Math 216: Foundations of algebraic geometry 2005-06

All notes in four big files

Here are the notes, as of June 28, 2007. You can check below to see which notes have been updated: they are in bold. I might not edit them any more, as I'll start new notes for the 2007-2008 class. Fall. Winter. Spring. Problem sets. Everything in one huge file.

Note to experts browsing the notes: They are quite rough, and roughly reflect what I did in class. At some point I intend to reorder and improve them, but this particular file will stay frozen. Later corrections are below, in bold.

In short, this not a course to take casually. But if you have the interest and time and energy, I will do my best to make this rewarding.

References: I hope to type up each day's notes fairly soon after the lecture (usually within a week). You should also take notes yourself, and not count on these. The posted lecture notes will be rough, so I would recommend having another source you like, for example Mumford's Red Book of Varieties and Schemes (the original edition is better, as Springer introduced errors into the second edition by retyping it), and Hartshorne's Algebraic Geometry. Both books are on reserve at the library. Hartshorne should be available at the bookstore. For background on commutative algebra, I'd suggest consulting Eisenbud's Commutative Algebra with a view toward algebraic geometry or Atiyah and MacDonald's Commutative Algebra. For background on abstract nonsense, Weibel's Introduction to Homological Algebra is good to have handy. Justin Walker also points out that Freyd's Abelian Categories is available online (free and legally) here.

Homework: Unlike most advanced graduate courses, there will be homework. It is important --- this material is very dense, and the only way to understand it is to grapple with it at close range.

Notes: Notes for the classes in ps and pdf formats will be posted here. Caution: All of these notes are quite rough, and just approximate transcriptions of my lecture notes. I encourage you to take notes yourselves, and not just rely on these. However, if you feel like pointing out improvements, I would appreciate it, as these notes are crudely extracted from a larger set of notes that I hope to make available eventually. Note that I give the dates of the last important update.

Fall quarter

• Class 1 (Mon. Sept. 26): about this course; what is algebraic geometry; motivation and program; crash course in category theory, Yoneda's Lemma. (ps, pdf). Small update: Jan. 30, 2007.
• Class 2 (Wed. Sept. 28): More examples of things defined using universal properties: inverse limits, direct limits, adjoint functors, groupification. Sheaves: the motivating example of differentiable functions. Definition of presheaves and sheaves. (ps, pdf). Small update: Jan. 31, 2007. Note: in Section 3.2, W should be required to contain x. (Thanks to Owen Jones.)
• Class 3 (Mon. Oct. 3): Presheaves and sheaves. Morphisms thereof. Sheafification. (ps, pdf). Small update: Jan. 31, 2007.
  
  • Problem Set 1 out (due Mon. Oct. 17). (ps, pdf). Last update: Oct. 18.
• Class 4 (Fri. Oct. 14): Understanding sheaves via stalks. Understanding sheaves via "sheaves on a nice base of a topology". Affine schemes Spec R: the set. (ps, pdf). Small update: Jan. 31, 2007.
• Class 5 (Mon. Oct. 17): Spec R: the set, and the Zariski topology. (ps, pdf). Small update: Jan. 31, 2007.
  • Problem Set 2 out (due Mon. Oct. 24). (ps, pdf). Last update: Oct. 20.
• Class 6 (Wed. Oct. 19): the structure sheaf, and schemes in general. (ps, pdf). Small update: Jan. 30, 2007.
• Class 7 (Fri. Oct. 21): playing with the structure sheaf. (ps, pdf). Small update: Jan. 28, 2007.
• Class 8 (Mon. Oct. 24): irreducible, connected, quasicompact, reduced, dimension. (ps, pdf). Small update: Jan. 31, 2007.
  • Problem Set 3 out (due Mon. Oct. 31). (ps, pdf). Last important update: Oct. 25.
• Class 9 (Wed. Oct. 26): Dimension continued, Krull's Principal Ideal theorem, height, affine communication lemma, properties of schemes: locally Noetherian, Noetherian, finite type A-scheme, locally of finite type A-scheme, normal. (ps, pdf). Small update: Jan. 31, 2007.
• Class 10 (Mon. Oct. 31): (locally) finite type A-scheme, projective schemes over A or k. (ps, pdf). Small update: Jan. 31, 2007. The first expression in Exercise 1.2 should be wz-xy. (Thanks Martin Olsson!)
  • Problem Set 4 out (due Mon. Nov. 7). (ps, pdf). Last update Nov. 5.
• Class 11 (Wed. Nov. 2): smoothness=regularity=nonsingularity, Zariski tangent space and related notions, Nakayama's lemma. (ps, pdf). Small update: Jan. 31, 2007.Problem 1.7(c) is wrong (thanks Dave Savitt!)
• Class 12 (Fri. Nov. 4): Jacobian criterion, Euler test, characterizations of discrete valuation rings = dimension 1 Noetherian regular local rings. (ps, pdf). Small update: Jan. 31, 2007.
• Class 13 (Mon. Nov. 7): discrete valuation rings (conclusion), cultural facts to know about regular local rings, the distinguished affine base of the topology, two definitions of quasicoherent sheaf. (ps, pdf). Small update: Jan. 30, 2007.
  • Problem Set 5 out (due Mon. Nov. 14). (ps, pdf). Minor update Dec. 19. For #4, same correction as for class 10 above. #16(c) is wrong, see class 11 note above.
• Class 14 (Wed. Nov. 9): quasicoherence is affine-local, locally free sheaves and vector bundles, invertible sheaves and line bundles, torsion-free sheaves, quasicoherent sheaves of ideals and closed subschemes. (ps, pdf). Small update: Jan. 31, 2007.
• Class 15 (Mon. Nov. 14): quasicoherent sheaves form an abelian category; finite type and coherent sheaves; support; rank; quasicoherent sheaves of ideals and closed subschemes. (ps, pdf). Last small update: late June 2007.
• Class 16 (Wed. Nov. 16): effective Cartier divisors, quasicoherent sheaves on projective A-scheme correspond to graded modules, line bundles O(n) on projective A-schemes, Serre's theorem. (ps, pdf). Small update: Jan. 31, 2007. Note: the last problem on p. 1 is false.
  • Problem Set 6 out (due Wed. Nov. 30). (ps, pdf). Slight revision Dec. 11.
• Class 17 (Wed. Nov. 30): associated points; more on normality; invertible sheaves and divisors take 1. (ps, pdf). Last small update: late June 2007.
  • Problem Set 7 out (due Wed. Dec. 7). (ps, pdf). Minor update Jan. 23.
• Class 18 (Fri. Dec. 2): invertible sheaves and divisors; morphisms of schemes. (ps, pdf). Last small update: late June 2007.
• Class 19 (Wed. Dec. 7): maps to affine schemes; surjective, open immersion, closed immersion, quasicompact, locally of finite type, finite type, affine morphism, finite, quasifinite. Images of morphisms: constructible sets, and Chevalley's theorem (finite type morphism of Noetherian schemes sends constructibles to constructibles). (ps, pdf). Last small update: late June 2007.
• Class 20 (Fri. Dec. 9): pushforwards and pullbacks of quasicoherent sheaves (ps, pdf). Last small update: late June 2007.
  • Problem Set 8 out ("due" Mon. Dec. 12, although I'll accept it --- and earlier sets --- until Friday Dec. 16). (ps, pdf). Minor update Dec. 19.
• Bonus handout: Proofs of "Hartogs Theorem", Krull's principal ideal theorem, and the fact that the intersection of all the powers of the maximal ideal in a Noetherian local ring is 0. [I would like a more correct name than "Hartogs Theorem", yet one that is still memorable. Perhaps "R1 implies regular", where "R1" means "regular in codimension 1"? Not great...] (ps, pdf). Posted Dec. 15. Minor update Dec. 16.

• Class 21 (Tues. Jan. 10): integral extensions, going-up theorem, Noether normalization, proof that transcendence degree = Krull dimension, proof of Chevalley theorem, invertible sheaves and morphisms to (quasi)projective schemes. (ps, pdf). Small update: Jan. 28, 2007.
• Class 22 (Thurs. Jan. 12): Morphisms to (quasi)projective schemes, and invertible sheaves; fibered products. (ps, pdf). Small update: Jan. 28, 2007.
• Class 23 (Tues. Jan. 17): Fibers of morphisms. Properties preserved by base change: open immersions, closed immersions, Segre embedding. Other schemes defined by universal property (take 1): reduction, normalization. (ps, pdf). Tiny update October 26, 2006.
  • Problem Set 9 out (due Tues. Jan. 24). (ps, pdf). Updated January 25.
• Class 24 (Thurs. Jan. 19): normalization (in a field extension), "sheaf Spec", "sheaf Proj", projective morphisms. (ps, pdf). Last small update: late June 2007.
• Class 25 (Tues. Jan. 24): separatedness, definition of variety (ps, pdf). Minor update June 20.
  • Problem Set 10 out (due Thurs. Feb. 2). (ps, pdf). Tiny update October 26, 2006.
• Class 26 (Thurs. Jan. 26): proper morphisms (ps, pdf). Minor update May 28 (thanks Brian Osserman!).
• Class 27 (Tues. Jan. 31): Proper morphisms (continued), scheme-theoretic closure and image (left over), rational maps, examples of rational maps (ps, pdf). Minor update March 8.
  • Problem Set 11 out (due Thurs. Feb. 9). (ps, pdf). Updated February 27.
• Class 28 (Thurs. Feb. 2): curves (ps, pdf). Small update: Jan. 28, 2007.
• Class 29 (Tues. Feb. 7): curves continued; Cech cohomology. (ps, pdf). Last small update: late June 2007.
  • Problem Set 12 out (due Thurs. Feb. 16). (ps, pdf). Minor update March 8.
• Class 30 (Thurs. Feb. 9): cohomology continued; Hilbert functions and polynomials. (ps, pdf). Last small update: late June 2007.
• Class 31 (Tues. Feb. 14): Hilbert polynomials and Hilbert functions; higher direct image sheaves. (ps, pdf). Last small update: late June 2007.
  • Problem Set 13 out (due Thurs. Feb. 23). (ps, pdf). Minor update March 8.
• Class 32 (Thurs. Feb. 16): applications of higher pushforwards; crash course in spectral sequences. (ps, pdf). Updated June 26.
• Class 33 (Tues. Feb. 21). The Leray sepctral sequence. Beginning "fun with curves": the Riemann-Hurwitz formula (ps, pdf). Updated June 26.
  • Problem Set 14 out (due Thurs. Mar. 2). (ps, pdf). Updated Feb. 27.
• Class 34 (Thurs. Feb. 23). More fun with curves: Serre duality, criterion for closed immersion, a series of useful remarks, curves of genus 0 and 2 (ps, pdf). Last small update: late June 2007.
• Class 35 (Tues. Feb. 28): hyperelliptic curves; curves of genus at least 2; elliptic curves take 1 (ps, pdf). Last small update: late June 2007.
  • Crib sheet from last day. (ps, pdf).
  • Problem Set 15 out (due Thurs. Mar. 9). Updated March 6. (ps, pdf).
• Class 36 (Thurs. Mar. 2): elliptic curves (continued); group schemes; the Picard variety; "the moduli space of genus g curves has dimension 3g-3". (ps, pdf). Small update: Jan. 30, 2007.
  • Problem Set 16 out around March 7 (due Thurs. Mar. 16). (ps, pdf).

• Class 37 (Tues. Apr. 4): introduction to differentials (ps, pdf). Last small update: late June 2007.
• Class 38 (Thurs. Apr. 6): differentials for schemes; examples (ps, pdf). Last small update: late June 2007.
• Classes 39 and 40 (Tues. Apr. 11 and Thurs. Apr. 13): the Euler exact sequence. Discussion of nonsingular varieties over algebraically closed fields: Bertini's theorem, the Riemann-Hurwitz formula, and the (co)normal exact sequence for nonsingular subvarieties of nonsingular varieties. (ps, pdf). Last small update: late June 2007.
  • Problem Set 17 out (due Thurs. Apr. 20). (ps, pdf). Minor update May 15.
• Classes 41 and 42 (Tues. Apr. 18 and Thurs. Apr. 20): flatness, tor, ideal-theoretic characterization of flatness, for coherent modules over Noetherian local ring flat=free, flatness over nonsingular curves. (ps, pdf). Last small update: late June 2007.
• Classes 43 and 44 (Tues. Apr. 25 and Thurs. Apr. 27): constancy of Euler characteristic in flat families. The semicontinuity theorem and consequences. Glimpses of the relative Picard scheme. (ps, pdf). Last small update: late June 2007.
  • Problem Set 18 out (due Thurs. May 4). (ps, pdf). Updated June 26.
• Classes 45 and 46 (Tues. May 2 and Thurs. May 4): Grauert's theorem and cohomology and base change theorem, and applications. The rigidity lemma. Proof of Grauert's theorem. Dimensions behave well for flat morphisms. Associated points go to associated points. (ps, pdf). Last small update: late June 2007.
• Classes 47 and 48 (Tues. May 9 and Thurs. May 11): Local criteria for flatness, (relatively) base-point-free, (relatively) ample, very ample, every ample on a proper has a tensor power that is very ample, Serre's criterion for amplitude, Riemann-Roch for generically reduced curves. (ps, pdf). Last small update: late June 2007.
  • Problem Set 19 out (due Thurs. May 18). (ps, pdf).
• Classes 49 and 50 (Tues. May 16 and Tues. May 23): blowing up a scheme along a closed subscheme. (ps, pdf). Updated June 8.
  • Problem Set 20 out (due Thurs. May 18). (ps, pdf). Updated June 26.
• Classes 51 and 52 (Thur. May 25 and Tues. May 30): smooth, etale, unramified. (ps, pdf). Updated June 8.
  • Problem Set 21 out (due Thurs. June 8). (ps, pdf).
• Classes 53 and 54 (Thurs. June 1 and Tues. June 6): Serre duality. (ps, pdf). Updated June 19.
• Here is another proof of Serre duality for curves, due to Serre. It requires as input the residue theorem.

Many thanks to Justin Walker for improving the class notes immeasurably!

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