2005/08/12 by Rodney Y. Sharp, Sharp, Rodney Y.
Computer Science · Mathematics · #13A15 #13A35 #13D45 #13E05 #13E10 #13H10 #16S36 (Primary) 13C15 (Secondary) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras #math.AC #msc:13A15 #msc:13A35 #msc:13C15 #msc:13D45 #msc:13E05 #msc:13E10 #msc:13H10 #msc:16S36
paper · pdf · doi:10.48550/arxiv.math/0508214
This is to appear in the Michigan Mathematical Journal
arxiv created 2005/08/12 · openalex publication_date 2005/08/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with the tight closure of an ideal I in a commutative Noetherian ring R of prime characteristic p. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not an element of R belongs to the tight closure of I. The situation in this respect is much improved by Hochster's and Huneke's test elements for tight closure, which exist when R is a reduced algebra of finite type over an excellent local ring of characteristic p. More recently, Hochster and Huneke have introduced the concept of test exponent for tight closure: existence of these test exponents would mean that one would have to perform just one single check to determine whether or not an element of R belongs to the tight closure of I. However, to quote Hochster and Huneke, 'it is not at all clear whether to expect test exponents to exist; roughly speaking, test exponents exist if and only if tight closure commutes with localization'. The main purpose of this paper is to provide a short direct proof that test exponents exist for parameter ideals in a reduced excellent equidimensional local ring of characteristic p.