Stable categories of higher preprojective algebras
Stable categories of higher preprojective algebras
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DOI：
10.1016/j.aim.2013.03.013
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发表时间：
2009-12
期刊：
影响因子：
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通讯作者：
O. Iyama;Steffen Oppermann
中科院分区：
文献类型：
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作者：
O. Iyama;Steffen Oppermann
摘要引入全局维数为n的代数的(n+ 1)-预射影代数，证明了如果一个代数是n-表示有限的，那么它的(n+ 1)-预射影代数是自内射的。在这种情况下，我们证明了(n+ 1)-预射影代数的稳定模范畴是(n+ 1) -Calabi-Yau，更准确地说，它是原代数的稳定n- auslander代数的(n+ 1)-Amiot聚类范畴。特别地，这个稳定的类别包含一个(n+ 1)-簇倾斜的对象。我们证明了即使(n+ 1)-预投影代数不是自内射，在某些假设下（对于n∈{1,2}总是满足），上述结果对于Cohen-Macaulay模的稳定范畴仍然成立。
Abstract We introduce (n+ 1)-preprojective algebras of algebras of global dimension n. We show that if an algebra is n-representation-finite then its (n+ 1)-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+ 1)-preprojective algebra is (n+ 1)-Calabi–Yau, and, more precisely, it is the (n+ 1)-Amiot cluster category of the stable n-Auslander algebra of the original algebra. In particular this stable category contains an (n+ 1)-cluster tilting object. We show that even if the (n+ 1)-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1, 2}) the results above still hold for the stable category of Cohen–Macaulay modules.