Théorie des Intersections et Théorème de Riemann-Roch

Jussila, O.; Grothendieck, Alexander; Raynaud, Michel; Kleiman, Steven L.; Illusie, Luc; Berthelot, Pièrre

doi:10.1007/bfb0066283

Cited by 327 publications

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“…Theorems 1-3 have well-known analogs in algebraic K-theory proved by Thomason in [30] based on the work of Waldhausen [31] and Grothendieck et al [4]. In fact, our Theorems 10, 14, 11, 15 and 16-special cases of which are Theorems 2, 3 and 1-are generalizations of the corresponding theorems in Thomason's work.…”

Section: Theorem 3 (Zariski Excision) Let J : U ⊂ X Be Quasi-compact mentioning

confidence: 58%

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

Schlichting

2009

Invent. math.

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We prove localization and Zariski-Mayer-Vietoris for higher Grothendieck-Witt groups, alias hermitian K-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove Additivity, Fibration and Approximation theorems for the hermitian K-theory of exact categories with weak equivalences and duality.

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Section: Theorem 3 (Zariski Excision) Let J : U ⊂ X Be Quasi-compact mentioning

confidence: 58%

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

Schlichting

2009

Invent. math.

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“…The "Leray-Hirsch" in the context of K-theory of complex vector bundles that we need is Theorem 2.7.8, [5]. For part (iv) we use a result of Grothendieck [7] to prove the analogue of Leray-Hirsch theorem. We do not know if parts (iii) and (iv) of the main theorem remain valid without the hypothesis that ( * ) hold.…”

Section: Introductionmentioning

confidence: 99%

Cohomology of toric bundles

Sankaran¹,

Uma²

2003

Comment. Math. Helv.

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Abstract. Let p : E−→B be a principal bundle with fibre and structure group the torus T ∼ = (C * ) n over a topological space B. Let X be a nonsingular projective T -toric variety. One has the X-bundle π :. This is a Zariski locally trivial fibre bundle in case p : E−→B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H * (B; Z)-algebra, (ii) the topological K-ring of K * (E(X)) as a K * (B)-algebra when B is compact. When p : E−→B is algebraic over an irreducible, nonsingular, noetherian scheme over C, we describe (iii) the Chow ring of A * (E(X)) as an A * (B)-algebra, and (iv) the Grothendieck ring K 0 (E(X)) of algebraic vector bundles on E(X) as a K 0 (B)-algebra.

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“…If we choose U such that U ∩ supp(T ) ∩ supp(T ) = ∅, the equations Hom (1) (T, T ) = Ext 1 (1) (T, T ) = 0 follow immediately. For Hom (1) …”

Section: Proof If We Associate To a Weil Divisormentioning

confidence: 99%

“…(1) (X) and (1) (X) determines X as an object in Var (1) . At the end of this section we give a short discussion of the case c > 1 and prove the non-existence of a Serre functor on the quotient category if dim(X) ≥ 2.…”

Section: Birational Geometry and D Bmentioning

confidence: 99%

Quotient categories, stability conditions, and birational geometry

Meinhardt

Partsch

2014

Geom Dedicata

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Abstract. This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, we will describe the space of stability conditions on its derived category in the case c=1. Moreover, we describe all exact equivalences between these quotient categories in this particular case which is closely related to classification problems in birational geometry.

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Théorie des Intersections et Théorème de Riemann-Roch

Cited by 327 publications

References 0 publications

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

Cohomology of toric bundles

Quotient categories, stability conditions, and birational geometry

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