Tight Closure of powers of ideals and tight Hilbert polynomials

Abstract

Let $(R,\mathfrak{m})$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^{\ast }$ denote its tight closure. We introduce the tight Hilbert function $H_I^{\ast }(n)=\ell (R/(I^n{)}^{\ast })$ and the corresponding tight Hilbert polynomial $P_I^{\ast }(n)$ where $I$ is an $\mathfrak{m}$-primary ideal. It is proved that $F$-rationality can be detected by the vanishing of the first coefficient of $P_I^{\ast }(n).$ We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

24 pages, to appear in Mathematical Proceedings of the Cambridge Philosophical Society