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Algebra retracts and Stanley-Reisner rings  [PDF]
Neil Epstein,Hop D. Nguyen
Mathematics , 2013, DOI: 10.1016/j.jpaa.2014.01.006
Abstract: In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field $k$ are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner rings over a field $k$ are again Stanley-Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over $k$ descend along graded algebra retracts.
Frobenius and Cartier algebras of Stanley-Reisner rings  [PDF]
Josep Alvarez Montaner,Alberto F. Boix,Santiago Zarzuela
Mathematics , 2011,
Abstract: We prove that the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring as well as its Matlis dual notion of Cartier algebra can be only principally generated or infinitely generated. As a consequence we are able to show that the set of F-jumping numbers of generalized test ideals associated to complete Stanley-Reisner rings form a discrete set.
Buchsbaum Stanley--Reisner rings with minimal multiplicity  [PDF]
Naoki Terai,Ken-ichi Yoshida
Mathematics , 2003,
Abstract: In this paper, we study non-Cohen--Macaulay Buchsbaum Stanley--Reisner rings with linear free resolution. In particular, for given integers $c$, $d$, $q$ with $c \ge 1$, $2 \le q \le d$, we give an upper bound $h_{c,d,q}$ on the dimension of the unique non-vanishing homology $\widetilde{H}_{q-2}(\Delta;k)$ of a $d$-dimensional Buchsbaum ring $k[\Delta]$ with $q$-linear resolution and codimension $c$. Also, we discuss about existence for such Buchsbaum rings with $\dim_k \widetilde{H}_{q-2}(\Delta;k) = h$ for any $h$ with $0 \le h \le h_{c,d,q}$, and prove an existence theorem in the case of $q=d=3$ using the notion of Cohen--Macaulay linear cover. On the other hand, we introduce the notion of Buchsbaum Stanley--Reisner rings with minimal multiplicity of type $q$, which extends the notion of Buchsbaum rings with minimal multiplicity defined by Goto. As an application, we give many examples of Buchsbaum Stanley--Reisner rings with $q$-linear resolution.
Stanley--Reisner rings with large multiplicities are Cohen--Macaulay  [PDF]
Naoki Terai,Ken-ichi Yoshida
Mathematics , 2005,
Abstract: We prove that certain class of Stanley--Reisner rings having sufficiently large multiplicities are Cohen--Macaulay using Alexander duality.
Cotangent cohomology of Stanley-Reisner rings  [PDF]
Klaus Altmann,Jan Arthur Christophersen
Mathematics , 2000,
Abstract: 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 algebro geometric objects assigned to X.
On the structure of Stanley-Reisner rings associated to cyclic polytopes  [PDF]
Janko Boehm,Stavros Argyrios Papadakis
Mathematics , 2009,
Abstract: We study the structure of Stanley-Reisner rings associated to cyclic polytopes, using ideas from unprojection theory. Consider the boundary simplicial complex Delta(d,m) of the d-dimensional cyclic polytope with m vertices. We show how to express the Stanley-Reisner ring of Delta(d,m+1) in terms of the Stanley-Reisner rings of Delta(d,m) and Delta(d-2,m-1). As an application, we use the Kustin-Miller complex construction to identify the minimal graded free resolutions of these rings. In particular, we recover results of Schenzel, Terai and Hibi about their graded Betti numbers.
Stellar subdivisions and Stanley-Reisner rings of Gorenstein complexes  [PDF]
Janko Boehm,Stavros Argyrios Papadakis
Mathematics , 2009,
Abstract: Unprojection theory aims to analyze and construct complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of non-acyclic Gorenstein simplicial complexes correspond to unprojections of type Kustin-Miller. As an application, we inductively calculate the minimal graded free resolutions of Stanley-Reisner rings associated to stacked polytopes, recovering results of Terai, Hibi, Herzog and Li Marzi.
Topology of polyhedral products and the Golod property of Stanley-Reisner rings  [PDF]
Kouyemon Iriye,Daisuke Kishimoto
Mathematics , 2013,
Abstract: The polyhedral product is a space constructed from a simplicial complex and a collection of pairs of spaces, which is connected with the Stanley Reisner ring of the simplicial complex via cohomology. Generalizing the previous work Grbic and Theriault, Grujic and Welker, and the authors, we show a decomposition of polyhedral products for a large class of simplicial complexes including the ones whose Alexander duals are shellable or sequentially Cohen-Macaulay. This implies the property, called Golod, of the corresponding Stanley-Reisner rings proved by Herzog, Reiner and Welker.
Addendum to "Frobenius and Cartier algebras of Stanley-Reisner rings" [J. Algebra 358 (2012) 162-177]  [PDF]
Josep Alvarez Montaner,Kohji Yanagawa
Mathematics , 2013,
Abstract: We give a purely combinatorial characterization of complete Stanley-Reisner rings having principally generated (equivalently, finitely generated) Cartier algebras.
Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids  [PDF]
Trygve Johnsen,Hugues Verdure
Mathematics , 2011, DOI: 10.1007/s00200-012-0183-7
Abstract: To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid.
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