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Abstract In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R , and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R . The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.
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This content will become publicly available on August 3, 2026
Rees algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings
Let I = {In} be a Q-divisorial filtration on a two dimensional normal excellent local ring (R, mR). Let R[I] = ⊕n≥0In be the Rees algebra of I and τ : ProjR[I]) → Spec(R) be the natural morphism. The reduced fiber cone of I is the R- algebra R[I]/ p mRR[I], and the reduced exceptional fiber of τ is Proj(R[I]/ p mRR[I]). In [7], we showed that in spite of the fact that R[I] is often not Noetherian, mRR[I] always has only finitely many minimal primes, so τ −1 (mR) has only finitely many ir- reducible components. In Theorem 1.2, we give an explicit description of the scheme structure of Proj(R[I]). As a corollary, we obtain in Theorem 1.3 a new proof of a theorem of F. Russo, showing that Proj(R[I]) is always Noetherian and that R[I] is Noetherian if and only if Proj(R[I]) is a proper R-scheme. In Corollary 1.4 to Theorem 1.2, we give an explicit description of the scheme structure of the reduced exceptional fiber Proj(R[I]/ p mRR[I]) of τ , in terms of the possible values 0, 1 or 2 of the analytic spread l(I) = dim R[I]/mRR[I]. In the case that l(I) = 0, τ −1 (mR) is the emptyset; this case can only occur if R[I] is not Noetherian. At the end of the introduction, we give a simple example of a graded filtration J of a two dimensional regular local ring R such that Proj(R[J ]) is not Noetherian. This filtration is necessarily not divisorial
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- PAR ID:
- 10595960
- Publisher / Repository:
- Taylor and Francis
- Date Published:
- Journal Name:
- Communications in Algebra
- Edition / Version:
- 1
- Volume:
- 53
- Issue:
- 8
- ISSN:
- 0092-7872
- Page Range / eLocation ID:
- 3364 to 3387
- Subject(s) / Keyword(s):
- Commutative Algebra, Commutative Algebra
- Format(s):
- Medium: X Size: unknown Other: unknown
- Size(s):
- unknown
- Sponsoring Org:
- National Science Foundation
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