Wednesday, April 13, 2016

For the last three weeks of the course, we will shift our attention from the general theory of algebraic geometry to specific examples coming from elliptic curves and linear algebraic groups.

For elliptic curves, we will use Milne’s book on elliptic curves.

For the theory of linear algebraic groups, we will use Humphreys book.

We will start with Humphrey’s book, moving quickly through a review of algebraic geometry from the first section of the book.  Skim through the first 12 pages, which should all be familiar.

  • Let k be a field. Check that in the category of affine k-schemes, the (categorical) product of two affine lines (A^1) over k is the plane A^2.
  • Read the construction of page 13 (Prop. 1.7) that identifies the product of two projective spaces with a closed subset of a larger projective space.  We can use this to give a provisional definition of the product of projective spaces.
  • page 14. Learn the definition of the Grassmannian and flag variety.
  • Skim to section 2.5.
  • We will discuss the “Hausdorff axiom” in Section 2.5.

Mon 4/4, Wed 4/6, Fri 4/8

The meetings on Mon, Wed, and Friday will be guided by Shekiaya, Bill, and Elise.  Spend a few minutes at the end of each meeting to make a group decision about whether to move forward to new material next time or to spend longer on current material.  Do not feel you need to rush to cover a certain  amount of material.  It is better to understand as you go.

Monday will cover the algebraic interlude (Section 7.2: Lying Over and Nakayama).

After that, we will return to 7.1, then continue with 7.3, and so on.

Monday

Read the definitions of integral ring homomorphism and integral extension of rings.  Read Lemma 7.2.1.

  • Exercise 7.2.A.
  • Exercise 7.2.B.
  • Exercise 7.2.C.
  • Exercise 7.2.D.

Understand the statement of the Lying over theorem (7.2.5) and its geometric translation in 7.2.6.  The proof is optional.

  • Exercise 7.2.F (The going-up theorem is an important exercise).

Read the two versions of Nakayama’s lemma.  As time permits,  work versions 3 and 4 of Nakayama’s lemma in Exercises 7.2.G and 7.2.H.  As time permits, work 7.2.I and 72.J.

 

Wednesday

Understand Grothendieck’s lesson in 7.0.1 about replacing properties of objects with properties of morphisms.   Read 7.1.1. about the three properties to expect of reasonable morphisms.  Learn the definition of open embedding and open subscheme.  Make sure that you are clear about the distinction between “local on the target”, “local on the source”, and “affine-local on the source” (in Definition 7.1.2).

  • Exercise 7.1.A.  Open embeddings are satisfy the first two properties of being a reasonable morphism.
  • Exercise 7.1.B.  Open embeddings satisfy the third property as well.  We have not covered fibered products of schemes.  For this exercise, you need to construct the fiber product in a special case.  I suggest that you spend time working through this fibered product.
  • Exercise 7.1.C.  Open embeddings and locally Noetherian schemes.

Move forward to Section 7.3 on a gazillion finiteness conditions on morphisms.  This is a long section that will spill over into Friday.

Read 7.3.1.  What is the definition of a quasicompact morphism?  Can you state what a quasicompact scheme (in the old terminology) is in terms of a morphism?  With each new property of morphism ask yourself if the definition is (affine) local on the source or target.  I suggest skipping the parts about quasiseparated morphisms unless there is time to spare at the end of class.

  • Exercise 7.3.A.
  • Exercise 7.3.B.a.
  • Exercise 7.3.C.a

Read the definition of an affine morphism.  I suggest reading over the proofs on pages 209-210 at home, but not to spend time in class with them unless there are specific questions.

 

Friday

Section 7.3 is called a “gazillion finiteness conditions” on morphisms. Give yourself time to sort out all of these different conditions.  Read 7.3.8.  What is a finite morphism?  What is the difference between a finite B-algebra and a finitely generated B-algebra?

  • Exercise 7.3.G. Finiteness is affine-local on the target.
  • Work through the three examples in class of finite morphisms on page 211, especially example 3.  Try to understand the third example both intuitively and scheme-theoretically.
  • Exercise 7.3.H.  Finite morphisms to a point.
  • Exercise 7.3.I. Finite morphisms compose.
  • Exercise 7.3.J. Finite morphisms are projective.
  • Exercise 7.3.K. Finite morphisms have finite fibers.

Move forward with definitions about and exercises on integral morphisms, morphisms of finite type, and quasi-finite morphisms as time permits.

 

Friday April 1

We will continue with exercises about morphisms of projective schemes

 

  • Work through example 6.4.1.
  • Work through example 6.4.2.
  • Exercise 6.4.C.  Different maps of graded rings can give the same morphism of schemes.
  • Exercise 6.4.D.
  • 6.4.E
  • 6.4.F.
  • 6.4.G.

Read the definition of a rational map, a dominant rational map, birational rational map, and what it means for schemes to be birational to each other.

  • Exercise 6.5.B.

 

 

Wednesday March 30, 2016

Z-valued points

Section 6.3.7 introduces a new notion of point (the Z-valued points).  Vakil notes that the “terminology ‘Z-valued point’ is unfortunate” because a scheme already has a set of points and Z-valued points are not points on the scheme.  Remember: algebraic geometry is a world where functions are not functions and points are not points.

  • 6.3.L.a A morphism induces a map of Z-valued points.
  • 6.3.8. Check the statement that A-valued points of an affine scheme are given by solutions to equations f1(x1,…,xn) = … = fr(x1,…,xn) = 0 in A.
  • 6.3.M. Points that take values in a local ring.
  • 6.3.N. Morphisms into projective space.

Read to the end of 6.3.

Maps of projective schemes

  • Exercise 6.4.A going from morphisms of graded rings to morphisms of schemes.
  • Work through example 6.4.1.
  • Work through example 6.4.2.
  • Exercise 6.4.C.  Different maps of graded rings can give the same morphism of schemes.
  • Exercise 6.4.D.

 

Monday March 28, 2016

This meeting will continue with morphisms of schemes from Chapter 6.

Read carefully the proof of 6.3.2 (the key proposition: every morphism of locally ringed spaces between affine schemes is induced by a ring homomorphism).  Someone will be asked to present the proof in class.  We will break it into several steps. In particular,

  • The map of points is determined by its map on global sections.
  • The map of sections is determined by the map of global sections.

 

We will continue with various exercises:

  • 6.3.D.  The category of rings and the opposite category of affine schemes are equivalent.
  • 6.3.C.  A morphism of schemes is a morphism of ringed spaces that is locally a morphism of affine schemes.

Read section 6.3.4 and the definition of S-schemes.   We have already seen a different definition of A-schemes.  Show that this new definition is compatible with the old (exercise 6.3.G).

  • 6.3.E.  Practice with morphisms.
  • 6.3.F. Morphisms X -> Spec A.
  • 6.3.H.
  • 6.3.I.
  • 6.3.J.