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

TL;DR

This paper explores categories of monomorphisms between (Gorenstein) projective modules over a local ring, revealing their Frobenius structure, connections to D-branes, and implications for ring regularity and singularity categories.

Contribution

It introduces and analyzes Frobenius categories of monomorphisms with cokernels annihilated by a non-zero divisor, establishing their relation to D-branes and singularity categories, and characterizing ring regularity.

Findings

Categories Mon(w;P) and Mon(w; G) are Frobenius with the same projectives.

Stable category Mon(w; P) is triangle equivalent to D-branes of type B.

Fully faithful functor from Mon(w; G) to the singularity category of R.

Abstract

Let (S; n) be a commutative noetherian local ring and let w in n be non-zero divisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by w. It is shown that these categories, which will be denoted by Mon(w;P) and Mon(w; G), are both Frobenius categories with the same projective objects. It is also proved that the stable category Mon(w;P) is triangle equivalent to the category of D-branes of type B, DB(w), which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories Mon(w;P) and Mon(w; G) are closely related to the singularity category of the factor ring R = S/(w). Precisely, there is a fully faithful triangle functor from the stable category Mon(w; 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(w;P), guarantees the regularity of the ring S.