The stable category of monomorphisms between (Gorenstein) projective modules with applications
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916393254912000 |
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| author | Bahlekeh, Abdolnaser Fotouhi, Fahimeh Sadat Hamlehdari, Mohammad Amin Salarian, Shokrollah |
| author_facet | Bahlekeh, Abdolnaser Fotouhi, Fahimeh Sadat Hamlehdari, Mohammad Amin Salarian, Shokrollah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16514 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The stable category of monomorphisms between (Gorenstein) projective modules with applications Bahlekeh, Abdolnaser Fotouhi, Fahimeh Sadat Hamlehdari, Mohammad Amin Salarian, Shokrollah Representation Theory Commutative Algebra 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. |
| title | The stable category of monomorphisms between (Gorenstein) projective modules with applications |
| topic | Representation Theory Commutative Algebra |
| url | https://arxiv.org/abs/2405.16514 |