授業の目的 【日本語】 Goals of the Course(JPN)   

授業の目的 【英語】 Goals of the Course   The course will have three main parts: Part 1: An introduction to schemes and to geometric properties of morphisms of schemes. Part 2: An introduction to (coherent) cohomology of schemes. Part 3: Applications of cohomology, including the RiemannRoch theorem for curves. 


到達目標 【日本語】 Objectives of the Course(JPN))   

到達目標 【英語】 Objectives of the Course   

授業の内容や構成 Course Content / Plan   The following is a preliminary outline of the lectures:
1) The Zariski space of a ring
2) Sheaves on a topological spaces
3) The prime spectrum of a ring
4) Schemes
5) Maps of schemes
6) Quasicoherent sheaves
7) Vector bundles, affine maps, and closed immersions
8) Line bundles and projective space
9) More on line bundles
10) More on projective space
11) Fiber products and proper maps
12) Cohomology
13) Finiteness of coherent cohomology
14) Curves over a field
15) Smooth proper curves over a field; RiemannRoch
16) Applications of RiemannRoch 


履修条件 Course Prerequisites   Some mathematical maturity is necessary. It is also good to have some familiarity with commutative algebra and category theory, but it is possible to learn this material as the course progresses. 


関連する科目 Related Courses   Algebra geometry, algebraic topology, representation theory. 


成績評価の方法と基準 Course Evaluation Method and Criteria   Grades are assigned based on solutions to weekly problem sets. 


不可(F)と欠席(W)の基準 Criteria for "Fail (F)" & "Absent (W)" grades   A sufficient score on weekly problem sets is necessary to pass the course. 


参考書 Reference Book   David Mumford. The Red Book of Varieties and Schemes. Second, Expanded Edition. Lecture Notes in Mathematics, Vol. 1358. SpringerVerlag, 1974. 


教科書・テキスト Textbook   Lecture notes will be uploaded weekly to the course homepage on NUCT. 


課外学習等（授業時間外学習の指示） Study Load(Selfdirected Learning Outside Course Hours)   Expect to spend at least three hours per week to read the course notes and complete the problem set. 


注意事項 Notice for Students   The theory of schemes uses several basic concepts in mathematics, including sheaf theory and cohomology. These concepts are not easy to master, so you should not expect to be entirely comfortable with these concepts after completing this course. 


他学科聴講の可否 Propriety of Other department student's attendance   

他学科聴講の条件 Conditions for Other department student's attendance   

レベル Level   

キーワード Keyword   Scheme, sheaf, cohomology. 


履修の際のアドバイス Advice   The theory of schemes uses several basic concepts in mathematics, including sheaf theory and cohomology. These concepts are not easy to master, so you should not expect to be entirely comfortable with these concepts after completing this course. 


授業開講形態等 Lecture format, etc.   Lectures will be uploaded to Google Drive for ondemand viewing. (You do not need a Google account to be able to view the lectures.) Links for the lectures and lecture notes are posted on NUCT along with weekly problem sets. Solutions to problem sets should be uploaded to NUCT. LaTeX'd solutions are preferable, but scanned PDF is acceptable. 


遠隔授業（オンデマンド型）で行う場合の追加措置 Additional measures for remote class (ondemand class)   You should have access to NUCT. 

