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

Author:

Bahlekeh Abdolnaser1,Fotouhi Fahimeh Sadat2,Hamlehdari Mohammad Amin3,Salarian Shokrollah4

Affiliation:

1. Department of Mathematics , Gonbad-Kavous University , Postal Code: 4971799151 , Gonbad-e-Kavous , Iran

2. School of Mathematics , Institute for Research in Fundamental Science (IPM) , P.O.Box: 19395-5746 , Tehran , Iran

3. Department of Pure Mathematics , Faculty of Mathematics and Statistics , University of Isfahan , P.O.Box: 81746-73441 , Isfahan , Iran

4. Department of Pure Mathematics , Faculty of Mathematics and Statistics , University of Isfahan , P.O.Box: 81746-73441 , Isfahan ; and School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O.Box: 19395-5746, Tehran , Iran

Abstract

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.

Funder

Iran National Science Foundation

Publisher

Walter de Gruyter GmbH

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