The families of matroid independence complexes and pure shifted simplicial complexes are two remarkable classes of simplicial complexes that share several structural properties, but for which theorems are typically proved in different ways. In a groundbreaking paper Kook, Reiner and Stanton [4] proved that matroid complexes are Laplacian integral. Afterwards Duval and Reiner [1] proved the anal...

Via the BGG-correspondence a simplicial complex ∆ on [n] is transformed into a complex of coherent sheaves L̃(∆) on the projective space P. In general we compute the support of each of its cohomology sheaves. When the Alexander dual ∆ is Cohen-Macaulay there is only one such non-zero cohomology sheaf. By considering when this sheaf can be an a’th syzygy sheaf in a locally free resolution, we get...

We consider the dg-category of twisted complexes over simplicial ringed spaces. It is clear that a map $$f:({\mathscr {U}},{\mathscr {R}})\rightarrow ({\mathscr {V}}, {\mathscr {S}})$$ between spaces induces dg-functor $$f^*:\mathrm{Tw}({\mathscr {S}})\rightarrow \mathrm{Tw}({\mathscr {U}}, {R}})$$ where $$\mathrm{Tw}({\mathscr denotes on $$({\mathscr . prove for homotopic maps f and g, there e...

Among shellable complexes a certain class has maximal modular homology, and these are the so-called saturated complexes. We extend the notion of saturation to arbitrary pure complexes and give a survey of their properties. It is shown that saturated complexes can be characterized via the p -rank of incidence matrices and via the structure of links. We show that rank-selected subcomplexes of sat...

We first introduce the notion of a completion. Completions are inductive properties which may be expressed in a declarative way and which may be combined. We show that completions may be used for describing structures or transformations which appear in combinatorial topology. We present two completions, 〈CUP〉 and 〈CAP〉, in order to define, in an axiomatic way, a remarkable collection of acyclic...