Sleek algebraic geometry is equipped upon primary 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 comparable sequence, Translations of Mathematical Monographs). within the current ebook, Ueno turns to the speculation of sheaves and their cohomology. Loosely talking, a sheaf is a fashion of maintaining a tally of neighborhood info outlined on a topological house, corresponding to the neighborhood holomorphic services on a fancy manifold or the neighborhood sections of a vector package. to review schemes, it truly is priceless to check the sheaves outlined on them, specially the coherent and quasicoherent sheaves. the first software in realizing sheaves is cohomology. for instance, in learning ampleness, it really is often necessary to translate a estate of sheaves right into a assertion approximately its cohomology.

The textual content covers the $64000 themes of sheaf thought, together with sorts of sheaves and the basic operations on them, corresponding to ...

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

For the mathematician unusual with the language of schemes and sheaves, algebraic geometry can look far away. in spite of the fact that, Ueno makes the subject look common via his concise kind and his insightful factors. He explains why issues are performed this fashion and vitamins his reasons with illuminating examples. consequently, he's in a position to make algebraic geometry very available to a large viewers of non-specialists.

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