Vijaylaxmi Trivedi
TIFR
March 10, 2016
Hilbert-Kunz density function and Hilbert-Kunz multiplicity: 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 $p > 0$).
TIFR
March 10, 2016
Hilbert-Kunz density function and Hilbert-Kunz multiplicity: 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 $p > 0$).
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.
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.