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Expansion of a simplicial complex. (English) Zbl 1338.13041

Let \(\Delta\) be a simplicial complex. There have been a number of work investigating how combinatorial modification of \(\Delta\) affects algebraic properties and invariants of its Stanley-Reisner ideal and ring. For instance, [J. Biermann et al., J. Commut. Algebra 7, No. 3, 337–352 (2015; Zbl 1328.05207)] and references therein. Motivated by this problem, the paper under review introduces a notion of expansion for simplicial complexes. This notion is the higher dimensional analog of the well-known construction for graphs (where a simplicial complex is viewed as a hypergraphs with edges being its facets).
The main results of the paper show that the vertex decomposability and shellability of \(\Delta\) are preserved after an expansion. Furthermore, the projective dimension and the regularity of (the Stanley-Reisner ring of) expansions of \(\Delta\) are given in terms of those of \(\Delta\).

MSC:

13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13D05 Homological dimension and commutative rings
13C14 Cohen-Macaulay modules

Citations:

Zbl 1328.05207

Software:

Macaulay2
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References:

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