Symbolic Analytic Spread: Upper Bounds and Applications

Dao, Hailong; Montaño, Jonathan

doi:10.1017/s147474802000016x

Cited by 7 publications

(5 citation statements)

References 28 publications

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“…Let A be the bigraded A 0,0 -subalgebra A := A 0,0 [A 1,0 t 1 , A 0,1 t 2 ] of M . Now we have a natural surjection A i 1,0 A j 0,1 ⊗ R S → B i,j for all i, j ≥ 0 by (16). Thus the natural homomorphism A ⊗ R S → B is surjective.…”

Section: Nef Divisorsmentioning

confidence: 98%

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Analytic spread of filtrations on two dimensional normal local rings

Cutkosky¹

2022

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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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Section: Nef Divisorsmentioning

confidence: 98%

“…A survey of recent work on symbolic algebras is given in [15]. A different notion of analytic spread for families of ideals is given in [16]. A recent paper exploring ideal theory in two dimensional normal local domains using geometric methods is [31].…”

Section: Introductionmentioning

confidence: 99%

Analytic spread of filtrations on two dimensional normal local rings

Cutkosky¹

2022

Preprint

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“…, x d ] so that x α ∈ I σ if and only if h i (α) σ for all 1 i r. Then there exists a monomial ideal J and g ∈ N so that for any σ ∈ Q + we have I σ = J The connection with symbolic powers leads to a generalization of the splitting maps method found in [14,15] which guarantees the convergence of depths and normalized CastelnuovoMumford regularities (see Theorem 4.6 and Theorem 5. This computation also yields that the symbolic analytic spread (as discussed in [6]) can be computed via the symbolic polyhedron (as discussed in [3]) in the case of squarefree monomial ideals (see Remark 5.2).…”

Section: Introductionmentioning

confidence: 93%

Limit Behavior of the Rational Powers of Monomial Ideals

Lewis

2020

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We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized CastelnuovoMumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this we show the before-now unknown convergence of Stanley depths of integral closure powers. In addition, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

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“…A survey of recent work on symbolic algebras is given in [13]. A different notion of analytic spread for families of ideals is given in [14]. A recent paper exploring ideal theory in 2 dimensional normal local domains using geometric methods is [24].…”

Section: Introductionmentioning

confidence: 99%