Symbolic powers of monomial ideals and vertex cover algebras

By Jürgen Herzog, Takayuki Hibi, Ngô Viêt Trung. 2005

169 citations recorded by the source corpus.

Abstract We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen–Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.

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

View Paper: https://www.semanticscholar.org/paper/319dc297f8456caa37f6682dafcc5822c3cc4972

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In Algebra and Number Theory

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Citations in the corpus

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals (2009, background): “, Alexander duality, cover ideals. Version: October 22, 2018. 1 HYPERGRAPHS AND ASSOCIATED PRIMES 2 In this paper, we shall concentrate on the cover ideal J(H) and its powers; and thus, complementing [12, 13, 22], we study squarefree monomial ideals and their powers. Our work was initially inspired ”
Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems (2006, background): “Symbolic Rees algebras have a combinatorial interpretation [22].”
Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals (2009, methodology): “We call A(∆) the vertex cover algebra of ∆ [4].”
Cohen-Macaulayness of large powers of Stanley-Reisner ideals (2010, background): “Note that the equality between ordinary and symbolic powers of J(∆) were already studied in [9], [10].”
Combinatorial symbolic powers (2006, unknown): “is allows us to give explicit formulas for the symbolic powers of the edge ideals of perfect graphs. The study of the symbolic powers of edge ideals was initiated in [23] and further elaborated on in [2, 14, 17]. Our emphasis on perfect graphs, and the extensions to secant ideals of edge ideals in t”