The facet ideal of a simplicial complex

By Sara Faridi. 2002

210 citations recorded by the source corpus.

Abstract. To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By generalizing the notion of a tree from graphs to simplicial complexes, we show that ideals associated to trees satisfy sliding depth condition, and therefore have normal and Cohen-Macaulay Rees rings. We also discuss connections with the theory of Stanley-Reisner rings.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/46980346de3056bf6134855716334920905259d0

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Citations in the corpus

Monomial ideals via square-free monomial ideals (2005, methodology): “See [5] and [6] for more on this concept.”
Regularity of Squarefree Monomial Ideals (2013, result): “This construction is similar to that of facet ideals in [20].”
Resolutions of Facet Ideals (2003, background): “Faridi generalized in [ 3 ] and [4] the definition of tree to simplicial complexes of any dimension, and also introduced facet ideals to study”
Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems (2006, background): “The related notion of facet ideal has been studied by Faridi [16, 17] and Zheng [44].”
Splittable ideals and the resolutions of monomial ideals (2005, background): “The facet ideal was introduced by Faridi [6, 7] to generalize an edge ideal.”
Standard graded vertex cover algebras, cycles and leaves (2006, background): “This follows from Theorem 2.2 and Theorem 2.5 In [10, Corollary 3.12] Faridi proved that the Rees algebra of the facet ideal I(∆) of a tree ∆ is a normal Cohen-Macaulay domain.”
Simplicial matrix-tree theorems (2008, unknown): “, [4, 10, 15, 19, 28].”
Simplicial trees are sequentially Cohen–Macaulay ☆ (2003, background): “Simplicial trees, on the other hand, were introduced in [6] in the context of Rees rings, and their facet ideals were studied further in [7] for their Cohen–Macaulay properties, and in [14] for their resolutions.”
Regularity of powers of forests and cycles (2014, background): “[5, 16, 21, 22, 29]), we start with the case when G contains no cycles.”
A Note on Rees Algebras and the MFMC Property (2005, background): “Edge ideals are also called facet ideals [9].”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, background): “Facet ideals, which can be seen as a generalization of edge ideals, were introduced later by Faridi in [11].”
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers (2006, background): “We can translate Faridi’s definition into hypergraph language; we call the translated version of Faridi’s leaf a f -leaf to distinguish it from a v-leaf.”
Embedded Associated Primes of Powers of Square-free Monomial Ideals (2008, methodology): “as appeared in the literature (cf. [14, 15, 16]). It is the same as the construction of edge ideals of clutters (cf. [12]) and similar to the construction of facet ideals of simplicial complexes (cf. [7, 13]). We shall also use H(I) to denote the corresponding hypergraph of a square-free monomial id”
A Beginner’s Guide to Edge and Cover Ideals (2013, background): “Further properties have been developed by He and Van Tuyl [35], Bouchat, Hà, and O’Keefe [3], and Alilooee and Faridi [1].”
Combinatorial secant varieties (2005, background): “The graph-theoretic interpretation of secant ideals extends to arbitrary squarefree monomial ideals, by thinking of these as facet ideals as in [12].”
A CLASS OF HYPERGRAPHS THAT GENERALIZES CHORDAL GRAPHS (2008, unknown): “See for instance [5], [7], [10], [11], [12], [13], [14], [16], [19].”
Algebraic Properties of the Path Ideal of a Tree (2009, background): “If R = k x1 xn , then a graded R-module M is called sequentially Cohen–Macaulay (over k) if there exists a finite filtration of graded Rmodules 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M such that each Mi/Mi−1 for i = 1 r is Cohen–Macaulay, and the Krull dimensions of the quotients are increasing: dim M1/M0 dim”
Combinatorial symbolic powers (2006, background): “Thus, a strategy for constructing Gröbner bases (and hence generating sets) for the symbolic powers I(r) would be the following: (1) Compute in≺(I) and give a combinatorial description for its minimal generators.”