Study of perfect complexes over algebras and their properties

Artin环A上有限生成模的有界导范畴的Grothendieck群K0(D^b(Moda))是一个阶为n的自由阿贝尔群，其中n是非同构单模的个数.但无界派生范畴的Grothendieck群是什么还不得而知。利用作为完全复形推广的紧对象和生成元的概念，引入了经典生成元的加性T-集的概念，并证明了有限生成A-模的上(下)有界复形的Grothendieck群K0(D^-(Moda))(分别为K0(D^-(Moda)同构于(Z{T，T^&lt；-1&gt；-1&gt；因此，D^-(Moda)，D^(Moda)，D(Moda)的Grothendieck群是平凡的.设R是凝聚环，T是倾斜R-模.对于具有T-0-&gt；T_&lt；n^-&gt；&gt；...-&gt；T_&lt；1-&gt；&gt；T_&lt；0-&gt；&gt；M_-&gt；0，分辨率的R-模M，证明了T的Auslander条件等价于End_R(T)-模Hom_R(T，E^I(M))的平坦维度小于或等于In，其中0->；E^0(M)-&gt；E^1(M)-&gt；…是M的一个极小内射分解.我们研究了具有倾斜层的代数簇的情形，并利用它们研究了有限维代数的派生Picard群与模的派生范畴的自同构群的情形.我们证明了正则局部环的剩余域的第二合子模的Rees代数是一个Gorenstein分解环.我们在Noether局部环的Chow群或Grothendieck群上定义了数值等价的概念。在较弱的条件下，证明了Chow群模数等价是有限维q-向量空间。证明了在q-Gorenstein环上e的Hilbert-kunz函数的二项系数为零。

The Grothendieck group K_0(D^b(modA)) of a bounded derived category of finitely generated modules over an Artinian ring A is a free abelian group of rank n, where n is the number of non-isomorphic simple modules. But it is not known what is the Grothendieck group of a unbounded derived category. By using the notions of compact objects and generators which are generalizations of perfect complexes, we introduce the notion of an additive T-set of classical generators, and show that The Grothendieck group K_0(D^-(modA)) (resp., K_0(D^-(modA))) of bounded above (resp., bounded below) complexes of finitely generated A-modules is isomorphic to (Z{T, T^<-1>}/(1+T))^n. As a consequence, the Grothendieck groups of D^-(modA), D^+(modA), D(modA) are trivial.Let R be a coherent ring, and T a tilting R-module. For an R-module M which has a T-resolutibn 0->T_<n^->>...->T_<1->>T_<0->>M_->0, we show that the Auslander condition for T is equivalent to the condition that the flat dimension of an End_R(T)-module Hom_R(T, E^i (M)) is less than or equal to i+ n, where 0- >M_->E^0(M)->E^1(M)->...is an minimal injective resolution of M. We study the case of algebraic varieties which have tilting sheaves, and applying them investigate the case that derived Picard group of a finite dimensional algebra is isomorphic to the automorphism group of the derived category of modules.We show that the Rees algebra of the second syzygy module of the residue field of a regular local ring is a Gorenstein factorial domain. We define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional Q-vector space. We prove that the coefficient of the second term of the Hilbert-Kunz function of e vanishes over a Q-Gorenstein ring.