Math 130b: Algebraic Geometry, Winter 2012

This class meets MWF 1-2pm in 159 Sloan.

Office hours: Thursdays 3-5pm in 374 Sloan

Course Description: The course will follow Vakil's Foundations Of Algebraic Geometry notes (available at http://math.stanford.edu/~vakil/216blog/). We will begin by studying some category theory and sheaves. Then we will develop the basic properties and theory of schemes, morphisms of schemes, and quasi-coherent sheaves on schemes.

Grading: Grades will be based on weekly homework. We may have a final exam.

Exercises

There will be a problem set due in class on Friday every week. This will be a fast paced course in which it is important to stay on top of the most recent material, so no late homework will be accepted.

You are strongly encouraged to submit solutions in groups of 2, though solutions by individuals and groups of 3 will also be accepted. You're welcome to talk about the problems with people outside of your group. You are not required to work with the same person every week.

Due date	Homework	Description
Jan. 6	HW1
Jan. 13	HW2	Read Chapter 3 of Vakil (Dec. 20, 2011 version), do problems 3.2.F, 3.3.C, 3.3.I, 3.4.B, 3.5.G, and do the property P argument.
Jan. 20	HW3	Read sections 4.1-4.3 of Vakil (Dec. 20, 2011 version) and do problems (possible hints in parentheses) 3.6.E, 3.7.C (consider stalks of sheaf Hom, c.f. 3.3.C), 3.7.D (use Zorn's Lemma), 4.2.C, 4.2.K, 4.2.L. Also do the tensor products exercise.
Jan. 27	HW4	Read sections 4.4-5.4 of Vakil (Jan. 14, 2012 version) and do problems 4.6.A, 4.6.B, 4.6.C, 4.6.D, 4.6.L, 4.6.S, 4.7.E (equivalent definition of a prime ideal: IJ⊆p implies I⊆p or J⊆p), 5.1.D, 5.3.C, 5.3.E, 5.4.B. Also do the Noetherian induction problem.
Feb. 3	HW5	Do the following problems in Vakil (Jan. 14 version): 6.2.A, 8.3.Q (for (c), Hilbert basis theorem: 4.6.15), 10.3.B (read the text above the exercise), 10.4.B (a)-(g). Also show that a morphism being a closed immersion, finite, or integral is an affine local condition on the base.
Feb. 10	HW6	Do the following problems in Vakil (Jan. 14 version): 5.5.I, 7.3.M (for $x\in X$, Spec($\mathcal O_{X,x}$) is the intersection of all open sets containing $x$), 7.3.N (there's a slick way to do this if you do 9.2.O first), 7.4.A, 7.4.D, 8.3.J, 9.2.N, 9.2.O.
Feb. 17	HW7	Read section 10.7 and chapter 11 of the Jan. 14, 2012 version of Vakil's notes and do the following problems: 10.7.A, 10.7.B, 10.7.F, 11.1.G, 11.1.H, 11.1.J, 11.1.L. Also do four other problems (projective implies proper, and the valuative criteria).
Feb. 24	HW8	Read sections 12.1-12.3, 14.1-14.2 of the Jan. 14, 2012 version of Vakil's notes and do the following problems: 12.2.C ("variety" means "reduced, separated, and finite type"), 12.3.B, 14.1.B, 14.1.C, 14.1.D, 14.1.E, 14.1.F, 14.1.G, 14.1.H, 14.1.I. Also do the support and local freeness problems.
Mar. 2	HW9	Read chapter 14 of the Jan. 14, 2012 version of Vakil's notes and do the following problems: 14.3.D, 14.5.C, 14.5.D, 14.6.A, 14.7.H, 14.7.I, 14.7.J. Also do the three other problems.
Mar. 9	HW10	Read chapter 16 of the Feb. 25, 2012 version of Vakil's notes and do the following problems: 15.1.C, 16.1.C, 16.1.D, 16.2.A, 18.1.F, 18.1.G. Also do the locally closed immersion and projection formula exercises.
Mar. 15 at noon	HW11	Do the following problems from the Feb. 25, 2012 version of Vakil's notes: 15.2.K, 15.2.P, 15.2.R, 17.4.B, 17.6.B, 17.6.G.

Resources

• Vakil, Foundations Of Algebraic Geometry
• Grothendieck and Dieudonné, EGA I, II, III(1), III(2), IV(1), IV(2), IV(3), IV(4)

A detailed table of contents for EGA is available here. Getting to know your way around EGA is a good use of time for an algebraic geometer.

• Hartshorne, Algebraic Geometry