一个非平凡的Calabi-Yau DG代数

证明例1中的DG代数不仅是Koszul,同调光滑DG代数,而且还是一个Calabi-Yau DG代数.该例子说明一个Calabi-Yau DG代数的同调分次代数不一定具有Calabi-Yau性质,甚至可能不是同调光滑的;另外,该例子还说明一个Calabi-Yau DG代数忘掉微分后得到的分次代数不一定是分次Calabi-Yau代数.     

紧致DG模和Gorenstein DG代数

证明同调有界的连通微分分次代数（简称为DG代数）上的紧致DG模的ampli-tude与基代数的amplitude的差恰为该DG模的投射维数.由此可得非平凡的正则DG代数是同调无界的.对正则DG代数A,若它的同调代数H（A）是分次Koszul代数,则证明H（A）有有限的整体维数;如果把条件减弱为A是Koszul DG代数,则给出了一个H（A）的整体维数为无限的例子.对一般的正则DG代数A,给出了其为Gorenstein DG代数的一些等价刻画.对同调有限维的连通DG代数A,证明由紧致对象全体构成的三角范畴Dc（A）和Dc（Aop）存在Auslander-Reiten三角当且仅当A和Aop都是Gorenstein DG代数.当A是非平凡的正则DG代数,且H（A）是局部有限维时,Dc（A）不存在Auslander-Reiten三角.对正则DG代数A,转而讨论了Auslander-Reiten三角在Dlbf（A）以及Dlbf（Aop）上的存在性.     

一类自入射Koszul特殊双列代数的Hochschild同调群

基于Snashall与Taillefer构造的极小投射双模分解,用组合的方法,清晰地计算出一类自入射Koszul特殊双列代数∧_N的各阶Hochschild同调群的维数,从而以计算的方式直观地表明了韩阳的猜想对这类代数∧_N成立.     

有限复杂度的Koszul代数

首先给出了Koszul代数的张量积的复杂度,然后研究了Koszul遗传代数上的Koszul单列模,并证明了Koszul遗传代数上的Koszul模M的Koszul合成列在同构意义下是唯一的.     

分段Koszul代数, II

本文继续研究了分段Koszul 代数. 具体地, 给出了一些分段Koszul 代数的判定准则; 作为构造更多分段Koszul 代数例子的尝试, 讨论了分段Koszul 代数的“单点扩张” 和“H-Galois 分次扩张”, 其中H 是有限维的半单余半单Hopf 代数.     

拟分段Koszul代数

引入了拟分段Koszul代数的概念,它是分段Koszul代数的非分次推广.详细讨论了拟分段Koszul代数的Yoneda-Ext代数,给出了一些使诺特半完全代数成为拟分段Koszul代数的充要条件.     

一类量子Koszul 代数的Hochschild 上同调

本文利用组合的方法, 详细地计算了一类量子Koszul 代数Λ_q (q ∈ k \{0}) 的各阶Hochschild 上同调空间的维数, 清晰地刻划了代数Λ_q 的Hochschild 上同调的cup 积, 确定了代数Λ_q 的Hochschild上同调环HH^*(Λ_q) 模去幂零元生成的理想N 的结构, 证明了当q 为单位根时, HH^*(Λ_q)/N 作为代数不是有限生成的, 从而为Snashall-Solberg 猜想(即HH^*(Λ)/N 作为代数是有限生成的) 提供了更多反例.     

拟分段Koszul代数

引入了拟分段Koszul代数的概念,它是分段Koszul代数的非分次推广.详细讨论了拟分段Koszul代数的Yoneda-Ext代数,给出了一些使诺特半完全代数成为拟分段Koszul代数的充要条件.     

广义分段Koszul代数

广义分段Koszul代数(简称为K_p代数)一般是一类二次代数,其平凡模允许有非单纯的投射分解.利用Yoneda-Ext代数E(A)给出了分次代数A是K_p代数的一个充分条件,同时讨论了K_p代数的商代数是否继承K_p性质.     

广义分段Koszul代数

广义分段Koszul代数(简称为κ_p代数)一般是一类二次代数,其平凡模允许有非单纯的投射分解.利用Yoneda-Ext代数E(A)给出了分次代数A是κ_p代数的一个充分条件,同时讨论了κ_p代数的商代数是否继承κ_p性质.     

同调光滑连通上链DG代数的一个注记

本文给出了有关同调光滑连通上链微分分次(简称DG)代数的两个重要结论.具体地说,当A是同调光滑连通上链DG代数且其同调分次代数H(A)是诺特分次代数时,证明D_(fg)(A)中的任意Koszul DG A-模都是紧致的.另外,当A是Kozul连通上链DG代数且其同调分次代数H(A)是有平衡对偶复形的诺特分次代数时,证明A的同调光滑性质等价于D_(fg)(A)=D~c(A).     

Koszul Algebras and Their Ideals

We study associative graded algebras that have a “complete flag” of cyclic modules with linear free resolutions, i.e., algebras over which there exist cyclic Koszul modules with any possible number of relations (from zero to the number of generators of the algebra). Commutative algebras with this property were studied in several papers by Conca and others. Here we present a noncommutative version of their construction.We introduce and study the notion of Koszul filtration in a noncommutative algebra and examine its connections with Koszul algebras and algebras with quadratic Grobner bases. We consider several examples, including monomial algebras, initially Koszul algebras, generic algebras, and algebras with one quadratic relation. It is shown that every algebra with a Koszul filtration has a rational Hilbert series.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 47–60, 2005Original Russian Text Copyright © by D. I. PiontkovskiiSupported in part by the Russian Foundation for Basis Research under project 02-01-00468.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)     

Koszul Differential Graded Algebras and BGG Correspondence II

The concept of Koszul differential graded （DG for short） algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module Ak is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] （see also Proposition 2.2）, Ak is not compact if H（A） is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H（A） is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.     

Full Graphs and Koszul Algebras II

This paper continues the study of n-full graphs and their connection to certain Koszul algebras started in Green and Hartman (to appear). We provide constructive methods for creating new full graphs from old and study the associated Koszul algebras and the projective resolution of simple modules over such algebras.     

On the Koszul Modules of Exterior Algebras

We prove that the Koszul modules over an exterior algebra can be filtered by the cyclic Koszul modules. We also introduce the cyclic dimension vector as invariants for studying the Koszul modules over an exterior algebra.