By Kenji Ueno

ISBN-10: 0821813579

ISBN-13: 9780821813577

Smooth algebraic geometry is equipped upon primary notions: schemes and sheaves. the speculation of schemes was once defined in Algebraic Geometry 1: From Algebraic forms to Schemes, (see quantity 185 within the comparable sequence, Translations of Mathematical Monographs). within the current booklet, Ueno turns to the speculation of sheaves and their cohomology. Loosely conversing, a sheaf is a manner of maintaining a tally of neighborhood details outlined on a topological area, corresponding to the neighborhood holomorphic capabilities on a fancy manifold or the neighborhood sections of a vector package deal. to check schemes, it really is invaluable to check the sheaves outlined on them, specifically the coherent and quasicoherent sheaves. the first instrument in knowing sheaves is cohomology. for instance, in learning ampleness, it's usually worthy to translate a estate of sheaves right into a assertion approximately its cohomology.

The textual content covers the $64000 issues of sheaf idea, together with different types of sheaves and the basic operations on them, reminiscent of ...

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

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

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**Extra info for Algebraic geometry 2. Sheaves and cohomology**

**Example text**

1. , for A a noetherian subring of R such that A ⊗ Q is a ﬁeld, an object H of MHS is deﬁned as a triple H = (HA , W, F ) where HA is a ﬁnitely generated A-module, W is a ﬁnite increasing ﬁltration on HA ⊗ Q and F is a ﬁnite decreasing ﬁltration on HA ⊗ C such that W, F and F is a system of opposed ﬁltrations. 1. An ∞-mixed Hodge structure H is a triple (HA , W, F ) where HA is any A-module, W is a ﬁnite increasing ﬁltration on HA ⊗Q and F is a ﬁnite decreasing ﬁltration on HA ⊗ C such that W, F and F is a system of opposed ﬁltrations.

Exotic (1, 1)-classes Consider X singular. 4 for p = 1. Moreover we show that there are edge maps generalizing the cycle class maps constructed in the previous section. For X a proper irreducible C-scheme, consider the mixed Hodge structure on H 2+i (X, Z) modulo torsion. The extension (5) is the following 0 → H 1+i ((H 1 )• ) → W2 H 2+i (X)/W0 → H i ((H 2 )• ) → 0. (15) Since the complex (H 1 )• is made of level 1 mixed Hodge structures then H 2+i (X)h = H 2+i (X)e in our notation. If X is nonsingular then H 2+i (X) is pure and there are only two cases where this extension is non-trivial.

Let X be a proper smooth C-scheme. Note that we have N 1 H j (X) = ker(H j (X) → H 0 (X, H j )) = { Zariski locally trivial classes in H j (X)}. Thus H j (X, Q) ∩ F 1 H j (X) = gr 0N H j (X) ∩ F 1 ⊆ H 0 (X, H j ) ∩ F 1 . N 1 H j (X) We remark that H j /F 1 is the constant sheaf associated to H j (X, OX ). Thus F 1 ∩ H 0 (X, H j ) ∼ = ker(H 0 (X, H j ) → H j (X, OX )). If j = 1 then H 1 (X) = H 0 (X, H 1 ) from (13) and (8) is trivially an equality. If j = 2 then F 1 ∩ H 0 (X, H 2 ) = 0 from the exponential sequence.

### Algebraic geometry 2. Sheaves and cohomology by Kenji Ueno

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