One-sided Gorenstein subcategories

Weiling Song; Tiwei Zhao; Zhaoyong Huang

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 2, page 483-504
  • ISSN: 0011-4642

Abstract

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We introduce the right (left) Gorenstein subcategory relative to an additive subcategory 𝒞 of an abelian category 𝒜 , and prove that the right Gorenstein subcategory r 𝒢 ( 𝒞 ) is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When 𝒞 is self-orthogonal, we give a characterization for objects in r 𝒢 ( 𝒞 ) , and prove that any object in 𝒜 with finite r 𝒢 ( 𝒞 ) -projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in 𝒜 with finite 𝒞 -projective dimension to an object in r 𝒢 ( 𝒞 ) . As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in 𝒜 having enough injectives.

How to cite

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Song, Weiling, Zhao, Tiwei, and Huang, Zhaoyong. "One-sided Gorenstein subcategories." Czechoslovak Mathematical Journal 70.2 (2020): 483-504. <http://eudml.org/doc/297412>.

@article{Song2020,
abstract = {We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\mathcal \{C\}$ of an abelian category $\mathcal \{A\}$, and prove that the right Gorenstein subcategory $r\mathcal \{G\}(\mathcal \{C\})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\mathcal \{C\}$ is self-orthogonal, we give a characterization for objects in $r\mathcal \{G\}(\mathcal \{C\})$, and prove that any object in $\mathcal \{A\}$ with finite $r\mathcal \{G\}(\mathcal \{C\})$-projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $\mathcal \{A\}$ with finite $\mathcal \{C\}$-projective dimension to an object in $r\mathcal \{G\}(\mathcal \{C\})$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\mathcal \{A\}$ having enough injectives.},
author = {Song, Weiling, Zhao, Tiwei, Huang, Zhaoyong},
journal = {Czechoslovak Mathematical Journal},
keywords = {right Gorenstein subcategory; self-orthogonal subcategory; relative projective dimension; cotorsion pair; kernel; (weak) Auslander-Buchweitz context},
language = {eng},
number = {2},
pages = {483-504},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {One-sided Gorenstein subcategories},
url = {http://eudml.org/doc/297412},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Song, Weiling
AU - Zhao, Tiwei
AU - Huang, Zhaoyong
TI - One-sided Gorenstein subcategories
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 2
SP - 483
EP - 504
AB - We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\mathcal {C}$ of an abelian category $\mathcal {A}$, and prove that the right Gorenstein subcategory $r\mathcal {G}(\mathcal {C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\mathcal {C}$ is self-orthogonal, we give a characterization for objects in $r\mathcal {G}(\mathcal {C})$, and prove that any object in $\mathcal {A}$ with finite $r\mathcal {G}(\mathcal {C})$-projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $\mathcal {A}$ with finite $\mathcal {C}$-projective dimension to an object in $r\mathcal {G}(\mathcal {C})$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\mathcal {A}$ having enough injectives.
LA - eng
KW - right Gorenstein subcategory; self-orthogonal subcategory; relative projective dimension; cotorsion pair; kernel; (weak) Auslander-Buchweitz context
UR - http://eudml.org/doc/297412
ER -

References

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