Gorenstein子范畴与相对奇点范畴

Gorenstein Subcategories and Relative Singularity Categories

中文关键词:  阿贝尔范畴  自正交  Gorenstein子范畴  半对偶化双模

英文关键词:abelian category  self-orthogonal  Gorenstein subcategories  semidualizing bimodules

中文摘要:

      令A是阿贝尔范畴, T是A的一个自正交子范畴, 且T中每个对象均有有限投射维数和内射维数. 假设左Gorenstein子范畴lG(T)等于T的右正交类,且右Gorenstein子范畴rG(T)等于T的左正交类,我们证明了Gorenstein子范畴$G(T)$等于T的左正交类与T的右正交类之交,并且证明了它们的稳定范畴三角等价于A关于T的相对奇点范畴.作为应用,令$R$是有有限左自内射维数的左诺特环, $_RC_s$是半对偶化双模,且所有内射左$R$-模的平坦维数的上确界有限, 我们证明了 若$\mbox{}_RC$有有限内射（平坦）维数且$C$的右正交类包含$R$,则存在从$C$-Gorenstein投射模与关于$C$的Bass类的交到关于$C$-投射模的相对奇点范畴间的三角等价,推广了某些经典的结果.

英文摘要:

      Let $\mathscr{A}$ be an abelian category, $\mathscr{T}$ a self-orthogonal subcategory of $\mathscr{A}$ and each object in $\mathscr{T}$ admit finite projective and injective dimensions. If the left Gorenstein subcategory $l\mathcal{G}(\mathscr{T})$ equals to the right orthogonal class of $\mathscr{T}$ and the right Gorenstein subcategory $r\mathcal{G}(\mathscr{T})$ equals to the left orthogonal class of $\mathscr{T}$, we prove that the Gorenstein subcategory $\mathcal{G}(\mathscr{T})$ equals to the intersection of the left orthogonal class of $\mathscr{T}$ and the right orthogonal class of $\mathscr{T}$, and prove that their stable categories are triangle equivalent to the relative singularity category of $\mathscr{A}$ with respect to $\mathscr{T}$. As applications, let $R$ be a left Noetherian ring with finite left self-injective dimension and ${_{R}}C_{S}$ a semidualizing bimodule, and let the supremum of the flat dimensions of all injective left $R$-modules be finite. We prove that if $_{R}C$ has finite injective (or flat) dimension and the right orthogonal class of $C$ contains $R$, then there exists a triangle-equivalence between the intersection of $C$-Gorenstein projective modules and Bass class with respect to $C$, and the relative singularity category with respect to $C$-projective modules. Some classical results are generalized.