197,005 research outputs found
On Stanley-Reisner rings of reduction number one
No full textIn this paper we study a particular class of algebraic varieties, which are the finite unions of linear spaces. For a suitable choice of the system of coordinates these varieties are defined by squarefree monomials. Their coordinate rings are Stanley-Reisner rings of simplicial complexes. Each simplicial complex determines a simple, one-dimensional non directed graph. We give a combinatorial criterion on the graph which assures that the Stanley-Reisner ring has a system of parameters consisting of linear forms. The resulting class of Stanley-Reisner rings strictly includes those which are Cohen-Macaulay of minimal degree. These belong to the class of varieties classified by Eisenbud and Goto in 1984. An explicit constructive description of these varieties has been developed in a previous paper by the same authors
Frobenius and Cartier algebras of Stanley-Reisner rings
No full textWe study the generation of the Frobenius algebra of
the injective hull of a complete Stanley-Reisner ring over a
field with positive characteristic. In particular, by extending
the ideas used by M. Katzman to give a counterexample to a
question raised by G. Lyubeznik and K. E. Smith about the
finite generation of Frobenius algebras, we prove that the
Frobenius algebra of the injective hull of a complete
Stanley-Reisner ring can be only principally generated or
infinitely generated. Also, by using our explicit description
of the generators of such algebra and applying the recent work
by M. Blickle about Cartier algebras and generalized test
ideals, 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 subset inside the
non-negative real numbersPreprin
Cohen–Macaulayness of large powers of Stanley–Reisner ideals
No full textAbstractWe prove that for m⩾3, the symbolic power IΔ(m) of a Stanley–Reisner ideal is Cohen–Macaulay if and only if the simplicial complex Δ is a matroid. Similarly, the ordinary power IΔm is Cohen–Macaulay for some m⩾3 if and only if IΔ is a complete intersection. These results solve several open questions on the Cohen–Macaulayness of ordinary and symbolic powers of Stanley–Reisner ideals. Moreover, they have interesting consequences on the Cohen–Macaulayness of symbolic powers of facet ideals and cover ideals
Frobenius and Cartier algebras of Stanley-Reisner rings
No full textWe study the generation of the Frobenius algebra of
the injective hull of a complete Stanley-Reisner ring over a
field with positive characteristic. In particular, by extending
the ideas used by M. Katzman to give a counterexample to a
question raised by G. Lyubeznik and K. E. Smith about the
finite generation of Frobenius algebras, we prove that the
Frobenius algebra of the injective hull of a complete
Stanley-Reisner ring can be only principally generated or
infinitely generated. Also, by using our explicit description
of the generators of such algebra and applying the recent work
by M. Blickle about Cartier algebras and generalized test
ideals, 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 subset inside the
non-negative real number
Frobenius and Cartier algebras of Stanley-Reisner rings
No full textWe study the generation of the Frobenius algebra of the injective
hull of a complete Stanley–Reisner ring over a field with positive
characteristic. In particular, by extending the ideas used by
M. Katzman to give a counterexample to a question raised by
G. Lyubeznik and K.E. Smith about the finite generation of Frobenius
algebras, we prove that the Frobenius algebra of the injective
hull of a complete Stanley–Reisner ring can be only principally
generated or infinitely generated. Also, by using our explicit description
of the generators of such algebra and applying the recent
work by M. Blickle about Cartier algebras and generalized test ideals,
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 subset inside the non-negative real numbers.Peer Reviewe
Frobenius and Cartier algebras of Stanley-Reisner rings
No full textWe study the generation of the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring over a field with positive characteristic. In particular, by extending the ideas used by M. Katzman to give a counterexample to a question raised by . Lyubeznik and K.E. Smith about the finite generation of Frobenius algebras, we prove that the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring can be only principally generated or infinitely generated. Also, by using our explicit description of the generators of such algebra and applying the recent work by M. Blickle about Cartier algebras and generalized test ideals, we are able to show that the set of -jumping numbers of generalized test ideals associated to complete Stanley-Reisner rings form a discrete subset inside the non-negative real numbers
Frobenius and Cartier algebras of Stanley-Reisner rings
No full textWe study the generation of the Frobenius algebra of
the injective hull of a complete Stanley-Reisner ring over a
field with positive characteristic. In particular, by extending
the ideas used by M. Katzman to give a counterexample to a
question raised by G. Lyubeznik and K. E. Smith about the
finite generation of Frobenius algebras, we prove that the
Frobenius algebra of the injective hull of a complete
Stanley-Reisner ring can be only principally generated or
infinitely generated. Also, by using our explicit description
of the generators of such algebra and applying the recent work
by M. Blickle about Cartier algebras and generalized test
ideals, 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 subset inside the
non-negative real numbersPreprin
Algebra, Combinatorics, and Computation of Certain Tight Closure Invariants in Stanley-Reisner Rings
No full textTight closure was first introduced in the 1980's \cite{HHOrigin} by Hochster and Huneke to answer questions about invariant theory and the Brian\c{c}on-Skoda theorem. It has since come into its own as a fairly robust theory. The tight closure of an ideal is named as such because it is, in general, contained in, but not equal to, the integral closure of the same ideal, so it is a ``tighter'' closure operator than integral closure. Tight closure is notoriously difficult to compute for an arbitary ideal, but with certain rings, this task is less arduous. In this dissertation, we build a a bridge between tight closure theory and combinatorics by way of simplicial complexes and Stanley-Reisner rings. We discuss the specifics of tight closure theory and Stanley-Reisner rings and make special effort to focus on the standard results of both topics that will be most useful to our purposes. We discuss the analogous notions for -reductions and reductions of ideals for tight and integral closure repectively. When we focus our attention on the maximal ideal, , of the Stanley-Reisner ring that is generated by the variables of the ring, we observe that if is a reduction of , then it is also a -reduction of . We will determine the the minimal number of generaters of a -reduction of , called the -spread of , and the intersection of all minimally generated -reductions of , called the *\core of . These notions were introduced by Epstein \cite{nme*spread} and Fouli and Vassilev \cite{FoVa-core} respectively. We endeavor to describe both in terms of the Stanley-Reisner ring and the simplicial complex of the Stanley-Reisner ring. Finally, we examine *\core{\mathfrak{m}} in specific examples and in slightly more general cases of Stanley-Reisner rings. These include dimension 1 Stanley-Reisner rings, Stanley-Reisner rings with disconnected simplicial complexes, and Stanley-Reisner rings with a graph for a simplicial complex
Lettre annonçant la découverte par M. Reisner du temple funéraire de la pyramide de Mykérinos à Memphis
No full textRicci Seymour de. Lettre annonçant la découverte par M. Reisner du temple funéraire de la pyramide de Mykérinos à Memphis. In: Comptes rendus des séances de l'Académie des Inscriptions et Belles-Lettres, 52ᵉ année, N. 10, 1908. pp. 806-808
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