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Articles
Publication Date: 2025
Bulletin Of The Iranian Mathematical Society (10186301)51(2)
Let (S,n) be a commutative noetherian local ring and let ω∈n be non-zero divisor. This paper is concerned with the category of monomorphisms between finitely generated Gorenstein projective S-modules, such that their cokernels are annihilated by ω. We will observe that this category, which will be denoted by Mon(ω,G), is an exact category in the sense of Quillen. More generally, it is proved that Mon(ω,G) is a Frobenius category. Surprisingly, it is shown that not only the category of matrix factorizations embeds into Mon(ω,G), but also its stable category as well as the singularity category of the factor ring R=S/(ω), can be realized as triangulated subcategories of the stable category Mon̲(ω,G). © The Author(s) under exclusive licence to Iranian Mathematical Society 2025.
Bahlekeh, A.,
Fotouhi, F.S.,
Hamlehdari M.A.,
Salarian S.,
Salarian, S. Publication Date: 2024
Forum Mathematicum (9337741)37pp. 1185-1200
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.
Publication Date: 2023
Kyoto Journal of Mathematics (21543321)63(1)pp. 1-22
Let (R, m) be a d-dimensional commutative complete Noetherian local ring and A be a Noetherian R-algebra. Motivated by the notion of Cohen-Macaulay Artin algebras of Auslander and Reiten, we say that A is Cohen-Macaulay if there is a finitely generated A-bimodule w that is maximal Cohen-Macaulay over R such that the adjoint pair of functors (w circle times?' -, Hom?' (w, -)) induces quasi-inverse equivalences between the full subcategories of finitely generated A'-modules consisting of modules of finite projective dimension, P degrees degrees(A'), and the modules of finite injective dimension, I degrees degrees(A'), whenever A' = A, Aop. It is proved that such a module w is unique, up to isomorphism, as a A'-module. It is also shown that A is a Cohen-Macaulay algebra if and only if there is a semidualizing A-bimodule w of finite injective dimension and P degrees degrees(A') and I degrees degrees(A') are contained in the Auslander and Bass classes, respectively. We prove that Cohen-Macaulayness behaves well under reduction modulo system of parameters of R. Indeed, it will be observed that if A is a Cohen-Macaulay algebra, then for any system of parameters x = x1, ... , xd of R, the Artin algebra A/xA is Cohen-Macaulay as well. Assume that w is a semidualizing A-bimodule of finite injective dimension that is maximal Cohen-Macaulay as an R-module. It will turn out that A being a Cohen-Macaulay algebra is equivalent to saying that the pair (CM(A'),I degrees degrees(A')) forms a hereditary complete cotorsion theory and the pair (CM(A'op), P degrees degrees(A')) forms a Tor-torsion theory, where CM(A') is the class of all finitely generated A'-modules admitting a right resolution by modules in addw. Finally, it is shown that Cohen-Macaulayness ascends from R to RF and RQ, where F is a finite group and Q is a finite acyclic quiver.
