Study of Derived Equivalences

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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.