We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far reaching generalization of the classical Auslander-Buchsbaum formula for the depth of finitely generated modules of finite projective dimension. We extend also Iversen's amplitude inequality to unbounded complexes. As a corollary we deduce: Given a local homomorphism Q-->R, if there is a non-zero finitely generated R-module that has finite flat dimension both over Q and over R, then the flat dimension of R over Q is finite. This last result yields a module theoretic extension of a characterization of regular local rings in characteristic p due to Kunz and Rodicio
给定一个双复数$X$，存在$E_2$项为H $_I$（H $_{II}(X))$）或H $_{II}($ H $_I (X))$的谱序列。但如果$H_I(X)=H_{II}(X)=0$，则两个谱序列的所有项都为0。即使存在与$X$相关的感兴趣的非零（共）同源性，也可能发生这种情况。在处理Tate（co）同源性时，情况往往如此。因此，在这种情况下，谱序列可能不会给出关于感兴趣的（共）同源性的任何信息。本文给出了当H $_I(X)=$ H $_{II}(X)=0$时构造$X$同调群的一种不同方法.在此基础上，给出了Tate同调与上同调平衡的一个新的初等证明。
Asymptotic depth of invariant chains of edge ideals
University of Education, Hue University（顺化大学教育学院）；Faculty of Mathematics and Informatics, Hanoi University of Science and Technology（河内科学技术大学数学与信息学院）；Department of Mathematics, FPT University（FPT大学数学系）；Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学院数学研究所）；McMaster University（麦克马斯特大学）
