Lectures (from 16/10/20):
Friday, 14:00-16:00, MS Teams
This course is a continuation of the TCC course Introduction to Algebraic Geometry from Spring 2020.
Valuation theory: ordered groups and their rank, the embedding theorems of Hölder and Hahn. The field of Hahn series. Hilbert polynomials, the Hilbert scheme. Artin-Rees lemma, Krull's theorem, injecive test lemma. The dimension theorem, regular local rings. Abelian categories, injective and projective resolutions. Derived functors. Delta-functors, universality, cohomology and higher direct images of sheaves. Flabby sheaves, direct systems and limits. Noetherian topological spaces, irreducible components. Grothendieck's theorem on cohomological dimension. Cohomology of quasicoherent sheaves on affine schemes. Čech cohomology of quasicoherent sheaves on separated Noetherian schemes. Higher direct images of quasicoherent sheaves are quasicoherent, under mild assumptions.
Lecture 8: unfortunately since my computer crashed, the slides of this lecture are lost. Here are the notes I used to prepare for this class instead:
Lectures (from 10/01/20):
Monday, 14:00-16:00, TCC room
Y. Manin: Introduction to the Theory of Schemes, Springer-Verlag, 2018.
R. Hartshorne: Algebraic Geometry, Springer-Verlag, 1977.
J. Bochnak, M. Coste and M.-F. Roy: Real Algebraic Geometry, Springer-Verlag, 1998.
Rings and their spectra. Classical examples: \(C^*\)-algebras and Boolean algebras. Zariski topology, radicals, irreducible sets and generic points. Noetherianness and connectivity. Localisation of rings and modules, the structure sheaf, quasi-coherent sheaves. Orderings, prime cones, the real spectrum. Schemes and morphisms of schemes. Products of schemes, separated and proper morphisms, valuative criteria. Affine and projective morphisms. Kähler differentials. Tools from model theory: ultraproducts, Łoś lemma, applications to algebraic geometry. Basic notions of set theory, back and forth. Quantifier elimination for algebraically closed and real closed fields.