Local cohomology of Frobenius images over graded affine algebras (Q1570882)

The author proves an important special case of a conjecture formulated by \textit{M. Hochster} and \textit{C. Huneke} [J. Am. Math. Soc. 3, No. 1, 31-116 (1990; Zbl 0701.13002)] in connection with the problem of the localization of tight closure. The conjecture states that if \((R,{\mathfrak m})\) is a local ring of characteristic \(p>0\) and dimension \(d\) and \(I\) is an ideal primary to a prime \(P\) of height \(d-1\) then there exists \(b>0\) such that \(H^0_{\mathfrak m}(R/I^{[q]})\) is annihilated by \({\mathfrak m}^{bq}\) for every \(q=p^e\). Here \(I^{[q]}\) denotes the ideal generated by the \(q\)-th powers of the elements of \(I\). The author shows that a similar statement is true when \(R\) is a finitely generated graded algebra over a field \(k\) (with \(\text{char} k=p>0)\), \(I\) is a homogeneous ideal of \(R\) of height \(\dim R-1\) and \({\mathfrak m}\) is the homogeneous maximal ideal. In the non-graded case, the author proves the conjecture when \(R\) is normal of dimension 2 and gives an elementary new proof of an unpublished result of \textit{P. Roberts} asserting that, when \((R,{\mathfrak m})\) is the localization of a finitely generated \(k\)-algebra in a maximal ideal, there exists \(B>0\) such that \({\mathfrak m}^{Bq^d}\) annihilates \(H^0_{\mathfrak m}(R/I^{[q]})\) \((d=\dim R)\). The main results are proved, in fact, in a more general setting, replacing \(R/I^{[q]}\) by \(F^e(M)\) where \(M\) is a finitely generated \(R\)-module of dimension 1 and \(F\) is the Frobenius functor.