The geometry of toric schemes

Rohrer, Fred

doi:10.1016/j.jpaa.2012.08.001

Cited by 5 publications

(9 citation statements)

References 18 publications

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“…For example, the R-scheme X Σ (R) → Spec(R) is proper if and only if Σ is complete, or Σ = ∅, or R = 0. (More results of this type can be found in [12] and [13]. )…”

Section: Toric Schemesmentioning

confidence: 85%

“…In 3.1 we define and study the Cox ring associated with Σ. Beware that our notion of Cox ring differs from and must not be confused with the one of the same name introduced in [16] (although both notions are inspired by [7]). In 3.2 and 3.3, Cox schemes and toric schemes are defined as special cases of a far more general construction of schemes from certain projective systems of monoids (treated in detail in [25]). Then, the Cox scheme associated with Σ is shown to be isomorphic to the toric scheme associated with Σ if and only if Σ is not contained in a hyperplane of the ambient space.…”

Section: Cox Rings Cox Schemes and Toric Schemesmentioning

confidence: 99%

“…14 cf. [25] for an extensive study of geometric properties of these functors commutes. So, for σ ∈ Σ we get a closed immersion γ σ := Spec(…”

Section: Cox Ringsmentioning

confidence: 99%

“…We start by briefly describing the construction of toric schemes from fans. For unexplained terminology and notation from polyhedral geometry we refer the reader to [12], [11, Chapter II] (available from the author's homepage), or [13].…”

Section: Toric Schemesmentioning

confidence: 99%

“…[17], [20], [6], [19], [15]). The author's article [25] contains a systematic study of basic properties of toric schemes over arbitrary bases. Now, let X = X Σ (R) be the toric scheme over R associated with a rational fan Σ; for simplicity, we suppose that Σ is not contained in a hyperplane of its ambient space.…”

Section: Introductionmentioning

confidence: 99%

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Quasicoherent sheaves on toric schemes

Rohrer

2014

Expositiones Mathematicae

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Let X be the toric scheme over a ring R associated with a fan Sigma. It is shown that there are a group B, a B-graded R-algebra S and a graded ideal I of S such that there is an essentially surjective, exact functor ~ from the category of B-graded S-modules to the category of quasicoherent O_X-modules that vanishes on I-torsion modules and that induces for every B-graded S-module F a surjection Xi_F from the set of I-saturated graded sub-S-modules of F onto the set of quasicoherent sub-O_X-modules of ~F. If Sigma is simplicial, the above data can be chosen such that ~ vanishes precisely on I-torsion modules and that Xi_F is bijective for every F. In case R is noetherian, a toric version of the Serre-Grothendieck correspondence is proven, relating sheaf cohomology on X with B-graded local cohomology with support in I.Comment: To appear in Expositiones Mathematicae; corrected typo

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“…For example, the R-scheme X Σ (R) → Spec(R) is proper if and only if Σ is complete, or Σ = ∅, or R = 0. (More results of this type can be found in [12] and [13]. )…”

Section: Toric Schemesmentioning

confidence: 85%

Section: Cox Rings Cox Schemes and Toric Schemesmentioning

confidence: 99%

“…14 cf. [25] for an extensive study of geometric properties of these functors commutes. So, for σ ∈ Σ we get a closed immersion γ σ := Spec(…”

Section: Cox Ringsmentioning

confidence: 99%

Section: Toric Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasicoherent sheaves on toric schemes

Rohrer

2014

Expositiones Mathematicae

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The KH-theory of complete simplicial toric varieties and the algebraic K-theory of weighted projective spaces

Massey

2013

Journal of Pure and Applied Algebra

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We show that, for a complete simplicial toric variety X, we can determine its homotopy KH-theory entirely in terms of the torus pieces of open sets forming an open cover of X. We then construct conditions under which, given two complete simplicial toric varieties, the two spectra KH(X)⊗Q and KH(Y )⊗Q are weakly equivalent. We apply this result to determine the rational KH-theory of weighted projective spaces. We next examine K-regularity for complete toric surfaces; in particular, we show that complete toric surfaces are K0-regular. We then determine conditions under which our approach for dimension 2 works in arbitrary dimensions, before demonstrating that weighted projective spaces are not K1-regular, and for dimensions bigger than 2 are also not in general K0-regular.Date: November 10, 2018.

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