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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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