Tilting Theory and Functor Categories I. Classical Tilting
Tilting Theory and Functor Categories I. Classical Tilting
复制标题
DOI：
10.1007/s10485-013-9322-y
复制
发表时间：
2011-10
影响因子：
0.6
通讯作者：
R. Martínez-Villa;M. Ortiz-Morales
中科院分区：
文献类型：
--
作者：
R. Martínez-Villa;M. Ortiz-Morales
倾斜理论在有限和驯服表示类型的有限维代数的分类中，以及在许多其他数学分支中，一直是一个非常重要的工具。Happel（1988）和Cline等人（J Algebra 304：397-409 1986）证明了广义倾斜诱导了模范畴之间的导出等价，Rickard（J Lond Math Soc 39：436-456，1989）使用倾斜复形发展了导出范畴的一般Morita理论。另一方面，函子范畴被Auslander（I Commun Algebra 1（3）：177-268，1974），Auslander（1971）引入表示论，并用于他的第一个Brauer-Thrall猜想的证明（Auslander 1978），后来在他与I. Reiten对稳定等价性的研究（Auslander and Reiten，Adv Math 12（3）：306-366，1974），Auslander and Reiten（1973）和许多其他应用。最近，函子范畴在Martínez-Villa和Solberg（J Algebra 323（5）：1369-1407，2010）中被用来研究有限维代数的Auslander-Reiten分支。本文的目的是将倾斜理论推广到任意函子范畴，考虑到函子范畴Mod（modΛ）的应用，其中Λ是有限维代数。
Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel (1988) and Cline et al. (J Algebra 304:397–409 1986) proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard (J Lond Math Soc 39:436–456, 1989) to develop a general Morita theory of derived categories. On the other hand, functor categories were introduced in representation theory by Auslander (I Commun Algebra 1(3):177–268, 1974), Auslander (1971) and used in his proof of the first Brauer–Thrall conjecture (Auslander 1978) and later on, used systematically in his joint work with I. Reiten on stable equivalence (Auslander and Reiten, Adv Math 12(3):306–366, 1974), Auslander and Reiten (1973) and many other applications. Recently, functor categories were used in Martínez-Villa and Solberg (J Algebra 323(5):1369–1407, 2010) to study the Auslander–Reiten components of finite dimensional algebras. The aim of this paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod (modΛ), with Λ a finite dimensional algebra.