|
|
Pure Mathematics 2025
相对于余挠对的Gorenstein投射表示
|
Abstract:
设 Q 是一个有限无圈箭图,R是一个环,Rep(Q,R)是 Q 的左 R- 模表示范畴。本文描述了 Rep(Q,R) 中相对于余挠对的 Gorenstein 投射表示。具体地，我们证明了X∈Rep(Q,R) 是相对于(A,B)的 Gorenstein 投射表示当且仅当对任意的i∈？？0,是单同态,并且X(i),CokerφiX是相对于 (A,B)的Gorenstein投射模.
Let Q be a finite acyclic quiver, R a ring, and Rep(Q,R) the category of left R-module representations of Q. This paper characterizes Gorenstein projective representations in Rep(Q, R) relative to cotorsion pairs. Specifically, we prove that a representation X∈Rep(Q, R) is Gorenstein projective relative to (A, B) if and only if} for every vertex i∈Q0, the canonical morphism
| [1] | Enochs, E.E. and Herzog, I. (1999) A Homotopy of Quiver Morphisms with Applications to
Representations. Canadian Journal of Mathematics, 51, 294-308.
https://doi.org/10.4153/cjm-1999-015-0 |
| [2] | Enochs, E. and Estrada, S. (2005) Projective Representations of Quivers. Communications in
Algebra, 33, 3467-3478. https://doi.org/10.1081/agb-200058181 |
| [3] | Eshraghi, H., Hafezi, R. and Salarian, S. (2013) Total Acyclicity for Complexes of Represen-
tations of Quivers. Communications in Algebra, 41, 4425-4441.
https://doi.org/10.1080/00927872.2012.701682 |
| [4] | Xu, A. (2017) Gorenstein Modules and Gorenstein Model Structures. Glasgow Mathematical
Journal, 59, 685-703. https://doi.org/10.1017/s0017089516000483 |
| [5] | Holm, H. and Jorgensen, P. (2019) Cotorsion Pairs in Categories of Quiver Representations.
Kyoto Journal of Mathematics, 59, 575-606. https://doi.org/10.1215/21562261-2018-0018 |
| [6] | Asadollahi, J., Eshraghi, H., Hafezi, R. and Salarian, S. (2011) On the Homotopy Categories
of Projective and Injective Representations of Quivers. Journal of Algebra, 346, 101-115.
https://doi.org/10.1016/j.jalgebra.2011.08.028 |
| [7] | Enochs, E.E. and Jenda, O.M.G. (2000) Relative Homological Algebra. In: De Gruyter Expositions
in Mathematics, Vol. 30, Walter de Gruyter & Co. |
