2024/09/14 by Das, Suprajo, Roy, Sudeshna, Trivedi, Vijaylaxmi · 1 citation
#13A30 #13D40 #14C17 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13H15 #Secondary 14C20
paper · doi:10.48550/arxiv.2409.09346
Let R=⊕m≥ 0Rm be a standard graded equidimensional ring over a field R0, and I⊆ J be two non-nilpotent graded ideals in R. Then we give a set of numerical characterizations of the integral dependence of I and J in terms of certain multiplicities. A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants. In particular, we show the following: let S=R[y], I = IS and J = JS and \bf d be the maximum of the generating degrees of both I and J. Let c>\bf d be any given integer. Then I = J\iff e(S[It]_Δ(c,1)) = e(S[Jt]_Δ(c,1)), where e(S[It]_Δ(c,1)) denotes the Hilbert-Samuel multiplicity of the standard graded domain S[It]_Δ(c,1) = ⊕n≥ 0(In)cntn. Further, if I is of finite colength in R then e(S[It]_Δ(c,1)) = cde(R) - e(I,R). If R is also a domain, then other numerical criteria are the following: I = J amp; \iff ε(I)=ε(J) and ei(R[It]) = ei(R[Jt]) for all 0≤ i lt;dim(R/I), where ε(I) denotes the epsilon multiplicity of I, and ei(R[It])'s are the mixed multiplicities of the Rees algebra R[It]. The relation between ei(S[It]) and the polar multiplicities of I_≥ \bf d provides another criterion in terms of polar multiplicities of I_≥ \bf d. The first two characterizations generalize Rees's classical result for ideals of finite colengths. Apart from several well-established results, the proofs of these results use the theory of density functions, which was developed in arXiv:2311.17679.