Speaker: Vijaylaxmi Trivedi
Date: March 10, 2016
Affiliation: TIFR

Abstract: In this talk we recall a well-studied $\mathrm{char}~p$ invariant {it Hilbert-Kunz multiplicity}, $e_{HK}(R, I)$, for a local ring/standard graded ring $R$ with respect to an {\bf m}-primary/graded ideal of finite colength $I$. This could be considered as an analogue of Hilbert-Samuel function and Hilbert-Samuel multiplicity (but specific to characteristic $p > 0$). <p> We give a brief survey of some of the results on this invariant and try to convey why $e_{HK}$ is a `better' and a `worse' invariant than Hilbert-Samuel multiplicity of a ring. <P. In the graded case (based on the recent work), for a pair $(R, I)$ we introduce a new invariant, the {it Hilbert-Kunz density function}, which is a limit of a uniformly convergent sequence of real valued compactly supported, piecewise linear and continous functions. We express $e_{HK}(R, I)$ as an integral of this function. <p> We prove that this function (unlike $e_{HK}$) satisfies a multiplication formula for the Segre product of rings. As a consequence some known result for $e_{HK}$ of rings hold for $e_{HK}$ of their Segre products. <p> We discuss a few other applications of this function, like asymptotic behaviour of $e_{HK}(R, I^k)$ as $k \to \infty$, $e_{HK}$ of the Segre product of rings and a possible approach for $e_{HK}$ in characteristic 0.