Schemes

Professor A.J. de Jong, Columbia university, Department of Mathematics.

In the Spring of 2020 I am will teach the course on schemes. Stay tuned!

Tuesday and Thursday, 10:10 -- 11:25 AM in room 507 math.

The TA is Carl Lian.

Grades will be based on the weekly problem sets and a final exam.

Lectures

  1. First lecture was about prerequisites, the spectrum of a ring as a locally ringed space, affine schemes, schemes. We gave two examples of non-affine schemes. We discussed the structure sheaf on affine 2 space A^2_k over a field k.
  2. Second lecture discussed the universal property of an affine scheme in the category of locally ringed spaces. We proved it. We discussed an easy case of glueing of topological spaces / ringed spaces / locally ringed spaces, namely the case where we are just glueing two pieces. We discussed the affine line with 0 doubled and the affine plane with 0 doubled.
  3. We discussed Proj of a graded ring and the sheaf of modules associated to a graded module. We defined projective space over a ring and we defined projective schemes over a ring.
  4. We discussed quasi-coherent modules on affine schemes and on general schemes. The category of quasi-coherent modules on Spec(A) is equivalent to the category of A-modules via the global sections functor. For a ring map A → B the pushforward and pullback functors on quasi-coherent modules correspond to the (adjoint) restriction and tensor product functors on modules. For a morphism of schemes pullback preserves quasi-coherent modules but in general pushforward does not. However, at the very end of the lecture we proved that pushforward along a quasi-compact and quasi-separated morphism of schemes does preserve quasi-coherency.
  5. We discussed fibre products of schemes, affine morphisms as well as closed immersions, finite morphisms, and integral morphisms.
  6. We discussed separation axioms for schemes. We discussed properties of morphisms of schemes which are "local on the base", "preserved by base change", or satisfy some "permanence" property. We defined universally closed morphisms. We proved that projective space P^n_S is universally closed over its base S. We defined a proper morphism as one which is of finite type, separated, and universally closed. Thus P^n_S is proper over S. We finished with a proof of the fact that a morphism A^1_S → P^n_S over S is never a closed immersion.
  7. We discussed invertible modules, the Picard group, the open defined by a section of an invertible module, the graded ring associated to an invertible module, the graded module associated to a sheaf of modules, the comparison map being an isomorphism for quasi-coherent modules, and the fact that quasi-coherent modules on Proj(A) are always associated to the graded module you get from them. References:
    • Invertible modules: 0AFW 01CR
    • Graded ring and graded module 01CV
    • Picard group 01CX
    • Open X_s associated to a section of invertible module 01CY
    • Sections of quasi-coherent modules over X_s 01PW
    • The isomorphism of a quasi-coherent module F over X = Proj(A) with the module associated to the graded module associated to F 0AG5
  8. We discussed ample invertible modules.
    • We defined them as in the Stacks project, see 01PR
    • We defined finite type modules on ringed spaces, see 01B4
    • We stated a proposition that an invertible module L is ample if and only if for every finite type quasi-coherent mdoule F there is an n > 0 such that F ⊗ L^n is globally generated, see 01Q3
    • We sketched ingredients in the proof of the proposition
    • We defined Noetherian schemes, see 01OU
    • We defined coherent modules on Noetherian schemes, see 01XY 01Y7
    • We formulated a goal wrt cohomology of coherent modules on projective schemes over Noetherian rings.
  9. We will discuss quasi-projectivity and the relationship of (quasi-)projectivity with the existence of ample invertible modules. Then we will start talking about cohomology of sheaves of modules.

Homework: (make sure you refresh the page)

  1. Due Thursday, January 30 in lecture: do 8 exercises from Section Tag 0280
  2. Due Thursday, February 6 in lecture: do 4 of the exercises from Section Tag 0280 and try to do Exercise Tag 02A1
  3. Due Thursday, February 13 in lecture: 029Q 069T 028Z and do 2 exercises from Section 0293
  4. Due Thursday, February 20 in lecture: 02A3 02AL 0DT4 029U 0D8V

Books to read/browse. Some of these are more advanced.

  1. Algebraic Geometry by Hartshorne
  2. EGA by Grothendieck
  3. the Stacks project online
  4. Basic Algebraic Geometry by Shavarevich
  5. Algebraic Geometry by Goertz and Wedhorn
  6. Algebraic Geometry and Arithmetic Curves by Qing Liu
  7. Red Book by Mumford
  8. Curves and their Jacobians by Mumford
  9. Abelian Varieties by Mumford
  10. Lectures on Curves on an Algebraic Surface by Mumford
  11. ACGH: Geometry of Algebraic Curves by Arbarello, Cornalba, Griffiths, and Harris
  12. Algebraic Curves by Fulton
  13. Algebraic Varieties by Kempf
  14. Undergraduate algebraic geometry by Reid
  15. Algebraic Geometry by Harris
  16. Principles of Algebraic Geometry by Griffiths and Harris
  17. Introduction to Algebraic Geometry by Cutkosky
  18. The Arithmetic of Elliptic Curves by Silverman
  19. Rational curves on algebraic varieties by Kollar
  20. Neron Models by Bosch, Lutkebohmert, and Raynaud