Abstract
This paper is concerned with existence of big tight closure test elements for a commutative Noetherian ring R of prime characteristic p. Let R ∘ denote the complement in R of the union of the minimal prime ideals of R. A big test element for R is an element of R ∘ which can be used in every tight closure membership test for every R-module, and not just the finitely generated ones. The main results of the paper are that, if R is excellent and satisfies condition ( R 0 ) , and c ∈ R ∘ is such that R c is Gorenstein and weakly F-regular, then some power of c is a big test element for R if (i) R is a homomorphic image of an excellent regular ring of characteristic p for which the Frobenius homomorphism is intersection-flat, or (ii) R is F-pure, or (iii) R is local. The Gamma construction is not used.
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