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

Abstract

Let ( S , 𝔫 ) {(S,{\mathfrak{n}})} be a commutative noetherian local ring and let ω ∈ 𝔫 {\omega\in{\mathfrak{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 ) {{\mathsf{Mon}}(\omega,\mathcal{P})} and 𝖬𝗈𝗇 ⁢ ( ω , 𝒢 ) {{\mathsf{Mon}}(\omega,\mathcal{G})} , are both Frobenius categories with the same projective objects. It is also proved that the stable category 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} is triangle equivalent to the category of D-branes of type B, 𝖣𝖡 ⁢ ( ω ) {\mathsf{DB}(\omega)} , which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} and 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} are closely related to the singularity category of the factor ring R = S / ( ω ) {R=S/({\omega)}} . Precisely, there is a fully faithful triangle functor from the stable category 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} to 𝖣 𝗌𝗀 ⁡ ( R ) {\operatorname{\mathsf{D_{sg}}}(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 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} , guarantees the regularity of the ring S.