Minors and resolutions of non-commutative schemes

TL;DR

This paper develops the theory of minors for non-commutative schemes, aiming to advance non-commutative resolutions of singularities, including categorical resolutions for non-commutative curves and their realization as derived categories of quasi-hereditary algebras.

Contribution

It introduces the concept of minors in non-commutative schemes and constructs categorical resolutions, connecting them to derived categories of quasi-hereditary algebras.

Findings

Constructed categorical resolutions for non-commutative curves.

Showed that in the rational case, these resolutions can be realized as derived categories of quasi-hereditary algebras.

Established foundational theory linking non-commutative minors to resolutions of singularities.

Abstract

In this article we develop the theory of minors of non-commutative schemes. This study is motivated by applications in the theory of non-commutative resolutions of singularities of commutative schemes. In particular, we construct a categorical resolution for non-commutative curves and in the rational case show that it can be realized as the derived category of a quasi-hereditary algebra.