Face Rings Research Articles

Differential operators, retracts, and toric face rings

Differential operators, retracts, and toric face rings

Persistent Tor-algebra for protein-protein interaction analysis.

Protein-protein interactions (PPIs) play crucial roles in almost all biological processes from cell-signaling and membrane transport to metabolism and immune systems. Efficient characterization of PPIs at the molecular level is key to the fundamental understanding of PPI mechanisms. Even with the gigantic amount of PPI models from graphs, networks, geometry and topology, it remains as a great challenge to design functional models that efficiently characterize the complicated multiphysical information within PPIs. Here we propose persistent Tor-algebra (PTA) model for a unified algebraic representation of the multiphysical interactions. Mathematically, our PTA is inherently algebraic data analysis. In our PTA model, protein structures and interactions are described as a series of face rings and Tor modules, from which PTA model is developed. The multiphysical information within/between biomolecules are implicitly characterized by PTA and further represented as PTA barcodes. To test our PTA models, we consider PTA-based ensemble learning for PPI binding affinity prediction. The two most commonly used datasets, i.e. SKEMPI and AB-Bind, are employed. It has been found that our model outperforms all the existing models as far as we know. Mathematically, our PTA model provides a highly efficient way for the characterization of molecular structures and interactions.

Cohomology ring of tree braid groups and exterior face rings

For a tree T T and a positive integer n n , let B n T B_nT denote the n n -strand braid group on T T . We use discrete Morse theory techniques to show that the cohomology ring H ∗ ( B n T ) H^*(B_nT) is encoded by an explicit abstract simplicial complex K n T K_nT that measures n n -local interactions among essential vertices of T T . We show that, in many cases (for instance when T T is a binary tree), H ∗ ( B n T ) H^*(B_nT) is the exterior face ring determined by K n T K_nT .

Monomial non-Golod face rings and Massey products

Monomial non-Golod face rings and Massey products

Stanley–Reisner rings for symmetric simplicial complexes, $G$-semimatroids and Abelian arrangements

We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset under a mild condition on the group action.We also identify a class of actions on simplicial complexes that preserve the homotopical Cohen–Macaulay property under quotients. When the acted-upon poset is the independence complex of a semimatroid, the h-polynomial of the ring of invariants can be read off the Tutte polynomial of the associated group action. Moreover, in this case an additional condition on the action ensures that the quotient poset is Cohen–Macaulay in characteristic 0 and every characteristic that does not divide an explicitly computable number. This implies the same property for the associated Stanley–Reisner rings. In particular, this holds for independence posets and rings associated to toric, elliptic and, more generally, (p,q) -arrangements. As a byproduct, we prove that posets of connected components (also known as posets of layers) of such arrangements are Cohen–Macaulay with the same condition on the characteristic.

Комплекс Аксай-1 (к проблеме изучения древних антропоморфных изваяний и оленных камней Центрального Казахстана)

The article introduces the Aksai-1 complex of three “cult stones” (Aksai valley, Karaganda region, Kazakhstan), which includes an ancient anthropomorphic statue featuring a human face image at the top of a monolith, and a deer stone of the common Eurasian type. An analysis of the positioning relative to each other of these objects suggests that they were mounted together only randomly. The statue of the Aksai-1 complex is analogous to one of the two other large anthropomorphic statues found in the Aksai valley. That other statue is located at the Aksai burial ground near the barrow with stone ridges). The authors question the direct connection between this statue and the barrow with the stone ridges. The third Aksai anthropomorphic statue referred to as “Akbikesh” stands out not only for its huge size and a peculiar stylization of the face image, but also for the presence of additional details — rings pecked on the sides of the head. The question of the cultural and chronological attribution of the Aksai anthropomorphic statues remains unresolved. Their belonging to the same artistic tradition is also unclear. It can be assumed that the “menhir-like” shape bring these statues closer to the “cult stones” of the Late Bronze Age. The latter tradition, according to A. Z. Baysenov, continued in the Early Iron Age. The likelihood of their coexistence with deer stones cannot be excluded, anyway, this is possible for the case of “Akbikesh”, whose composition combine a face image and rings similar to the common attribute of deer stones.

Ext and local cohomology modules of face rings of simplicial posets

There are a large number of theorems detailing the homological properties of the Stanley–Reisner ring of a simplicial complex. Here we attempt to generalize some of these results to the case of a simplicial poset. By investigating the combinatorics of certain modules associated with the face ring of a simplicial poset from a topological viewpoint, we extend some results of Miyazaki and Gräbe to a wider setting.

On toric face rings I

We introduce the class of weakly log canonical singularities, a natural generalization of semi-log canonical singularities. Toric varieties (associated to toric face rings, possibly non-normal or reducible) which have weakly (semi-) log canonical singularities are classified. In the toric case, we discuss residues to lc centers of codimension one or higher.

Face Rings of Cycles, Associahedra, and Standard Young Tableaux

We show that $J_n$, the Stanley-Reisner ideal of the $n$-cycle, has a free resolution supported on the $(n-3)$-dimensional simplicial associahedron $A_n$. This resolution is not minimal for $n \geq 6$; in this case the Betti numbers of $J_n$ are strictly smaller than the $f$-vector of $A_n$. We show that in fact the Betti numbers $\beta_{d}$ of $J_n$ are in bijection with the number of standard Young tableaux of shape $(d+1, 2, 1^{n-d-3})$. This complements the fact that the number of $(d-1)$-dimensional faces of $A_n$ are given by the number of standard Young tableaux of (super)shape $(d+1, d+1, 1^{n-d-3})$; a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of $J_n$ that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.

Volume Polynomials and Duality Algebras of Multi-Fans

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(\Delta)$. We study the structure and properties of $V_\Delta$ and $\mathcal{A}^*(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik--Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.

Dualizing complexes of seminormal affine semigroup rings and toric face rings

We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen–Macaulay semigroup ring by the “shape” of the canonical module. We also characterize the seminormality of a toric face ring in terms of the dualizing complex. A toric face ring is a simultaneous generalization of Stanley–Reisner rings and affine semigroups.

Gas-Lubricated Mechanical Face Rings for CO2 Reduction

Gas-Lubricated Mechanical Face Rings for CO2 Reduction

On the Koszul property of toric face rings

Toric face rings are a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field $k$. Generalizing works of Laudal, Sletsj\o{}e and Herzog et al., graded Betti numbers of $k$ over the toric face rings are computed, and a characterization of Koszul toric face rings is provided. We investigate a conjecture suggested by R\"{o}mer about the sufficient condition for the Koszul property. The conjecture is inspired by Fr\"{o}berg's theorem on the Koszulness of quadratic squarefree monomial ideals. Finally, it is proved that initially Koszul toric face rings are affine monoid rings.

Gas-Lubricated Mechanical Face Rings for CO2 Reduction

Gas-Lubricated Mechanical Face Rings for CO2 Reduction

The geometry and combinatorics of cographic toric face rings

In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures of a graph: the cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally cyclic orientations. We describe the properties of the cographic ring and, in particular, relate the invariants of the ring to the invariants of the corresponding graph. Our study of the cographic ring fits into a body of work on describing rings constructed from graphs. Among the rings that can be constructed from a graph, cographic rings are particularly interesting because they appear in the study of compactified Jacobians of nodal curves.

Seminormality and local cohomology of toric face rings

We characterize the toric face rings that are normal (respectively seminormal). Extending results about local cohomology of Brun, Bruns, Ichim, Li and Römer of seminormal monoid rings and Stanley toric face rings, we prove the vanishing of certain graded parts of local cohomology of seminormal toric face rings. The combinatorial formula we obtain generalizes Hochsterʼs formula. We also characterize all (necessarily seminormal) toric face rings that are F-pure or F-split over a field of characteristic p>0. An example is given to show that F-injectivity does not behave well with respect to face projections of toric face rings. Finally, it is shown that weakly F-regular toric face rings are normal affine monoid rings.

Formula omitted] factors of Koszul algebras and applications to face rings

Generalizing the notion of a Koszul algebra, a graded k-algebra A is K2 if its Yoneda algebra ExtA(k,k) is generated as an algebra in cohomology degrees 1 and 2. We prove a strong theorem about K2 factor algebras of Koszul algebras and use that theorem to show the Stanley–Reisner face ring of a simplicial complex Δ is K2 whenever the Alexander dual simplicial complex Δ⁎ is (sequentially) Cohen–Macaulay.

Moment-angle complexes from simplicial posets

Abstract We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.

F-vectors of simplicial posets that are balls

Results of R. Stanley and M. Masuda completely characterize the h-vectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vectors. We also present new methods for constructing poset balls with specific h-vectors. Combining this work with a new result of S. Murai we are able to give a complete characterization of the h-vectors of simplicial poset balls in all even dimensions, as well as odd dimensions less than or equal to five.

Higher Cohen-Macaulay property of squarefree modules and simplicial posets

Recently, G. Fløystad studied higher Cohen-Macaulay property of certain finite regular cell complexes. In this paper, we partially extend his results to squarefree modules, toric face rings, and simplicial posets. For example, we show that if (the corresponding cell complex of) a simplicial poset is l l -Cohen-Macaulay, then its codimension one skeleton is ( l + 1 ) (l+1) -Cohen-Macaulay.