Study of Derived Equivalences

我们证明了在派生范畴中有自射自同态环的表示有限自射artin代数上的部分倾斜复形表现为倾斜复形的直接和，并且证明了对于任意两个派生等价表示有限自射artin代数，它们之间存在一个派生等价，该等价是由两项倾斜复形引起的派生等价的迭代。证明了推导出的等价表示-有限自射代数具有相同的中山置换，且具有循环中山置换的自射代数与具有自射自同态环上的每一个倾斜复同构于一个单项复，即两个具有循环中山置换的自射代数之间的每一个推导等价是一个森田等价。我们给出了Auslander公式的推广，该公式在满足Auslander条件的左右诺瑟环上的每个有限生成的左模上推导出了一个很好的滤波。此外，我们还确定了该过滤中模块的具体形式。引入具有结构系统的全矩阵代数的概念，证明了Gorenstein平纹阶的某些因子代数是具有Frobenius结构系统的全矩阵代数，当矩阵的大小小于等于7时，反之成立，而当矩阵的大小大于7时，反之不成立。我们证明了上有界的派生范畴的Grothendieck群。（下有界和无界）左环上有限生成左模的复形是平凡的。

We show that a partial tilting complex over a representation-finite selfinjective artin algebra appears as a direct summand of a tilting complex whenever it has a selfinjective endomorphism ring in the derived category, that for any two derived equivalent representation-finite selfinjective artin algebras there there exists a derived equivalence between them which is an iteration of derived equivalences induced by two-term tilting complexes, that the derived equivalent representation-finite selfinjective artin algebras have the same Nakayama permutation, and that every tilting complex over a selfinjective artin algebra with a cyclic Nakayama permutation and with a selfinjective endomorphism ring is isomorphic to a one-term complex, i.e., every derived equivalence between two selfinjective artin algebras with a cyclic Nakayama permutation is a Morita equivalence.We provide a generalization of the Auslander formula which induces a nice filtration on each finitely generated left module over a left and right noetherian ring satisfying the Auslander condition. Furthermore, we determine the concrete form of modules in this filtration.We introduce the notion of full matrix algebras with structure systems and show that certain factor algebras of Gorenstein tiledorders are full matrix algebras with Frobenius structure systems, that the converse is also true if the size of the matrix is smaller the or equal to seven and that the converse is not true if the size of the matrix is greater then seven.We show that the Grothendieck groups of derived categories of bounded above (resp., bounded below and unbounded) complexes of finitely generated left modules over a left artinian ring are trivial.