Bockstein homomorphism

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:

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