Monday March 21 and Wednesday March 23

On Monday, we will go over many definitions in detail.

On Wednesday, we will continue with exercises related to those definitions.

(Monday) Study the definitions of 5.3.6 and 5.3.7 carefully.  Some courses spend the entire semester on varieties (over a field k, an algebraically closed field, or the field of complex numbers).  In the classical setup, the only points that are considered are the closed points.  (See Exercise 5.3.E.)

We will continue our discussion of affine, quasi-projective, and projective A-schemes on Wednesday.

  • Show that a projective A-scheme X is indeed an A-scheme.  In particular, the sheaf restriction morphisms are all A-algebra homomorphisms.
  • Show that a quasi-projective k-scheme has finite type over k.   (The starting point for this exercise is the definition of projective k-scheme as Proj S_*, for some finitely generated S_* over k=S_0.)
  • (Monday) 5.3.D.  Work this example in detail.  Why are all f.gen. k-algebras of the form k[x1,…,xn]/I?
  • (Wednesday) 5.3.D.b.
  • 5.3.E. (Use 3.6.9 – Hilbert’s Nullstellensatz.)
  • 5.3.J.

(Monday) Read pages 160-161 including the optional exercise 5.3.F and the proof of 5.3.3.

(Monday) learn the definitions of normal schemes and factorial schemes.

  • 5.4.A.
  • 5.4.B.
  • 5.4.C.
  • 5.4.E.
  • 5.4.F.