Work

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

Year: 2024

Type: article

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 )

Source: Forum Mathematicum

DOI: 10.1515/forum-2023-0317

Authors: Abdolnaser Bahlekeh, Fahimeh Sadat Fotouhi, Mohammad Amin Hamlehdari, Shokrollah Salarian

Institutions: Gonbad Kavous University, University of Isfahan, Institute for Research in Fundamental Sciences

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Iran National Science Foundation

Award: G6345885809