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The \(S_ 2\)-closure of a Rees algebra. (English) Zbl 0777.13006

In this article, the authors show the following:
Theorem 1.3. Let \(A\) be a Noetherian ring with canonical module \(\omega_ A\), and suppose that \(A\) is generically a Gorenstein ring. Then \(B=\text{Hom}_ A(\omega_ A,\omega_ A)\) is the minimal extension of \(A\) with the property \((S_ 2)\).
Theorem 1.6. Let \(A\) be a Noetherian integral domain and let \(B\) be a finite extension of \(A\) with the same field of fractions. Then \(A=B\) if and only if \(\dim_ AB<\infty\).
Theorem 2.2. Let \(R\) be a Noetherian ring with \((S_{k+1})\) and let \(I\) be an ideal of \(R\) such that height \(I\geq k+1\). Then \(R[It]\) has \((S_{k+1})\) if and only if \(gr_ I(R)\) has \((S_ k)\).
Theorem 2.5. Let \(R\) be a Cohen-Macaulay ring and let \(I\) be an equimultiple ideal of codimension \(g\geq 1\). If \(R[It]\) has the \((S_ 2)\) condition then all the powers \(I^ n\) are unmixed ideals.
Theorem 2.6. Let \((R,{\mathfrak m})\) be a Gorenstein local ring of Krull dimension \(d\) and let \(I\) be an \({\mathfrak m}\)-primary, perfect, Gorenstein ideal. If the associated graded ring \(gr_ I(R)\) satisfies the condition \((S_{d-2})\), then \(I\) is a complete intersection.

MSC:

13A50 Actions of groups on commutative rings; invariant theory
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13D30 Torsion theory for commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14E05 Rational and birational maps

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References:

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