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Squarefree vertex cover algebras. (English) Zbl 1295.13007

Let \(\Delta\) be a simplicial complex and \(K\) a fixed field. J. Herzog et al. [Adv. Math. 210, No. 1, 304–322 (2007; Zbl 1112.13006)] introduced the vertex cover algebra \(A(\Delta)=\bigoplus_{k\geq 0}J_k(\Delta)t^k\subset K[x_1,\dots,x_n][t]\). Here \(J_k(\Delta)\) is a \(K\)-vector space spanned by all the monomials \(x^{ {\mathbf c}}\in K[x_1,\dots,x_n]\) where \({\mathbf c}\) is a \(k\)-cover of \(\Delta\). The vertex cover algebra was introduced in some sense for a better understanding of the symbolic power property, since \(J_k(\Delta)\) is the \(k\)-th symbolic power of \(J_1(\Delta)\), the cover ideal of \(\Delta\).
Due to the complexity of \(A(\Delta)\), the paper under review introduces a “simpler” subalgebra \[ B(\Delta)=\bigoplus_{k\geq 0}L_k(\Delta)t^k, \] which is generated by all the elements \(x^{ {\mathbf c}}t^k\) where \({\mathbf c}\) is a squarefree \(k\)-cover of \(\Delta\). A duality result was exhibited when the complex \(\Delta\) is pure.
The paper then focuses on the following two questions:
(1) When is \(B(\Delta)\) standard graded and when does this imply that \(A(\Delta)\) is standard graded?
(2) When do we have that \(B(\Delta)=A(\Delta)\)?
Among others, special cases are discussed regarding these questions in terms of (hyper)-cycles of \(\Delta\). For instance, Theorem 2.12 of this paper says that “Let \(\Delta\) be a simplicial complex satisfying the strict intersection property and suppose that no two cycles of \(G_\Delta\) have precisely two edges in common. Then \(B(\Delta)=A(\Delta)\) if and only each connected component of \(G_\Delta\) is a bipartite graph or an odd cycle.”
This interesting result brings our attention to a similar result regarding the linear type problem of hyper-edge ideals of complexes. The linear type problems are considered for understanding when the symmetric algebra and Rees algebra of an ideal of the ring coincide. R. H. Villarreal [Commun. Algebra 23, No. 9, 3513–3524 (1995; Zbl 0836.13014)] showed that the edge ideal of a connected simple graph is of linear type if and only if this graph contains at most one odd cycle. The strict intersection property in Theorem 2.12 seems to be related to the simplicial cycle, a notion introduced in [M. Caboara et al., J. Symb. Comput. 42, No. 1–2, 74–88 (2007; Zbl 1124.05094)]. Interestingly, simplicial cycles can be applied for producing higher dimensional results when studying the linear type problem of hyper-edge ideals of complexes, following Villarreal.
The classes of complexes considered so far in the paper for which \(B(\Delta)=A(\Delta)\), happened to have the property that \(A(\Delta)\) is generated in degree at most \(2\). Thus, the paper under review turns to the class of shifted complexes. In these cases, the generators have higher degrees. The tools used extensively here come from C. A. Francisco et al. [J. Algebra 332, No. 1, 522–542 (2011; Zbl 1258.13019)]. The cases considered in this direction in this paper are mostly principal Borel.

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
05C65 Hypergraphs

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References:

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