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## Bokstein homomorphism as a universal object

Advances in Mathematics. 2018. Vol. 324. P. 267-325.

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).