Study of Derived Equivalences

证明了表示-有限自射Artin代数上的部分倾斜复形是倾斜复形的直积，只要它在派生范畴中有一个自射自同态环；对于任意两个导出等价的表示-有限自射Artin代数，它们之间存在一个导出等价，它是由二项倾斜复形引起的导出等价的迭代；导出等价的表示-有限自射复形具有相同的Nakayama置换；具有循环Nakayama置换和自同态环的自投射Artin代数上的每个倾斜复形都同构于一个单项复形，即两个具有循环Nakayama置换的自射Artin代数之间的每个导出等价都是Morita等价.我们给出了Auslander公式的一个推广，该公式在满足Auslander条件的左和右Notherian环上的每个有限生成的左模上都有很好的滤子.引入了带结构系统的全矩阵代数的概念，证明了某些Gorenstein阶因子代数是具有Frobenius结构系统的全矩阵代数，如果矩阵的大小小于或等于7，则逆也成立，如果大于7，则逆不成立.我们证明了左Artin环上有限生成的左模的上有界(上有界，下有界和无界)复形的派生范畴的Grothendieck群是平凡的.

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.