It has been estimated that, at the present stage of our knowledge,
one could give a 200 semester course on commutative algebra and
algebraic geometry without ever repeating himself. So any
introduction to this subject must be highly selective.
--E. Kunz, Introduction to commutative algebra and algebraic geometry,
1978.
Overview
This semester, Math 673 will be an introduction to schemes, with an
emphasis on arithmetic applications. Schemes provide a unified
framework for extensions of classical algebraic geometry, such as:
- Working over fields like Q or Fp,
instead of insisting on an algebraically closed field;
- Tangent spaces, deformations, and other infinitesimal calculations;
- Reducing an abstract variety modulo p;
- Relating generic behavior to special behavior;
- Exploiting the parallels between number fields and function fields
of curves;
- etc., etc., etc.
One of the main motivations for constructing the theory of schemes was
to prove Weil's conjectures on the number of solutions to systems of
equations over finite fields. The course will conclude with an
overview of Deligne's proof of these conjectures and some of their
applications.
Textbook