


到達目標 Target to be Reached

The goal of this course is to gain a very beginning knowledge and skill of Scheme Theory in Algebraic Geometry.
A successful student, even whose research area is not in Algebraic Geometry, will be able to apply the technique obtained from this course to her/his research.


授業内容 Course Content

Algebraic Geometry has a long history, which goes back to the 17th century more or less. In middle of the last century, Algebraic Geometry started to change in appearance drastically. In this trend, the notion ``Scheme" was introduced in this subject. In the course, I will teach the very beginning part of this language.


授業計画 Course Planning

The contents of this lecture are included in that of Hartshorne's book [Chapter II Sections 14].
1. Presheaf and Sheaf on a topological space
2. Morphism of (pre)sheaves
3. Sheafification of a presheaf
4. Behavior of a sheaf under a continuous map of topological space
5. Spectrum of a commutative ring, and its topology
6. Ringed space, and Morphism of ringed spaces
7. Affine scheme, and examples
8. Proj of a graded ring
9. Comparison of schemes with classical geometry
10. First geometric properties of schemes
11. Morphism of schemes
12. Examples  to illustrate previous two lectures
13. Separated morphism
14. Valuation criterion of separated morphisms
15. Proper morphism, and its characterization by valuation rings


授業運営 Course Management

Follow the standard way of advanced math lectures.


評価方法 Evaluation Method

By an oral exam.


オフィスアワー Office Hour (s)

Every Wednesday, 9:0010:00



参考書 Book (s) for Reference

R. Hartshorne,Algebraic Geometry,Springer


