Quasi-abelian hearts of twin cotorsion pairs on triangulated categories
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866909293677117440 |
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| author | Shah, Amit |
| author_facet | Shah, Amit |
| contents | We prove that, under a mild assumption, the heart H of a twin cotorsion pair ((S,T),(U,V)) on a triangulated category C is a quasi-abelian category. If C is also Krull-Schmidt and T=U, we show that the heart of the cotorsion pair (S,T) is equivalent to the Gabriel-Zisman localisation of H at the class of its regular morphisms.
In particular, suppose C is a cluster category with a rigid object R and [X_R] the ideal of morphisms factoring through X_R=Ker(Hom(R,-)), then applications of our results show that C/[X_R] is a quasi-abelian category. We also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of C. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_05423 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Quasi-abelian hearts of twin cotorsion pairs on triangulated categories Shah, Amit Category Theory Representation Theory 18E30, 16G20, 18E05, 18E35, 18E40 We prove that, under a mild assumption, the heart H of a twin cotorsion pair ((S,T),(U,V)) on a triangulated category C is a quasi-abelian category. If C is also Krull-Schmidt and T=U, we show that the heart of the cotorsion pair (S,T) is equivalent to the Gabriel-Zisman localisation of H at the class of its regular morphisms. In particular, suppose C is a cluster category with a rigid object R and [X_R] the ideal of morphisms factoring through X_R=Ker(Hom(R,-)), then applications of our results show that C/[X_R] is a quasi-abelian category. We also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of C. |
| title | Quasi-abelian hearts of twin cotorsion pairs on triangulated categories |
| topic | Category Theory Representation Theory 18E30, 16G20, 18E05, 18E35, 18E40 |
| url | https://arxiv.org/abs/1807.05423 |