Numerical criteria for integral dependence

Abstract

We study multiplicity based criteria for integral dependence of modules or of standard graded algebras, known as 'Rees criteria'. Rather than using the known numerical invariants, we achieve this goal with a more direct approach by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths; this multiplicity is a non-negative real number that can be irrational.

Generalized multiplicities of edge ideals

“…The vanishing of the ε-multiplicity of an ideal is captured by the analytic spread of the ideal. Indeed, as in the case of j -multiplicity, the ε-multiplicity of I is not zero if and only if the analytic spread of I is maximal [23,41]. In particular, by Proposition 6.1 we obtain the following result.…”

“…More recently, Achilles and Manaresi introduced the concept of j -multiplicity [1], and Ulrich and Validashti proposed the notion of ε-multiplicity [41], extending the classical Hilbert-Samuel multiplicity to arbitrary ideals in a general algebraic setting. These invariants have been proven useful in commutative algebra and algebraic geometry for their connections to the theory of integral closures and Rees valuations, the study of the associated graded algebras, intersection theory, equisingularity and local volumes of divisors [11,23,24,30,41]. Recently, Jeffries and Montaño showed that these numbers measure certain volumes defined for arbitrary monomial ideals, similar to the zero-dimensional case [21].…”

Relative multiplicities of graded algebras

“…We also prove that being weakly birational is equivalent to the equality of the above two relative multiplicities. One may be tempted to define a relative multiplicity based on the simpler function Λ(n) := λ R (Γ m (B n /A n )) instead [13]. However, this function need not be polynomial eventually [2].…”

Prime filtrations of the powers of an ideal

“…Using the multiplicity estimates of the filtrations constructed by Theorem 5, we give a different proof of the existence of ǫ-multiplicity introduced in [11]. Here ℓ(M) denotes the length of M. …”

“…We also give an estimate on the number of times that a given prime appears in these special filtrations, and we use it to bound the length of the local cohomology modules, reproving a result of Ulrich and Validashti [11].…”