. Atiyah and Wall's article on group cohomology claims that one-cycles (not cocycles) can also be understood as functions $G \rightarrow A$. $\endgroup$ – D_S May 28 '18 at 18:11
  • $\begingroup$ Ok I asked because you wrote "one-cocycle". I don't know then, is it a compactly supported function $G\to A$ ? $\endgroup$ – Roland May 28 '18 at 21:00
  • $\begingroup$ Sorry, I have fixed it $\endgroup$ – D_S May 28 '18 at 21:29
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First, notice that $\mathbb{Z}[G \times G] = \mathbb{Z} [G]\otimes_\mathbb{Z} \mathbb{Z}[G]$, so that we can identify $K_1$ with $\mathbb{Z}[G] \otimes_\mathbb{Z} A$. Now, elements of this last group are formal sums $\sum_{g \in G} a_g g$ with $a_g \in A$, only finitely many of which are non zero. We can think of the coefficients $a_g$ as specifying a function $a : G \to A$. Now, tranlasting the cycle condition into a condition about the function $a : G \to A$ should give us the desired description.

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