Algebraic Geometry 2: Overview

In the first semester of algebraic geometry, we outlined the foundations of the modern theory, the theory of schemes, giving basic definitions and constructions. We showed how the category of schemes enlarges the (opposite) category of commutative rings, gave tools for the construction of schemes by gluing, and showed the existence of arbitrary fiber products. We also discussed ``global'' properties of morphisms: finite, finite type, separable and proper morphisms, open and closed immersions, projective morphisms, and the construction of projective morphisms using the functor Proj. The second semester will deal with local properties of morphisms and with cohomology of coherent sheaves.

Course Outline

As a list of topic headings, we will cover:

A. Local properties of morphisms

  • The module of relative differentials and smooth morphisms
  • Flatness and flat morphisms
  • Dimension, upper-semicontinuity, Chevalley's theorem

B. Cohomology of coherent and quasi-coherent sheaves

  • Some homological algebra, derived categories, derived functors, cohomology
  • The categories of quasi-coherent sheaves and coherent sheaves
  • Vanishing of cohomology for affine schemes
  • Leray covering theorem and Čech cohomology
  • Cohomology of tautological sheaves on projective space
  • Direct image sheaves, finiteness theorems
  • Base-change and upper semi-continuity
  • Ext and Tor
  • Serre duality

C. Applications

  • Riemann-Roch theorems for curves and surfaces
  • Some geometric applications of Riemann-Roch

We will be using Hartshorn [H] as our main text, supplemented by Griffiths-Harris [GH], and Serre [FAC]

[H] Hartshorne, Robin. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp. ISBN: 0-387-90244-9

[GH] Griffiths, Phillip; Harris, Joseph. Principles of algebraic geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley \& Sons, Inc., New York, 1994. xiv+813 pp. ISBN: 0-471-05059-8

[FAC] Serre, Jean-Pierre. Faisceaux algébriques cohérents. Ann. of Math. (2) 61, (1955). 197–278.