Friday March 18, 2016

We will start with reducedness and integrality

  • 5.2.A. reducedness can be checked on stalks.
  • 5.2.B A reduced implies Spec A is reduced.   P^n_k is reduced
  • 5.2.E X quasicompact and f vanishes at every point, then f^n = 0, some n.
  • 5.2.F.  X is integral iff it is irreducible and reduced.
  • 5.2.G.  An affine scheme is integral iff the ring A is an integral domain.
  • 5.2.H.  An integral scheme has a function field.

Read Prop 5.3.1 and Prop 5.3.2 (affine communication lemma).  Read 5.3.6 and 5.3.7.  We finally reach the important definitions of affine varieties and quasi-projective varieties over a field k.

  • 5.3.D.
  • (If time permits: 5.3.F. analytification of complex varieties.)

 

Wednesday March 16, 2016

We will start on Chapter 5 with topological properties of schemes.

  • 5.1.A Show that P^n_k is irreducible.
  • 5.1.B Give a bijection between irreducible closed subsets and points.
  • 5.1.C. If X has a finite cover by Spec(Noetherian rings), then it is a Noetherian topological space.
  • 5.1.D A scheme is quasicompact iff it is a finite union of affine open subschemes.
  • 5.1.E Quasicompact schemes have closed points.
  • 5.1.G Affine schemes are quasiseparated.
  • 5.1.H Projective A-schemes are quasicompact and quasiseparated.

As time permits, we will work some problems on reducedness and integrality

  • 5.2.A. reducedness can be checked on stalks.
  • 5.2.B A reduced implies Spec A is reduced.   P^n_k is reduced
  • 5.2.E X quasicompact and f vanishes at every point, then f^n = 0, some n.
  • 5.2.F.  X is integral iff it is irreducible and reduced.

Friday March 4, 2016

The main point of this lecture will be to understand projective space (and projective schemes) from the point of view of homogeneous ideals.  Review the theory of graded rings and homogeneous ideals from Algebra 2, if it is not fresh in your mind.  In particular, review the definition of a Z-graded ring, homogeneous ideal.

If S_* is  a graded ring, then we will construct a scheme Proj(S_*) — first as a set, then as a topological space, and finally as a scheme.  Read 4.5.7 very carefully.  Pages 149 and 150 are also very important.

  • Check that the Proj construction applied to the graded ring k[x0,…,xn] and with coordinate charts indexed by f = xi, gives the construction of P^n discussed earlier.
  • Let g be a homogeneous polynomial in k[x0,…,xn] (k algebraically closed).  Apply the Proj construction to k[x0,…,xn]/(g), with coordinate charts indexed by f = xi.  Show that the set of closed points of the resulting scheme can be identified with the set of nonzero solutions [a0:…:an] of g(a0,…,an) = 0.
  • Exercise 4.5.E.
  • How are the definitions of V(T) and D(f) modified in the projective setting?
  • Check that the sets V(I) satisfy the axioms of a closed set of a topological space.
  • 4.5.G.  Show that the sets D(f) form a base of the Zariski topology.
  • Mimic proofs in the affine case, to do Exercises 4.5.H and 4.5.I.
  • We will continue to do exercises on page 150 until time runs out.