Openness of the F-rational locus and smooth base change
Throughout, for simplicity, \(R\) will denote a commutative noetherian reduced ring of prime characteristic \(p\). We say that \(R\) is \(F\)-rational if every ideal generated by parameters in \(R\) is tightly closed, and that \(R\) is strongly \(F\)-rational if for each nonzero divisor \(x\) in \(R\) there exists an integer \(q_0\) such that the \(R\)-linear map \(R \otimes R/ I\to R^{1/q} \otimes R/I\), \(1\otimes 1 \mapsto x^{1/q} \otimes 1\), is injective for any ideal \(I\) and for any \(q = p^e \geq q_0\). The main purpose of the paper is to prove the openness of the \(F\)-rational locus in the case that \(R\) is finitely generated over an excellent local ring. For this, the author studies strongly \(F\)-rational rings in section 1, develops the theory of \(\Gamma\)-construction due to Hochster and Huneke in section 2, and proves that \(F\)-rationality is preserved under smooth base change (theorem 3.1). -- The author also shows that certain \(\Gamma\)-constructions of an \(F\)-rational ring are \(F\)-rational, that \(F\)-rationality implies strong \(F\)-rationality, and that a test element for strong parameter ideals exists, under the same condition in the main result. (By parameters in \(R\) we mean a sequence of elements \(x_1, \ldots, x_k\) in \(R\) such that for each prime ideal \(P\) of \(R\) containing them, their images in \(R_P\) form part of a system of parameters in \(R_P\). If this condition holds modulo every minimal prime ideal of \(R\), then we say \(x_1, \ldots, x_k\) form a sequence of strong parameters in \(R\) (section 0). In general, strong \(F\)-rationality implies \(F\)-rationality (proposition 1.4)).
scientific article; zbMATH DE number 763201
scientific article