A criterion for integral dependence of modules

Abstract

Abstract. Let R be a universally catenary locally equidimensional Noetherian ring. We give a multiplicity based criterion for an arbitrary finitely generated R-module to be integral over a submodule. Our proof is self-contained and implies the previously known numerical criteria for integral dependence of ideals and modules.

Relative multiplicities of graded algebras

“…If E = F , then j(U |E, N) is equal to j(U , N) in the sense of Ulrich and Validashti [12]. Similar to the relative multiplicities of graded algebras in Remark 3.1, the set of primes q of R for which j t (U q |E q , N q ) = 0 is finite.…”

“…See for instance the discussion at the end of Section 6. Finally, the example discussed in Remark 2.4 in [12] yields the following instance of the relative j-multiplicities of algebras. …”

“…defined by Ulrich and Validashti in [12]. In particular, if we set A := R[E] and B := Sym(F ), where E is a submodule of a free module F , then j( A|B) coincides with the j-multiplicity j(E) of E in [12].…”

“…We can restate Theorem 2.5 of [12] in terms of relative multiplicities. In the above inequality, M 0 B denotes the B-module generated by the degree zero part of M. Unfortunately, the equality may not hold in general.…”

Multiplicities and Rees valuations

“…We would also like to point out that in the theorem below, we only need to check equality of -multiplicity locally at finitely many prime ideals (c.f. [18]). (i) J is a reduction of I.…”