Multiplicities of graded families of ideals on Noetherian local rings
2026-04-09 16:00 - 17:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Dale Cutkosky, University of Missouri
Title: Multiplicities of graded families of ideals on Noetherian local rings
Abstract: Let $R$ be an arbitrary $d$-dimensional Noetherian local ring with maximal ideal $m_R$. In this talk, we give a generalization of the multiplicity $e(I)$ of an $m_R$-primary ideal $I$ of $R$ to a multiplicity $e(mathcal I)$ of a graded family of $m_R$-primary ideals $mathcal I$ in $R$. This multiplicity gives the classical multiplicity $e(I)$ if $mathcal I={I^n}$ is the $I$-adic filtration, and agrees with the volume, $lim_{nrightarrow infty}d!frac{ell(R/I_n) }{n^d}$ for $R$ such that $dim N(hat R)>d$, the required condition for the volume of graded families of $m_R$-primary ideals to exist as a limit. We will show that many of the classical theorems for the multiplicity of an ideal generalize to this multiplicity, including mixed multiplicities, the Rees theorem and the Minkowski inequality and equality. We give proofs which are independent of the theory of volumes and Okounkov bodies for all of our results, with the one exception being the proof of the Minkowski equality. We do this by interpreting the multiplicity of graded families of $m_R$-primary ideals as an intersection product on the family of $R$-schemes which are obtained by blowing up $m_R$-primary ideals in $R$.
