Stanley-Reisner Ring Research Articles

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.

Graded Betti Numbers of Balanced Simplicial Complexes

We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and Isubseteq S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.

The cohomology rings of smooth toric varieties and quotients of moment-angle complexes

Partial quotients of moment-angle complexes are topological analogues of smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring of such a partial quotient in terms of a torsion product involving the corresponding Stanley-Reisner ring. We show that their formula gives the correct cup product if 2 is invertible in the chosen coefficient ring, but not in general. We rectify this by defining an explicit deformation of the canonical multiplication on the torsion product.

CONSTRUCTING NONPROXY SMALL TEST MODULES FOR THE COMPLETE INTERSECTION PROPERTY

AbstractA local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.

Systems of parameters and the Cohen–Macaulay property

We recall numerical criteria for Cohen–Macaulayness related to system of parameters and introduce monomial ideals of König type which include the edge ideals of König graphs. We show that a monomial ideal is of König type if and only if its corresponding residue class ring admits a system of parameters whose elements are of the form \(x_i-x_j\). This provides an algebraic characterization of König graphs. We use this special parameter systems for the study of the edge ideal of König graphs and the study of the order complex of a certain family of posets. Finally, for any simplicial complex \(\Delta \) we introduce a system of parameters for \(K[\Delta ]\) with a universal construction principle, independent of the base field and only dependent on the faces of \(\Delta \). This system of parameters is an efficient tool to test Cohen–Macaulayness of the Stanley–Reisner ring of a simplicial complex.

Construction of strictly closed rings

The notion of strict closedness of rings was given by J. Lipman in connection with a conjecture of O. Zariski. The present purpose is to give a practical method of construction of strictly closed rings. It is also shown that the Stanley-Reisner rings of simplicial complexes (resp. $F$-pure rings satisfying the condition $(S_2)$ of Serre) are strictly closed (resp. weakly Arf) rings.

F-invariants of Stanley-Reisner rings

In prime characteristic there are important invariants that allow us to measure singularities. For certain cases, it is known that they are rational numbers. In this article, we show this property for Stanley-Reisner rings in several cases.

$b$-vectors of chordal graphs

The b-vector (b1,b2…,bd) of a graph G is defined in terms of its clique vector (c1,c2…,cd) by the equation ∑i=1dbi(x+1)i−1= ∑i=1dcixi−1, where d is the largest cardinality of a clique in G. We study the relation of the b-vector of a chordal graph G with some structural properties of G. In particular, we show that the b-vector encodes different aspects of the connectivity and clique dominance of G. Furthermore, we relate the b-vector with the Betti numbers of the Stanley–Reisner ring associated to clique simplicial complex of G.

Higher weight spectra of codes from Veronese threefolds

We study binary linear codes C obtained from the quadric Veronese embedding of P 3 in P 9 over F 2 . We show how one can find the higher weight spectra of these codes. Our method will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C .

Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.

$g$-vectors of manifolds with boundary

We extend several g-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain Kühnel-type bounds on the Betti numbers as well as on certain weighted sums of Betti numbers of manifolds with boundary. Our main tool is the completion Δ ^ of a manifold with boundary Δ; it is obtained from Δ by coning off the boundary of Δ with a single new vertex. We show that despite the fact that Δ ^ has a singular vertex, its Stanley–Reisner ring shares a few properties with the Stanley–Reisner rings of homology spheres. We close with a discussion of a connection between three lower bound theorems for manifolds, PL-handle decompositions, and surgery.

On the Frobenius complexity of Stanley–Reisner rings

The Frobenius complexity of a local ring R measures asymptotically the abundance of Frobenius operators of order e on the injective hull of the residue field of R. It is known that, for Stanley–Reisner rings, the Frobenius complexity is either −∞ or 0. This invariant is determined by the complexity sequence of the ring of Frobenius operators on the injective hull of the residue field. We will show that is constant for generalizing work of Àlvarez Montaner, Boix, and Zarzuela. Our result settles an open question mentioned by Àlvarez Montaner.

Pure resolutions, linear codes, and Betti numbers

We consider the minimal free resolutions of Stanley-Reisner rings associated to linear codes and give an intrinsic characterization of linear codes having a pure resolution. We use this characterization to quickly deduce the minimal free resolutions of Stanley-Reisner rings associated to MDS codes as well as constant weight codes. We also deduce that the minimal free resolutions of Stanley-Reisner rings of first order Reed-Muller codes are pure, and explicitly describe the Betti numbers. Further, we show that in the case of higher order Reed-Muller codes, the minimal free resolutions are almost always not pure. The nature of the minimal free resolution of Stanley-Reisner rings corresponding to several classes of two-weight codes, besides the first order Reed-Muller codes, is also determined.

Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

Let ${\mathcal R}_\mathbb{K}[H]$ be the Hibi ring over a field $\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $\omega$ the canonical ideal of ${\mathcal R}_\mathbb{K}$[H]. We show the powers $\omega^{(n)}$ of $\omega$ in the group of divisors $\mathrm{Div}({\mathcal R}_\mathbb{K}[H])$ is identical with the ordinary powers of $\omega$, describe the $\mathbb{K}$-vector space basis of $\omega^{(n)}$ for $n \in \mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n \geq 0} \omega^n/\mathfrak{m} \omega^n$ and $\bigoplus_{n \geq 0} (\omega^{(-1)})^n/\mathfrak{m} (\omega^{(-1)})^n$ of $\omega$ and $\omega^{(-1)}$ are sum of the Ehrhart rings, defined by sequences of elements of $P$ with a certain condition, which are polytopal complex version of Stanley–Reisner rings. Moreover, we show that the analytic spread of $\omega$ and $\omega^{(-1)}$ are maximum of the dimensions of these Ehrhart rings. Using these facts, we show that the question of Page about Frobenius complexity is affirmative: $\lim_{p \to \infty} \mathrm{cx}_F ({\mathcal R}_\mathbb{K}[H]) = \dim(\bigoplus_{n \geq 0} \omega^{(-n)}/\mathfrak{m} \omega^{(-n)}) - 1$, where $p$ is the characteristic of the field $\mathbb{K}$.

Hodge-theoretic mirror symmetry for toric stacks

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.

Cohomologically Full Rings

Abstract Inspired by a question raised by Eisenbud–Mustaţă–Stillman regarding the injectivity of maps from ${\operatorname{Ext}}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings, which we call cohomologically full rings. This class of rings includes many well-known singularities: Cohen–Macaulay rings, Stanley–Reisner rings, F-pure rings in positive characteristics, and Du Bois singularities in characteristics $0$. We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. When $R$ is a standard graded algebra over a field of characteristic $0$, we show under certain conditions that being cohomologically full is equivalent to the intermediate local cohomology modules being generated in degree $0$. Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras.

Higher Weight Spectra of Veronese Codes

We study ${q}$ -ary linear codes ${C}$ obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of ${C}$ over all field extensions of $\mathbb {F}_{q}$ . Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code ${C}$ .

Equivariant $$\varvec{K}$$-theory of quasitoric manifolds

Let \(X(Q,\Lambda )\) be a quasitoric manifold associated to a simple convex polytope Q and characteristic function \(\Lambda \). Let \(T\cong ({S}^1)^n\) denote the compact n-torus acting on \(X=X(Q,\Lambda )\). The main aim of this article is to give a presentation of the T-equivariant K-ring of X, as a Stanley–Reisner ring over \(K^*(pt)\). We also derive the presentation for the ordinary K-ring of X.

Tight closure of powers of ideals and tight hilbert polynomials

AbstractLet (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex Δ . Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of Δ . Along the way, we also show that if a monomial ideal has at least one generator of degree 2, then the linear strand of its minimal free resolution can be written using only ± 1 coefficients.