The blowup closure of a set of ideals with applications to \(TI\) closure.
The author introduces in this paper a new closure operation on sets of ideals in a commutative Noetherian ring of characteristic \(p\), called the blowup closure. Let \(R\) be a Noetherian commutative ring of prime characteristic \(p\). Let us denote by \(J^*\) the tight closure of a given ideal \(J\) in a Noetherian ring of prime characteristic. Let \(I_1,\dots, I_n\) be ideals of \(R\). Then the blowup closure of \(I_1,\dots, I_n\), denoted by \((I_1, \dots, I_n)\widetilde \,\), is defined as the set of those \(x\in R\) such that for every affine open set of the blowup of the product ideal \(I=I_1\cdots I_n\), if \(S\) is the coordinate ring of that affine set, then \(x\in (JS)^*\), where \(J=I_1+\cdots+I_n\). The theory of this operation is developed in this paper. The blowup closure has the advantage that its study is reduced to the case where all the ideals are principal. It is proven in this paper that given a set of ideals in a Noetherian ring of characteristic \(p\), under mild conditions, the tight integral closure of this set [which is a notion defined by \textit{M. Hochster}, see: J. Algebra 230, No. 1, 184--203 (2000; Zbl 0966.13003)] agrees with the blowup closure of certain extensions of those ideals in an extension of the original ring. The author uses properties of blowup closure to settle open questions on tight integral closure posed in the cited article of Hochster. In particular, it is shown in the paper under review that, under mild conditions on the ring, tight integral closure persists under ring maps and that it commutes with localizations if and only if tight closure does. We remark that the TI closure of the title is the abbreviation of ``tight integral closure''. The last section of the paper is devoted to develop a theory of test elements for tight integral closure which is based on a result concerning blowup closure (theorem 3.3). Test elements are the key ingredients for tight closure methods and the existence of test elements is one of the most important results in tight closure theory [see \textit{M. Hochster} and \textit{C. Huneke}, Trans. Am. Math. Soc. 346, No. 1, 1--62 (1994; Zbl 0844.13002)]. As mentioned by the author, the contents of this paper is part of her Ph.D. thesis (Univ. Michigan).
scientific article; zbMATH DE number 1750365
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