Related papers: An Alexander-type duality for valuations
The class of simplicial complexes representing triangulations and subdivisions of Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for oriented matroid duality.
By using Alexander duality on simplicial complexes we give a new and algebraic proof of Dirac's theorem on chordal graphs.
Associated to each simplicial complex is a binary hierarchical model. We classify the simplicial complexes that yield unimodular binary hierarchical models. Our main theorem provides both a construction of all unimodular binary hierarchical…
We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of…
Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is…
Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is…
We prove a geometric refinement of Alexander duality for certain 2-complexes, the so-called gropes, embedded into 4-space. This refinement can be roughly formulated as saying that 4-dimensional Alexander duality preserves the disjoint Dwyer…
We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many…
Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Begueri, Bester and Kato. In this paper, we give another formulation and proof of…
In these notes a recently developed technique for the computation of line bundle-valued sheaf cohomology group dimensions on toric varieties is reviewed. The key result is a vanishing theorem for the contributing components which depends on…
We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I \subset \Bbbk[x_1, \ldots, x_n]$ with ${\rm deg}(\mathsf{m}) \le d$ for all $\mathsf{m} \in G(I)$, its dual $I^* \subset…
In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general…
An alternative proof of bornological Verdier duality for complex manifolds, as proven initially by Prosmans & Schneiders is given, using Schneider's theory of quasi-abelian homological algebra, and the theory of residues and duality.
This note contributes to the point calculus of persistent homology by extending Alexander duality to real-valued functions. Given a perfect Morse function $f: S^{n+1} \to [0,1]$ and a decomposition $S^{n+1} = U \cup V$ such that $M = \U…
In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral product complexes. Then we compute the universal algebra of polyhedral product complexes under certain split…
We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was…
We characterise integral Poincar\'e duality moment-angle complexes $\mathcal{Z}_{\mathcal{K}}$ in combinatorial terms of the Fan-Wang duality of the simplicial complex $\mathcal{K}$, and consequently in algebraic terms of the Gorenstein…
We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and…
For a simplicial complex $\Delta$, we introduce a simplicial complex attached to $\Delta$, called the expansion of $\Delta$, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the…
Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincar\'e sphere determines the topology of the link. It was shown that, except some explicitly described cases, the…