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Fundamental results on $$s$$-closures. (English) Zbl 07264670
Summary: This paper establishes the fundamental properties of the $$s$$-closures, a recently introduced family of closure operations on ideals of rings of positive characteristic. The behavior of the $$s$$-closure of homogeneous ideals in graded rings is studied, and criteria are given for when the $$s$$-closure of an ideal can be described exactly in terms of its tight closure and rational powers. Sufficient conditions are established for the weak $$s$$-closure to equal to the $$s$$-closure. A generalization of the Briançon-Skoda theorem is given which compares any two different $$s$$-closures applied to powers of the same ideal.
##### MSC:
 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 13A02 Graded rings 13B22 Integral closure of commutative rings and ideals 13H15 Multiplicity theory and related topics
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##### References:
 [1] Bruns, Winfried; Herzog, Jürgen, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39 (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0788.13005 [2] Epstein, Neil; Yao, Yongwei, Some extensions of Hilbert-Kunz multiplicity, Collect. Math., 68, 1, 69-85 (2017) · Zbl 1405.13011 [3] Hochster, Melvin; Huneke, Craig, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Am. Math. Soc., 3, 1, 31-116 (1990) · Zbl 0701.13002 [4] Huneke, Craig; Swanson, Irena, Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, vol. 336 (2006), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1117.13001 [5] Smith, Karen E., Tight closure in graded rings, J. Math. Kyoto Univ., 37, 1, 35-53 (1997) · Zbl 0902.13005 [6] Taylor, William D., Interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicity, J. Algebra, 509, 212-239 (2018) · Zbl 1406.13022
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