Bokstein homomorphism as a universal object

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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/p², 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 W₂(R) considered as a square-zero extension of R by the Frobenius twist R⁽¹⁾.