There is a notion of Bockstein homomorphism $\beta$. I am interested precisly in the case of sequence $$0 \rightarrow \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p^2 \mathbb{Z} \rightarrow \mathbb{Z}/p \rightarrow 0$$
Wikipedia says that $\beta^2 = 0$. So we can consider cohomology of $\beta$. The chains of this complex are cohomology $H^i ( X, \mathbb{Z}/p \mathbb{Z} ) $ and $\beta$ is the differential. Moreover this cohomology has structure of $dg$ algebra (usual cohomology has such structure and $\beta$ is a differential).
This cohomology is neither zero (for instance there are spaces with no odd cohomology as $\mathbb{CP^n}$) nor coincides with usual cohomology (as far as $\beta$ is not 0).
Question Did anybody consider this cohomology? Do you know any statments or references about it? Do they have a name?
This question is duplicated from math stack exchange, where it got no response. https://math.stackexchange.com/questions/1528408/bockstein-cohomology
