Cohomology of products and coproducts of augmented algebras

TL;DR

This paper investigates how cohomology functors behave with respect to products and coproducts of augmented algebras, revealing structural relationships and conditions under which Hochschild cohomology preserves these operations.

Contribution

It demonstrates that the ordinary cohomology functor exchanges coproducts and products, and characterizes the Hochschild cohomology structure of algebra products modulo an ideal.

Findings

Cohomology exchanges coproducts and products for augmented algebras.

Hochschild cohomology approximates sending coproducts to products when factors are self-injective.

The multiplicative structure of Hochschild cohomology of a product is described in terms of the factors' cohomology.

Abstract

We show that the ordinary cohomology functor from the category of augmented $k$-algebras to itself exchanges coproducts and products, and that Hochschild cohomology is close to sending coproducts to products if the factors are self-injective. We identify the multiplicative structure of the Hochschild cohomology of a product modulo a certain ideal in terms of the cohomology of the factors.