Università Roma Tre

Sheaf theory and its use in on schemes

Preseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks
and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products.
Algebraic sheaves on a scheme. Quasi-coherent sheaves and coherent sheaves.

Cohomology of sheaves

Homological algebra in the category of modules over a ring. Flasque sheaves.
The cohomology of the sheaves using canonical resolution with flasque sheaves.

Cohomology of quasi-coherent and coherent sheaves on a scheme.

Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.

Invertible sheaves and linear systems

Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems. Base points. Linear systems, ample
and very ample sheaves. Amplitude criterion.

Notes from Prof. Sernesi
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).