Normality of Rees Algebras of Monomial Ideals

Rees algebras of square-free monomial ideals

Journal of Commutative Algebra ◽

10.1216/jca-2015-7-1-25 ◽

2015 ◽

Vol 7 (1) ◽

pp. 25-53 ◽

Cited By ~ 1

Author(s):

Louiza Fouli ◽

Kuei-Nuan Lin

Keyword(s):

Monomial Ideals ◽

Rees Algebras

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Splittings and Symbolic Powers of Square-free Monomial Ideals

International Mathematics Research Notices ◽

10.1093/imrn/rnz138 ◽

2019 ◽

Author(s):

Jonathan Montaño ◽

Luis Núñez-Betancourt

Keyword(s):

Linear Optimization ◽

Monomial Ideals ◽

Sufficient Condition ◽

Graded Algebra ◽

Prime Characteristic ◽

Rees Algebra ◽

Rees Algebras ◽

Symbolic Powers ◽

Frobenius Map

Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

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Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

Forum Mathematicum ◽

10.1515/forum-2016-0007 ◽

2017 ◽

Vol 29 (4) ◽

Cited By ~ 2

Author(s):

Alberto Corso ◽

Uwe Nagel ◽

Sonja Petrović ◽

Cornelia Yuen

Keyword(s):

Blow Up ◽

Symmetric Matrix ◽

Monomial Ideals ◽

Edge Ideals ◽

Rees Algebras ◽

Determinantal Ideals ◽

Blowing Up ◽

Initial Ideals ◽

Vertex Decomposable ◽

Liaison Theory

AbstractWe investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo–Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind–Mertens formula for the content of the product of two polynomials.

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