Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism

$$\beta:H^n(G,\mathbb{F}_p)\to H^{n+1}(G,\mathbb{F}_p)$$

of the long exact sequence associated to the short exact sequence of $kG$-modules

$$0\to\mathbb{F}_p\xrightarrow{p}\mathbb{Z}/p^2\mathbb{Z}\to\mathbb{F}_p\to 0$$

My question is:

How do I view $\mathbb{F}_p$ and $\mathbb{Z}/p^2\mathbb{Z}$ as $kG$-modules in this short exact sequence?

It seems that $G$ should act trivially, so that for some $x\in\mathbb{F}_p$ (or $\mathbb{Z}/p^2\mathbb{Z}$) we have

$$\left(\sum\alpha_gg\right)\cdot x=\left(\sum\alpha_g\right)\cdot x=\sum(\alpha_g\cdot x)$$

At this point, I don't know how to define $\alpha_g\cdot x$ for $\alpha_g\in k$. It cannot be trivial, or else the second equality above would not always be true (if $\alpha_g\cdot x=x$ for all $\alpha_g$, then the second equality above would give $x=|G|x$). Even for the cohomology group $H^n(G,\mathbb{F}_p)$ to make sense, we must have some $kG$-module structure on $\mathbb{F}_p$. I'm comfortable defining a $\mathbb{Z}G$-module structure for $\mathbb{F}_p$ simply through the maps $\mathbb{Z}G\to\mathbb{Z}\to\mathbb{F}_p$, but I'm not sure what to do when working over an algebraically closed field $k$.

Edit: I'm interested in the Bockstein because I'm trying to follow a proof of the following theorem due to Quillen:

Suppose $k$ is a field of characteristic $p$, $M$ is a $kG$-module and $H$ is a normal subgroup of index $p$ in $G$. If $\zeta$ is an element of positive degree in $\mathrm{Ext}^*_{kG}(M,M)$ with $\mathrm{res}_{G,H}(\zeta)=0$, then $\zeta^2=\beta(x)\cdot\zeta'$ for some $\zeta'\in\mathrm{Ext}^*_{kG}(M,M)$, where $x$ is the inflation to $G$ of a non-zero element $\overline{x}\in H^1(G/H,\mathbb{F}_p)$.

Notice for the product $\beta(x)\cdot \zeta'$ to make sense, I should have $\beta(x)\in\mathrm{Ext}^2_{kG}(k,k)$, but according to Qiaochu's comment below, we have $\beta(x)\in\mathrm{Ext}^2_{\mathbb{Z}G}(\mathbb{Z},\mathbb{F}_p)$. It seems the map $\mathbb{F}_p\to k$ allows us to view $\beta(x)\in\mathrm{Ext}^2_{\mathbb{Z}G}(\mathbb{Z},k)$. Is there also a natural map $\mathrm{Ext}^2_{\mathbb{Z}G}(\mathbb{Z},k)\to\mathrm{Ext}^2_{kG}(k,k)$?

  • 1
    $\begingroup$ I don't understand your setup. What's the relationship between $k[G]$-modules and group cohomology with coefficients in $\mathbb{F}_p$? The latter is naturally a $\mathbb{Z}[G]$ or $\mathbb{F}_p[G]$-module, no more. $\endgroup$ – Qiaochu Yuan Apr 1 '14 at 4:35
  • $\begingroup$ @QiaochuYuan: The definition of group cohomology I'm working with is Ext in the category of $kG$-modules: $H^n(G,M):=\mathrm{Ext}^n_{kG}(k,M)$ where $k$ is the trivial $kG$-module, and $M$ is a $kG$-module. The references I've checked (Evens, Benson) just assume $k$ is commutative, or perhaps a Dedekind domain or Noetherian. Then they jump right into defining the Bockstein homomorphism, which leads me to believe they're making no further assumptions of $k$. Please note the edit in the question. $\endgroup$ – Jared Apr 1 '14 at 5:46

The answer to your question is that you don't view those things as $kG$-modules. To construct the Bockstein, you identify $\operatorname{Ext}_{\mathbb{F}_pG}(\mathbb{F}_p, \mathbb{F}_p)$ with $\operatorname{Ext}_{\mathbb{Z}G}(\mathbb{Z}, \mathbb{F}_p)$. You have a short exact sequence of $\mathbb{Z}G$-modules with trivial action $$ 0 \to \mathbb{F}_p \to \mathbb{Z}/p^2 \to \mathbb{F}_p \to 0$$ so applying $\operatorname{Hom}_{\mathbb{Z}G}(\mathbb{Z}, -)$ you get a long exact sequence whose connecting homomorphism maps $$\operatorname{Ext}^n_{\mathbb{Z}G}(\mathbb{Z}, \mathbb{F}_p) \to \operatorname{Ext}^{n+1}_{\mathbb{Z}G}(\mathbb{Z}, \mathbb{F}_p) $$ Identifying these groups with $\operatorname{Ext}^n_{\mathbb{F}_pG}(\mathbb{F}_p, \mathbb{F}_p)$ and $\operatorname{Ext}^{n+1}_{\mathbb{F}_pG}(\mathbb{F}_p, \mathbb{F}_p)$ you get the Bockstein. To understand the construction on a down-to-earth level, a good exercise is to check that if $f :G \to \mathbb{F}_p$ is a group homomorphism representing an element of $\operatorname{Ext}^1_{\mathbb{F}_pG} (\mathbb{F}_p, \mathbb{F}_p)$ then its Bockstein is the 2-cocycle for the bar resolution sending $[g|h]$ to 1 if $\widehat{f(g)} + \widehat{f(h)} \geq p$ and 0 otherwise. Here if $x \in \mathbb{F}_p$ then $\hat{x}$ is the canonical representative of $x$ in $\{0,1,\ldots,p-1\} \subset \mathbb{Z}$.

To extend to larger fields $k \supset \mathbb{F}_p$, use the fact that $\operatorname{Ext}_{kG}(k,k) = \operatorname{Ext}_{\mathbb{F}_pG}(\mathbb{F}_p,\mathbb{F}_p) \otimes_{\mathbb{F}_p} k$ and extend semilinearly from the map defined above. If you are reading Benson's book, see Representations and Cohomology II section 4.3.


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