A sufficient condition for the finiteness of Frobenius test exponents

Abstract:

The Frobenius test exponent $\operatorname {Fte}(R)$ of a local ring $(R,\mathfrak {m})$ of prime characteristic $p>0$ is the smallest $e_0 \in \mathbb {N}$ such that for every ideal $\mathfrak {q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak {q}^F$ has $(\mathfrak {q}^F)^{\left [p^{e_0}\right ]}=\mathfrak {q}^{\left [ p^{e_0}\right ]}$. We establish a sufficient condition for $\operatorname {Fte}(R)<\infty$ and use it to show that if $R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e., $H^j_\mathfrak {m}(R)/0^F_{H^j_\mathfrak {m}(R)}$ is finite length for $0 \le j < \dim (R)$, then $\operatorname {Fte}(R)<\infty$.