Monday Feb 29, Wednesday Mar 2

Exercise 4.3.A.

Read Section 4.4.

  • 4.4.1. Show that every function on A^2 minus is point extends to a global section of A^2.

For Wednesday March 2.

  • 4.4.4. Read how to glue two schemes together along open sets.  Use it to create a line with a doubled origin (4.4.5).
  • Go over the construction of the projective line very carefully (4.4.6), including the proof that the projective line is not affine.

We will start discussing projective space.

  • 4.4.9 Go through the construction of projective space, including 4.4.D.
  • Exercise 4.4.E.
  • Fun aside. 4.4.11. Interpret the Chinese remainder theorem in terms of schemes.
  • Read Section 4.5.

Friday Feb 26

This lecture will cover 4.3 (Definition of schemes).  Once schemes are defined, there are many exercises about the definition….

  • Read the definition of a scheme and the definition of a locally ringed spaces (4.3.6).
  • Exercise 4.3.A.  Ring isomorphisms correspond bijectively with isos of ringed spaces.
  • Exercise 4.3.B.
  • Exercise 4.3.C.
  • Exercise 4.3.D.
  • Exercise 4.3.E.b.  An infinite disjoint union of affine schemes is not an affine scheme.
  • Exercise 4.3.F.

 

Wednesday Feb 24

We will introduce the structure sheaf on Spec A in terms of its distinguished base.  Go through the proof that we have a sheaf on a base (Theorem 4.1.2.).  This construction of the structure sheaf will be the main point of the lecture.  There will be more lecturing than usual.

  • Go through 4.1.D carefully, constructing a sheaf for every A-module M.  Read 4.1.4.
  • Read Section 4.2 on visualizing schemes and come prepared to discuss the examples.

Monday February 22, 2016

This meeting will try to finish the chapter 3.  Read to the end of the chapter.  We will cover miscellaneous topics related to irreducible components, Noetherian topological spaces, generic points, and I(S) – the ideal of functions vanishing on S.

  1. Exercise 3.6.J.a.  The set of closed points are dense for Spec of a f.g. k-algebra.
  2. Exercise 3.6.K.  In the classical setting, functions are determined by their values on closed points.
  3. Ex 3.6.M.  An example of a generic point.
  4. Ex 3.6.N.  A generic point of a closed subset K is near every point of K, and not near any point in the complement of K.
  5. Ex 3.6.O.  Every topological space is the union of its irreducible components.

Study Proposition 3.6.15 (unique decomposition into irreducibles for Noetherian spaces) and its proof carefully.  This would be a good time to review basic facts about Noetherian rings, such as the Hilbert basis theorem.

  • Do the first part of Ex 3.6.S: If A is Noetherian then Spec A is Noetherian.
  • Ex. 3.7.A, 3.7.B, 3.7.C, 3.7.D.
  • Understand how these exercises give Theorem 3.7.1.
  • Ex. 3.7.E.

Friday Feb 19, 2016

We will spend much of the hour discussing properties of the distinguished open sets D(f).

  • 3.4.H.  The contravariant functor from Ring to Top.
  • 3.4.I.    The induced topology on A/I.
  • 3.5.A.  D(f) form a base for the topology on Spec A.
  • 3.5.B.  Covers by distinguished open sets.
  • 3.5.C.  quasi-compactness
  • 3.5.D.  3.5.E

We will also spend time on topological properties of Spec A: connectedness, irreducibility, and quasi-compactness.  Study these definitions and examples.

  • 3.6.B
  • 3.6.C
  • 3.6.D
  • 3.6.G

 

Wednesday Feb 17, 2016

We will continue to work on visualizing Spec A.

 

  1. Work exercise 3.2.K in detail. (Again, if you have done the ring theory bijection before, you do not need to rework this. But you should see how to picture this geometrically.)
  2. Exercise 3.3.L.
  3. Exercise 3.2.M. This exercise is easy but really essential. Recall that this fails if prime ideal is replaced by maximal ideal.
  4. Read 3.2.10 and work exercise 3.2.0.
  5. Exercise 3.2.P.
  6. Read 3.2.11 to see why what Grothendieck calls functions are not functions in the traditional set-theoretic sense. Exercise 3.2.T is fun if you have time.
  7. Read 3.3 to improve your visualization of points.
  8. (3.4.C) What is the definition of the Zariski topology on Spec A? Check that it is a topology.
  9. Exercise 3.4.D.  How is the vanishing set of an ideal I related to the vanishing set of its radical?
  10. (page 116) Describe the closed sets of the affine line A^1 and the affine plane A^2 explicitly.
  11. 3.4.H.  Interpret Spec as a contravariant functor Rings->Top.

Lecture Monday Feb 15

Sections 3.2- 3.3.  This lecture will continue from the previous lecture in visualizing the points of Spec R.   Read over the statement of the weak and strong versions of the Nullstellensatz.

  1. Let A=C([0,1]) be the ring of continuous functions on the unit interval [0,1].  Prove that every maximal ideal has the form M_x, for some x in [0,1], where M_x is the ideal of continuous functions vanishing at x.  Hint: [0,1] is compact.
  2. Let V be a finite dimensional vector space over the field of complex numbers C. Let T:V->V be a linear map.  Let A=C[T] be the commutative ring of linear maps V->V that are polynomials in T.  Describe the spectrum of A. (Hint: eigenvalues).
  3. Work exercise 3.2.J in detail.  (If you have already seen the bijection between ideals of A/I and ideals of A containing I, you do not need to rework this.  But you should see how to picture Spec A/I as a subset of Spec A.
  4. Work exercise 3.2.K in detail.  (Again, if you  have done the ring theory bijection before, you do not need to rework this.  But you should see how to picture this geometrically.)
  5. Exercise 3.3.L.
  6. Exercise 3.2.M.  This exercise is easy but really essential.  Recall that this fails if prime ideal is replaced by maximal ideal.
  7. Read 3.2.10 and work exercise 3.2.0.
  8. Exercise 3.2.P.
  9. Read 3.2.11 to see why what Grothendieck calls functions are not functions in the traditional set-theoretic sense.  Exercise 3.2.T is fun if you have time.
  10. Read 3.3 to improve your visualization of points.