相关论文: On Stanley-Reisner Rings with Minimal Betti Number…
We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex…
We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring,…
When a cone is added to a simplicial complex $\Delta$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex…
In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the…
The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology…
Nested set complexes appear as the combinatorial core of De Concini-Procesi arrangement models. We show that nested set complexes are homotopy equivalent to the order complexes of the underlying meet-semilattices without their minimal…
We study the minimal free resolution of the Veronese modules of the polynomial ring in n variables, by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We characterize when…
In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary…
We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti…
In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a…
We introduce a natural class of models of random chain complexes of real vector spaces that some classical ensembles of random matrices, the length $1$ case. We are interested here in the homological properties of these random complexes.…
Let $S=\mathbb{K}[x_1,\ldots,x_n]$ the polynomial ring over a field $\mathbb{K}$. In this paper for some families of monomial ideals $I \subset S$ we study the minimal number of generators of $I^k$. We use this results to find some other…
Using the concept of $d$-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of "chordal…
In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension…
The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are…
We present small triangulations of all connected sums of $\mathbb{CP}^2$ and $S^2 \times S^2$ with the standard piecewise linear structure. Our triangulations have $2\beta_2+2$ pentachora, where $\beta_2$ is the second Betti number of the…
Simplicial complexes X provide commutative rings A(X) via the Stanley-Reisner construction. We calculated the cotangent cohomology, i.e., T1 and T2 of A(X) in terms of X. These modules provide information about the deformation theory of the…
A Betti splitting $I=J+K$ of a monomial ideal $I$ ensures the recovery of the graded Betti numbers of $I$ starting from those of $J,K$ and $J \cap K$. In this paper, we introduce this condition for simplicial complexes, and, by using…
We show that birational smooth complex projective varieties with numerically effective canonical bundles along the exceptional loci have the same Betti numbers. In particular, birational smooth minimal models share the same Betti numbers.…
We study certain random simplicial complexes, called random quota complexes. A quota complex on $N+1$ weighted vertices is constructed by adding an $n$-simplex if the sum of the weights of the vertices is below a given quota, $q$. In this…