Bokstein homomorphism as a universal object

Abstract

We give a simple construction of the correspondence between square-zero extensions $R'$ of a ring $R$ by an $R$-bimodule $M$ and second MacLane cohomology classes of $R$ with coefficients in $M$ (the simplest non-trivial case of the construction is $R=M=Z/p$, $R'=Z/p^2$, thus the Bokstein homomorphism of the title). Following Jibladze and Pirashvili, we treat MacLane cohomology as cohomology of non-additive endofunctors of the category of projective $R$-modules. We explain how to describe liftings of $R$-modules and complexes of $R$-modules to $R'$ in terms of data purely over $R$. We show that if $R$ is commutative, then commutative square-zero extensions $R'$ correspond to multiplicative extensions of endofunctors. We then explore in detail one particular multiplicative non-additive endofunctor constructed from cyclic powers of a module $V$ over a commutative ring $R$ annihilated by a prime $p$. In this case, $R'$ is the second Witt vectors ring $W_2(R)$ considered as a square-zero extension of $R$ by the Frobenius twist $R^{(1)}$.