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The Brian\c{c}on-Skoda Theorem via weak functoriality of big Cohen-Macaulay algebras
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abstract
We prove that, given a sufficiently functorial assignment from rings to big Cohen-Macaulay algebras $R \mapsto B$, that the associated big Cohen-Macaulay closure operation on ideals $I \mapsto I B \cap R$ necessarily satisfies the Brian\c{c}on-Skoda type property. The proof combines arguments of Lipman-Teissier, Hochster, Ma, and Hochster-Huneke. Specializing to mixed characteristic, and utilizing a result of Bhatt on absolute integral closures, this recovers a slight strengthening of a result of Heitmann.
Forward citations
Cited by 2 Pith papers
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The Brian\c{c}on-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
For pseudo-rational and many Du Bois singularities, the full Briançon–Skoda containment J^{n+k-1} ⊆ J^k holds, and quasi-excellent finite-dimensional rings satisfy uniform Briançon–Skoda and uniform Artin–Rees.
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Factoring maps to big Cohen-Macaulay algebras through blowups
Functorial balanced big Cohen-Macaulay algebra assignments factor through RΓ(Y, O_Y) for every proper birational map Y → Spec R.
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