Chennai Mathematical Institute


3.30 pm, Seminar Hall
Hilbert-Kunz Density Function and Hilbert-Kunz Multiplicity

Vijaylaxmi Trivedi
TIFR, Mumbai.


In this talk we recall a well-studied char p invariant Hilbert-Kunz multiplicity, e_{HK}(R,I), for a local ring/standard graded ring R with respect to an 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).

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.

In the graded case (based on the recent work), for a pair (R,I) we introduce a new invariant, the 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.

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.

We discuss a few other applications of this function, like asymptotic behaviour of e_{HK}(R,I^k) as k -> \infty, e_{HK} of the Segre product of rings and a possible approach for eHK in characteristic 0.