Relationships Among Hilbert-Samuel Multiplicities, Koszul Cohomology, and Local Cohomology

commutative algebra;homological algebra;Hilbert-Samuel multiplicities;Koszul homology;Lech"s inequality;Mathematics;Science;Mathematics

Klein, PatriciaSmith, Karen E ;

University of Michigan

关键词: commutative algebra;    homological algebra;    Hilbert-Samuel multiplicities;    Koszul homology;    Lech";    s inequality;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/145974/triciajk_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

We consider relationships among Hilbert-Samuel multiplicities, Koszul cohomology, and local cohomology. In particular, we investigate upper and lower bounds on the ratio e(I,M)/l(M/IM) for m-primary ideals I of the local ring (R,m) and finitely generated quasi-unmixed R-modules M and, in joint work with Linquan Ma, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao, give a precise formula for the upper bound for all finitely-generated R-modules M and show that the ratio is bounded away from 0 whenever M is quasi-unmixed. We also, as independent work, give a characterization of quasi-unmixed R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. We show that if M is an equidimensional module over a complete local ring, then M is asymptotically Cohen-Macaulay if and only if the supremum of the set of lengths of lower Koszul cohomology modules on systems of parameters if finite if and only if M is Cohen-Macaulay on the punctured spectrum.