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