Regular Sequence And Depth

Depth is a fundamental and important notations in algrbra. Meanwhile, depth found its importance in algebraic geometry, since it can help to study the properties of algebraic varieties. Hence, it is of great value to give characterization and generalization of depth. After a short recall of basic algebraic notations and properties, we succeed to generalize the notation of depth in the paper. What's more, we give some characterizations of the generalized depth and study its properties carefully. Let R be a Nother ring, I be an ideal, M be a finitely generated module, then depth(I, M) is the minimal r such that H_l~┌(M)≠0. The literature [11] gave a generalization of depth by defining the notation filter regular sequence when R is a local Noether ring. More general definitions of regular sequence and depth are given in our paper when R is a general Noether ring. And some characterizations of them are showed with the help of the techniques of homological algebra such as local cohomology, long exact sequences and that of commutative algebra such as localization and associated primes. What's more, properties of the supports of some Extension groups and some local cohomology groups are showed. The main theorem3.10in [11] is a special case of our Theorem7in this paper.