Noncommutative matrix factorizations with an application to skew exterior algebras

TL;DR

This paper introduces noncommutative matrix factorizations to study noncommutative hypersurfaces, establishing invariance under twist, category equivalences, and applying these concepts to skew exterior algebras.

Contribution

It defines noncommutative matrix factorizations, proves their invariance under twist, establishes category equivalences, and applies the theory to skew exterior algebras.

Findings

Category of noncommutative graded matrix factorizations is invariant under twist

Category equivalences with totally reflexive modules are established

Indecomposable noncommutative graded matrix factorizations over skew exterior algebras are described

Abstract

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras.