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An improved Briançon-Skoda theorem with applications to the Cohen- Macaulayness of Rees algebras. (English) Zbl 0788.13001
We give an improved version of the Briançon-Skoda theorem for regular rings containing a field. If \((R,m)\) is such a ring, \(I\) is an ideal of analytic spread \(\ell\) and bigheight \(h\), and \(J\) is a reduction of \(I\) then we show that \(I^ \ell \subseteq J(I^{\ell-h})^{\text{un}}\) (where \(( )^{\text{un}}\) means: take the intersection of the minimal primary components). The proof uses tight closure in characteristic \(p\) and then standard techniques of reduction to characteristic \(p\) for local rings containing fields of characteristic 0.
We then apply this result to study the relationship between the Cohen- Macaulayness of \(R[It]\) and \(\text{Gr}_ I(R)\). In particular we show that if \(R\) is a regular local ring containing a field and \(I\) is an unmixed curve then \(R[It]\) is Cohen-Macaulay if and only if \(\text{Gr}_ I(R)\) is Cohen-Macaulay.
In a final section we continue the investigation of 4-generated unmixed ideals of height 2 in a 3-dimensional regular local ring begun by Vasconcelos.

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13C14 Cohen-Macaulay modules
13H05 Regular local rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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