Rings of Frobenius operators

<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>R</jats:italic> be a local ring of prime characteristic. We study the ring of Frobenius operators <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0305004114000176_inline1" /><jats:tex-math>${\mathcal F}(E)$</jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:italic>E</jats:italic> is the injective hull of the residue field of <jats:italic>R</jats:italic>. In particular, we examine the finite generation of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0305004114000176_inline1" /><jats:tex-math>${\mathcal F}(E)$</jats:tex-math></jats:alternatives></jats:inline-formula> over its degree zero component <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0305004114000176_inline2" /><jats:tex-math>${\mathcal F}^0(E)$</jats:tex-math></jats:alternatives></jats:inline-formula>, and show that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0305004114000176_inline1" /><jats:tex-math>${\mathcal F}(E)$</jats:tex-math></jats:alternatives></jats:inline-formula> need not be finitely generated when <jats:italic>R</jats:italic> is a determinantal ring; nonetheless, we obtain concrete descriptions of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0305004114000176_inline1" /><jats:tex-math>${\mathcal F}(E)$</jats:tex-math></jats:alternatives></jats:inline-formula> in good generality that we use, for example, to prove the discreteness of <jats:italic>F</jats:italic>-jumping numbers for arbitrary ideals in determinantal rings.</jats:p>