Sheaves on finite posets and modules over normal semigroup rings

Yanagawa, Kohji

doi:10.1016/s0022-4049(00)00095-5

Cited by 26 publications

(50 citation statements)

References 18 publications

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“…In particular, Corollary 6.3, which recaptures a result of [1], gives a formula on the Hilbert function of the local cohomology module H i m (R) at the maximal ideal m := (t a | 0 = a ∈ |M|). In [14], [16], the second author defined squarefree modules M over a normal semigroup ring k[M σ ], and gave corresponding constructible sheaves M + on the closed ball σ. We can extend this to a toric face ring R, that is, we define squarefree R-modules and associate constructible sheaves on X with them.…”

Section: Clearly Our I •mentioning

confidence: 99%

“…We denote, by Sq R, the full subcategory of mod ZM R consisting of squarefree R-modules. As in the case of affine semigroup rings or Stanley-Reisner rings (see [14], [15]), Sq R has nice properties. Since their proofs are also quite similar to these cases, we omit some of them.…”

mentioning

confidence: 99%

“…The category mod Λ is abelian and enough injectives, and any indecomposable injective object is isomorphic toĒ(σ) for some σ ∈ X . [14], [15]) There is an equivalence between Sq R and mod Λ. Hence Sq R is abelian, and enough injectives.…”

mentioning

confidence: 99%

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Dualizing Complex of a Toric Face Ring

Okazaki¹,

Yanagawa²

2009

Nagoya Mathematical Journal

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Abstract. A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Römer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a toric face ring R in a very concise way. Since R is not a graded ring in general, the proof is not straightforward. We also develop the squarefree module theory over R, and show that the Cohen-Macaulay, Buchsbaum, and Gorenstein* properties of R are topological properties of its associated cell complex.

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Section: Clearly Our I •mentioning

confidence: 99%

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confidence: 99%

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Dualizing Complex of a Toric Face Ring

Okazaki¹,

Yanagawa²

2009

Nagoya Mathematical Journal

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“…This multigraded framework slightly differs from the original situation for regular local rings stated in [13]. However, as stated in [27,Remark 5.14], if ∆ ∼ = ∆ ′ as simplicial complexes, then R/I ∆ and R ′ /I ∆ ′ have the same Lyubeznik numbers. In this sense, to study the Lyubeznik numbers of a quotient R/I ∆ of a Gorenstein normal simplicial semigroup ring R by a radical monomial ideal I ∆ , we may assume that R is a polynomial ring and R/I ∆ is a Stanley-Reisner ring.…”

Section: Definition 51 ([27]mentioning

confidence: 86%

“…As shown in [27], Sq R is an abelian category with enough injectives. For an indecomposable squarefree module M, it is injective in Sq R if and only if M ∼ = R/p F for some F ∈ L.…”

Section: Definition 51 ([27]mentioning

confidence: 98%

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals

Montaner¹,

Yanagawa²

2017

Nagoya Math. J.

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Abstract. In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a Z-graded ideal I ⊆ R = k[x 1 , . . . , x n ]. We also prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for the Lyubeznik table. For the case of squarefree monomial ideals we get more insight on the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of StanleyReisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field.

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Graded rings and equivariant sheaves on toric varieties

Perling

2003

Mathematische Nachrichten

172

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In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are equivalent to certain sets of filtrations of vector spaces. We systematically construct the theory from the point of view of graded ring theory and this way we considerably clarify earlier constructions of Kaneyama and Klyachko. We also connect the formalism to the theory of fine-graded modules over Cox' homogeneous coordinate ring of a toric variety. As an application we construct minimal resolutions of equivariant vector bundles of rank two on toric surfaces.

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Sheaves on finite posets and modules over normal semigroup rings

Cited by 26 publications

References 18 publications

Dualizing Complex of a Toric Face Ring

Dualizing Complex of a Toric Face Ring

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals

Graded rings and equivariant sheaves on toric varieties

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