Hội thảo Quốc tế về Tổ hợp, Đại số giao hoán, Hình học đại số và các Ứng dụng – Trường Đại học Công nghệ, ĐHQGHN – Univeristy of Engineering and Technology

Tiêu đề: Extensions of Regular Sequence Concept and Structure of Finitely Generated Modules

Tóm tắt: In this talk, we present our recent results in [LN], [LLN]. Let (R, 𝔪) be a Noetherian local ring, M a finitely generated R-module, and I an ideal of R. The concept of a regular sequence (first introduced by Serre under the name of R-sequence [Serr]) has been applied in many areas of mathematics. Note that M is Cohen-Macaulay if and only if every system of parameters (s.o.p.) of M is an M-sequence. Cuong-Schenzel-Trung [CST] introduced the class of generalized Cohen-Macaulay modules and defined the notion of filtered regular sequence (or f-sequence) as a natural extension of regular sequence, showing that if R is a quotient of a Cohen-Macaulay local ring, then M is generalized Cohen-Macaulay if and only if each s.o.p. of M is an f-sequence. The class of generalized Cohen-Macaulay modules and the notion of f-sequence have become fundamental in Commutative Algebra. The generalized regular sequence (g-sequence) was introduced in [Nh] as a further extension used to study the finiteness of associated primes of local cohomology modules. It was shown in [NM] that if R is a quotient of a Cohen-Macaulay local ring, then dimR Hi_𝔪(M) ≤ 1 for all i < dimR(M) if and only if each s.o.p. of M is a g-sequence. If M/IM ≠ 0 (resp. dimR(M/IM) > 0, dimR(M/IM) > 1) then each M-sequence (resp. f-sequence, g-sequence) in I can be extended to a maximal one, and all maximal sequences of each type have the same length. This length is denoted respectively by depth(I, M), f-depth(I, M), and g-depth(I, M), where:

depth(I, M) = inf{i ∈ ℕ | Hⁱ_I(M) ≠ 0} f-depth(I, M) = inf{i ∈ ℕ | Hⁱ_I(M) is not Artinian} g-depth(I, M) = inf{i ∈ ℕ | Supp_R Hⁱ_I(M) is infinite}

Regular, f-, and g-sequences are very useful for studying unmixed finitely generated modules, but they are less effective for modules that are not unmixed. For example, M is Cohen-Macaulay if and only if M/xM is Cohen-Macaulay for any regular element x of M. However, while M being sequentially Cohen-Macaulay implies M/xM is so, the converse does not hold generally. In this talk, we introduce some extensions of the regular sequence concept suitable for studying modules not necessarily unmixed. We clarify the structure of certain classes of modules that include all sequentially Cohen-Macaulay modules.

References:

1. [CST] N. T. Cuong, P. Schenzel, N. V. Trung, Verallgemeinerte Cohen-Macaulay Moduln, Math. Nachr. 85 (1978), 57-73.
2. [LT] R. Lü, Z. Tang, The f-depth of an ideal on a module, Proc. Amer. Math. Soc. 130 (2001), 1905-1912.
3. [Nh] L. T. Nhan, On generalized regular sequences and the finiteness for associated primes of local cohomology modules, Comm. Algebra 33 (2005), 793-806.
4. [NM] L. T. Nhan, M. Morales, Generalized f-modules and the associated primes of local cohomology modules, Comm. Algebra 34 (2006), 863-878.
5. [LN] N. X. Linh, L. T. Nhan, On sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules, J. Algebra 678 (2025), 635-653.
6. [LLN] N. X. Linh, N. T. H. Loan, L. T. Nhan, An extension of the class of sequentially Cohen-Macaulay modules, Preprint.
7. [Serr] J. P. Serre, Sur la dimension cohomologique des anneaux et des modules noetheriens, Proc. Intern. Symp. on Alg. Number Theory, Tokyo-Nikko (1955), 176-189.