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.
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.
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.
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.