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On the structure of Noetherian symbolic Rees algebras. (English) Zbl 0696.13014

Let A be a Noetherian unmixed local ring, \(I\subset A\) an ideal, \(S\subset A\) a multiplicative system such that \(I\cap S=\emptyset\) and \(I^{(n)}:=A\cap I^ nA_ S\), \(n\in {\mathbb{Z}}\). If \(\ell (I^{(n)})=ht(I^{(n)})\) for a certain \(n\geq 1\), \(\ell (I)\) denotes the analytic spread of I, then the ring \(R_ I:=\oplus_{n\geq 0}I^{(n)} \) is Noetherian. The converse is also true when \(A/I^{(n)}\) is Cohen- Macaulay for \(n>>0\). The first implication extends some results of D. Katz and L. J. Ratliff jun. [Commun. Algebra 14, 959-970 (1986; Zbl 0609.13011)], A. Ooishi [Hiroshima Math. J. 15, 581-584 (1985; Zbl 0617.13011)] and others. Some nice examples show how important the “unmixed” condition is for the above result.
Now, let A be a Noetherian normal local domain of dimension \(\geq 1\), \(I\subset A\) an ideal of height 1 and \(S:=A\setminus \cup p \), where the union is made over all \(p\in Spec(A)\), \(p\supset I\), \(ht(p)=1\). If the order t of the class of I in Cl(A) is finite then \(R_ I\) is Noetherian and the following assertions are equivalent:
(1) \(R_ I\) is a CM-ring;
(2) \(R_ I':=\oplus_{n\in {\mathbb{Z}}}I^{(n)} \) is a CM-ring;
(3) \(G_ I:=\oplus_{n\geq 0}I^{(n)}/I^{(n+1)} \) is a CM-ring,
(4) \(I^{(n)}\) is a maximal CM-module over A for \(0\leq n<t;\)
(5) \(I^{(n)}\) is a maximal CM-module for every \(n\in {\mathbb{Z}}.\)
Moreover when A is a factor ring of a regular local ring then
(a) \(R_ I\) is a Gorenstein ring iff \(R_ I\) is a CM-ring and \(Hom_ A(I,A)\) is the canonical module of A,
(b) the following assertions are equivalent:(1’) \(R_ I'\) is a Gorenstein ring, (2’) \(R_ I'\) is a CM-ring and A is Gorenstein, (3’) \(G_ I\) is a Gorenstein ring.

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
13E05 Commutative Noetherian rings and modules
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References:

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