Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities
Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities
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DOI：
10.1515/crelle-2015-0031
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发表时间：
2013-04
期刊：
影响因子：
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通讯作者：
O. Iyama;M. Wemyss
中科院分区：
文献类型：
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作者：
O. Iyama;M. Wemyss
本文定义并研究了一类三角范畴，其中的Hom-空间在某个基环上的Krull维数至多为1（因此它们有一个自然的两步滤子），并且滤子的每个因子满足某些Calabi-Yau型性质.如果C是这样一个范畴，我们说C是维数至多为1的卡-丘范畴。我们扩展的概念卡-丘还原到这个设置，并证明一般结果是一个类似的集群理论中的已知结果。这样的范畴自然出现在三维Gorenstein奇点的设置作为Cohen-Macaulay模的稳定范畴。我们解释之间的联系卡-丘减少$\underline{\rm CM} R$和部分cropant决议和Q-阶乘终止Spec R，我们表明在相当一般的假设下，卡-丘减少存在。在本文的剩余部分中，我们主要讨论完全局部cA_n奇点R。通过使用基于$\underline{\rm CM} R$的Calabi-Yau约化的纯代数论证，我们根据对称群给出了最大修改模的完整分类，推广和加强了[BIKR]和[DH]中的结果，其中我们不需要对基场进行任何限制。我们还描述了修改模块在任意（不一定不可分解）的直接和项的突变。作为域为复数时的推论，我们得到了Spec R的Q-阶乘终结化的导出范畴的许多自等价。
In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi-Yau type property. If C is such a category, we say that C is Calabi-Yau with dimension at most one. We extend the notion of Calabi-Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories of Cohen-Macaulay modules. We explain the connection between Calabi-Yau reduction of $\underline{\rm CM} R$ and both partial crepant resolutions and Q-factorial terminalizations of Spec R, and we show under quite general assumptions that Calabi-Yau reductions exist. In the remainder of the paper we focus on complete local cA_n singularities R. By using a purely algebraic argument based on Calabi-Yau reduction of $\underline{\rm CM} R$, we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [BIKR] and [DH], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when the field is the complex numbers, we obtain many autoequivalences of the derived category of the Q-factorial terminalizations of Spec R.