2023/05/21 by Linquan Ma, Ma, Linquan, Phạm Hùng Quý +3
Mathematics · #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2305.12469
Let (R, \mathfrakm) be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an \mathfrakm-primary ideal. We introduce versions of these invariants by restricting to various closures and ``cross-pollinate'' the two multiplicity theories by asking for analogues invariants already established in one of the theories. On the Hilbert--Samuel side, we prove that the analog of the Stückrad--Vogel invariant (that is, the infimum of the ratio between the multiplicity and colength) for integrally closed \mathfrakm-primary ideals is often 1 under mild assumptions. We also compute the supremum and infimum of the relative drops of multiplicity for (integrally closed) \mathfrakm-primary ideals. On the Hilbert--Kunz side, we study several analogs of the Lech--Mumford and Stückrad--Vogel invariants.