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:

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

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:

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