科研之友
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摘要
Let X := Spec(R) be an affine Noetherian scheme, and M subset of N be a pair of finitely generated R-modules. Denote their Rees algebras by R(M) and R(N). Let N-n be the nth homogeneous component of R(N) and let M-n be the image of the nth homogeneous component of R(M) in N-n. Denote by (M) over bar (n) be the integral closure of M-n in N-n. We prove that Ass(X) (N-n /(M) over bar (n)) and Ass(X) (N-n / M-n) are asymptotically stable, generalizing known results for the case where M is an ideal or where N is a free module. Suppose either that M and.Af are free at the generic point of each irreducible component of X or N is contained in a free R-module. When X is universally catenary, we prove a generalization of a classical result due to McAdam and obtain a geometric classification of the points appearing in Ass(X) (N-n / (M) over bar (n)). Notably, we show that if x is an element of Ass(X) (N-n / (M) over bar (n)) for some n, then x is the generic point of a codimension-one component of the nonfree locus of N / M or x is a generic point of an irreducible set in X where the fiber dimension Proj(R(M)) -> X jumps. We prove a converse to this result without requiring X to be universally catenary. Our approach is geometric in spirit. Also, we recover, strengthen, and prove a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules.
- 出版日期2018-8-15
- 单位东北大学
