A sufficient condition for the finiteness of Frobenius test exponents

Abstract

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