Derived equivalences for $\Phi$-Auslander-Yoneda algebras
Derived equivalences for $\Phi$-Auslander-Yoneda algebras
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DOI：
10.1090/s0002-9947-2013-05688-7
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发表时间：
2009-12
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影响因子：
--
通讯作者：
Weiqun Hu;Changchang Xi
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文献类型：
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作者：
Weiqun Hu;Changchang Xi
本文在$\mathbb N$中引入了参数集为$\Phi$的三角化范畴中的$\Phi$-Auslander-Yoneda代数，并给出了一种方法，从一个几乎$\nu$-稳定的导出等价出发，构造这些$\Phi$-Auslander-Yoneda代数（不一定是Artin代数）或它们的商代数之间的新的导出等价.（1）设$A$和$B$是表示有限的自内射Artin代数，$A$和$B$分别具有$_AX$和$_BY$可加生成元。如果$A$和$B$是导-等价的，则$X$和$Y$的$\Phi$-Auslander-Yoneda代数对每个容许集$\Phi$是导-等价的.特别地，Auslander代数$A$和$B$都是导出等价和稳定等价的. (2)对于自内射Artin代数A$和A$-模X$，对于每一个容许集$\Phi$，导出了$A\oplus X$和$A\oplus \Omega_A（X）$的$\Phi$-Auslander-Yoneda代数是等价的，其中$\Omega$是Heller环算子.出于这些导出的等价之间的$\Phi$-Auslander-Yoneda代数，我们考虑商代数的导出等价的构造，并显示，除其他外，两个基本自内射代数之间的导出等价可以转移到通过因式分解出socles获得的商代数之间的导出等价。
In this paper, we introduce $\Phi$-Auslander-Yoneda algebras in a triangulated category with $\Phi$ a parameter set in $\mathbb N$, and provide a method to construct new derived equivalences between these $\Phi$-Auslander-Yoneda algebras (not necessarily Artin algebras), or their quotient algebras, from a given almost $\nu$-stable derived equivalence. As consequences of our method, we have: (1) Suppose that $A$ and $B$ are representation-finite, self-injective Artin algebras with $_AX$ and $_BY$ additive generators for $A$ and $B$, respectively. If $A$ and $B$ are derived-equivalent, then the $\Phi$-Auslander-Yoneda algebras of $X$ and $Y$ are derived-equivalent for every admissible set $\Phi$. In particular, the Auslander algebras of $A$ and $B$ are both derived-equivalent and stably equivalent. (2) For a self-injective Artin algeba $A$ and an $A$-module $X$, the $\Phi$-Auslander-Yoneda algebras of $A\oplus X$ and $A\oplus \Omega_A(X)$ are derived-equivalent for every admissible set $\Phi$, where $\Omega$ is the Heller loop operator. Motivated by these derived equivalences between $\Phi$-Auslander-Yoneda algebras, we consider constructions of derived equivalences for quotient algebras, and show, among others, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factorizing out socles.