Liftable derived equivalences and objective categories
Liftable derived equivalences and objective categories
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DOI：
10.1112/blms.12364
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发表时间：
2019-02
影响因子：
0.9
通讯作者：
Xiaofa Chen;Xiao-Wu Chen
中科院分区：
文献类型：
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作者：
Xiaofa Chen;Xiao-Wu Chen
我们给出以下定理的两个证明和一个部分推广：如果有限维代数A导出等价于光滑投射概型，则A与另一个代数B之间的任何导出等价都是标准的，即通过双侧倾斜复形同构于导出张量函子。证明的主要内容如下：（1）在两个模范畴的导出范畴之间，可提升函子与标准函子重合;（2）模范畴与阿贝尔范畴之间的任何导出等价唯一分解为伪恒等式与可提升导出等价的合成;（3）某类投射概型上的凝聚层的导出范畴是三角目标的，即其上的任意三角自等价，它保持所有对象的同构类，必然同构于恒等函子。
We give two proofs of the following theorem and a partial generalization: if a finite‐dimensional algebra A is derived equivalent to a smooth projective scheme, then any derived equivalence between A and another algebra B is standard, that is, isomorphic to the derived tensor functor by a two‐sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition of a pseudo‐identity and a liftable derived equivalence; (3) the derived category of coherent sheaves on a certain class of projective schemes is triangle‐objective, that is, any triangle autoequivalence on it, which preserves the isomorphism classes of all objects, is necessarily isomorphic to the identity functor.