The stable category of monomorphisms between (Gorenstein) projective modules with applications

Abstract

Let (S, n) be a commutative noetherian local ring and let ω ∈ n be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by Mon(ω, P) and Mon(ω, G), are both Frobenius categories with the same projective objects. It is also proved that the stable category Mon(ω, P) is triangle equivalent to the category of D-branes of type B, DB(ω), which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories Mon(ω, P) and Mon(ω, G) are closely related to the singularity category of the factor ring R = S/(ω). Precisely, there is a fully faithful triangle functor from the stable category Mon(ω, G) to Dsg(R), which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to Mon(ω, P), guarantees the regularity of the ring S. © 2024 Walter de Gruyter GmbH. All rights reserved.