The course webpage for the 2009-10 version of the course is here. You should go there first. The course webpage for the 2007-08 version of the course is here.

*Here are the notes, as of June 28, 2007. You can check below to see
which notes have been updated: they are in bold.
Fall.
Winter.
Spring.
Problem sets.
Everything in one huge file.
*

*
These notes are quite rough,
and roughly reflect what I did in class. A new version is available from the 2007-08 class page, and a better version accompanies
the 2009-10 class, so
this particular file (and all remaining errors) will stay frozen.
Later corrections are below, in bold. I will eventually remove this webpage, as all material here is trumped by later pages.*

There are several types of courses that can go under the name of "introduction to algebraic geometry": complex geometry; the theory of varieties; a non-rigorous examples-based course; algebraic geometry for number theorists (perhaps focusing on elliptic curves); and more. There is a place for each of these courses. This course will deal with schemes, and will attempt to be faster and more complete and rigorous than most, but with enough examples and calculations to help develop intuition for the machinery. Such a course is normally a "second course" in algebraic geometry, and in an ideal world, people would learn this material over many years. This is not an ideal world. To make things worse, I am experimenting with the material, and trying to see if a non-traditional presentation will make it possible to help people learn this material better, so this year's course is only a first approximation.

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:**
The posted lecture notes are rough, so I 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.

- 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.
- 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.
- 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!) - 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.
- 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.
- Class 17 (Wed. Nov. 30): associated points; more on normality; invertible sheaves and divisors take 1. (ps, pdf). Last small update: late June 2007.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Class 36 (Thurs. Mar. 2):
elliptic curves (continued); group schemes; the Picard variety; "the moduli
space of genus
*g*curves has dimension 3*g*-3". (ps, pdf). Small update: Jan. 30, 2007.

- 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.
- 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.
- 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.
- Classes 49 and 50 (Tues. May 16 and Tues. May 23): blowing up a scheme along a closed subscheme. (ps, pdf). Updated June 8.
- Classes 51 and 52 (Thur. May 25 and Tues. May 30): smooth, etale, unramified. (ps, pdf). Updated June 8.
- 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!*

All content on this website (including course notes) is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 2.5 License.

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