Bockstein homomorphism

This article defines a stable cohomology operation

Definition

Given a short exact sequence of Abelian groups:

$0\to P\to Q\to R\to 0$

We can define a Bockstein homomorphism (denoted $\beta$) of type $(R,P)$ and degree $1$. This is the connecting homomorphism in the associated long exact sequence of cohomology, for the cochain complex:

$\to H^{i}(C;P)\to H^{i}(C;Q)\to H^{i}(C;R)\to H^{i+1}(C;P)\to \ldots$

Thus, $\beta _{i}$ is the map:

$H^{i}(C;R)\to H^{i+1}(C;P)$

The Bockstein homomorphism for a prime field of $p$ elements is defined as the Bockstein homomorphism for the short exact sequence:

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