Download PDF by Kenji Ueno: Algebraic geometry 2. Sheaves and cohomology

By Kenji Ueno

ISBN-10: 0821813579

ISBN-13: 9780821813577

Glossy algebraic geometry is equipped upon basic notions: schemes and sheaves. the idea of schemes was once defined in Algebraic Geometry 1: From Algebraic types to Schemes, (see quantity 185 within the related sequence, Translations of Mathematical Monographs). within the current ebook, Ueno turns to the idea of sheaves and their cohomology. Loosely talking, a sheaf is a fashion of maintaining a tally of neighborhood details outlined on a topological area, comparable to the neighborhood holomorphic capabilities on a posh manifold or the neighborhood sections of a vector package deal. to check schemes, it really is worthy to review the sheaves outlined on them, particularly the coherent and quasicoherent sheaves. the first software in figuring out sheaves is cohomology. for instance, in learning ampleness, it really is usually worthy to translate a estate of sheaves right into a assertion approximately its cohomology.

The textual content covers the $64000 themes of sheaf conception, together with forms of sheaves and the basic operations on them, resembling ...

coherent and quasicoherent sheaves. right and projective morphisms. direct and inverse photographs. Cech cohomology.

For the mathematician surprising with the language of schemes and sheaves, algebraic geometry can appear far-off. in spite of the fact that, Ueno makes the subject appear common via his concise variety and his insightful reasons. He explains why issues are performed this fashion and supplementations his factors with illuminating examples. accordingly, he's capable of make algebraic geometry very available to a large viewers of non-specialists.

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3 Q-Divisors and R-Divisors For questions of positivity, it is very useful to be able to discuss small perturbations of a given divisor class. The natural way to do so is through the formalism of Q- and R- divisors, which we develop in this section. As an application, we establish that amplitude is an open condition on numerical equivalence classes. 1. (Q-divisors). Let X be an algebraic variety or scheme. A Q-divisor on X is an element of the Q-vector space DivQ (X) =def Div(X) ⊗Z Q. 13) where ci ∈ Q and Ai ∈ Div(X).

9. Let B be a globally generated line bundle on a complete variety or scheme X. 2 The Classical Theory U =def 27 y ∈ X | B ⊗ my is globally generated is an open subset of X. (Since B is globally generated, it suffices by Nakayama’s Lemma to prove the openness of the set V =def y ∈ X | H 0 X, B −→ B ⊗ OX /m2y is surjective }. But this follows from the existence of a coherent sheaf P on X, whose fibre at y is canonically P(y) = B ⊗ OX /m2y , together with a map u : H 0 X, B ⊗ OX −→ P which fibre by fibre is given by evaluation of sections.

H 1,1 X; C . Finally we say a word about functoriality. Let f : Y −→ X be a morphism of complete varieties or projective schemes. If α ∈ Pic(X) is a class mapping to zero in N 1 (X), then it follows from the projection formula that f ∗ (α) is numerically trivial on Y . 5 a functorial induced homomorphism f ∗ : N 1 (X) −→ N 1 (Y ). 22. (Non-projective schemes). A, the integrality and projectivity hypotheses in the previous paragraph arise only in order to use the functorial properties of line bundles to discuss divisors.

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Algebraic geometry 2. Sheaves and cohomology by Kenji Ueno

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