Upper bound of multiplicity in prime characteristic

Abstract

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p$ of dimension $d$ with the embedding dimension $v$. Suppose the Frobenius test exponent for parameter ideals $Fte(R)$ of $R$ is finite, and let $Q=p^{Fte(R)}$. It is shown that $$e(R)\le Q^{v-d}\left(\genfrac{}{}{0ex}{}{v}{d}\right).$$ We also improve our bound for $F$-nilpotent rings. Our result extends the main results of Huneke and Watanabe and of Katzman and Zhang.

To appear in Forum Mathematicum