K\"unneth and extension theorems for Morita invariants of non-commutative schemes

TL;DR

This paper extends Morita invariants to non-commutative schemes using derived Morita theory, providing new formulas and insights into their algebraic properties and applications to Hochschild cohomology of complex categories.

Contribution

It develops K"unneth and extension theorems for Morita invariants in non-commutative schemes, broadening their applicability and understanding.

Findings

Extended identities for Morita invariants to non-commutative schemes.

Established a K"unneth formula for Hochschild cohomology in this context.

Linked derived Morita theory with dg-enhancements of schemes.

Abstract

In this article, we apply the derived Morita theory of dg-categories to show how to extend the domain of validity of many identities relating Morita invariants from associative dg-algebras toward non-commutative scheme. Doing so, we obtain that the dg-category of associative algebras can be used to test the exactness of any sequence and the commutativity of any diagram involving Morita invariants depending multilinearly on their arguments, under a mild condition of stability by cofibrant replacement. This gives a simple picture of how the underlying algebra of a non-commutative scheme captures its Morita invariant properties. As an application, we use a K\"unneth formula on non-commutative schemes to factorise the Hochschild cohomology of a product of quasi-phantom categories such as built by Orlov and Gorshinsky in [GO13]. This is an occasion to explore how the derived Morita theory interacts with the \v{C}ech dg-enhancement of a projective scheme and to shed an unifying light upon the landscape of dg-modules over dg-categories.