Integral closure and other operations on monomial ideals

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

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On the Stanley Depth of Powers of Monomial Ideals

Mathematics ◽

10.3390/math7070607 ◽

2019 ◽

Vol 7 (7) ◽

pp. 607 ◽

Cited By ~ 1

Author(s):

S. A. Seyed Fakhari

Keyword(s):

Polynomial Ring ◽

Upper Bound ◽

Integral Closure ◽

Monomial Ideals ◽

The Other ◽

Finitely Generated ◽

Stanley Depth ◽

Survey Article ◽

Symbolic Powers ◽

Other Hand

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.

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Limit behavior of the rational powers of monomial ideals

Journal of Algebra and Its Applications ◽

10.1142/s021949882350069x ◽

2021 ◽

Author(s):

James Lewis

Keyword(s):

Local Cohomology ◽

Integral Closure ◽

Monomial Ideals ◽

Limit Behavior ◽

Associated Primes ◽

Powers Of Ideals ◽

Symbolic Powers ◽

Local Cohomology Modules ◽

Cohomology Modules

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 Castelnuovo–Mumford 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. Additionally, 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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MULTIPLICITY AND ŁOJASIEWICZ EXPONENT OF GENERIC LINEAR SECTIONS OF MONOMIAL IDEALS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714001154 ◽

2015 ◽

Vol 91 (2) ◽

pp. 191-201 ◽

Cited By ~ 4

Author(s):

CARLES BIVIÀ-AUSINA

Keyword(s):

Integral Closure ◽

Monomial Ideals ◽

Homogeneous Polynomials ◽

Łojasiewicz Exponent ◽

Łojasiewicz Exponents ◽

Samuel Multiplicity ◽

Linear Sections ◽

Lojasiewicz Exponent

AbstractWe obtain a characterisation of the monomial ideals $I\subseteq \mathbb{C}[x_{1},\dots ,x_{n}]$ of finite colength that satisfy the condition $e(I)={\mathcal{L}}_{0}^{(1)}(I)\cdots {\mathcal{L}}_{0}^{(n)}(I)$, where ${\mathcal{L}}_{0}^{(1)}(I),\dots ,{\mathcal{L}}_{0}^{(n)}(I)$ is the sequence of mixed Łojasiewicz exponents of $I$ and $e(I)$ is the Samuel multiplicity of $I$. These are the monomial ideals whose integral closure admits a reduction generated by homogeneous polynomials.

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