Math 632: Algebraic Geometry

Professor: David E Speyer

Winter 2015

Course meets: Monday, Wednesday, Friday 3-4; 3866 East Hall.

Something great to do after class: Note that the Algebraic Geometry Seminar meets at 4 on Wednesday's (4096 East Hall)! Stay and enjoy the talks! All of you are ready to benefit from at least the first hour of the main Algebraic Geometry seminar. This is one of three Algebraic Geometry seminars at Michigan this term: Baby Algebraic Geometry will meet Monday 5-6 (4096 East Hall) and Student Algebraic Geometry will meet Thursday 4-5 (4096 East Hall).

This paragraph by Ravi Vakil fully agrees with my experience:
Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)

Gathering time It is extremely difficult to learn this material alone. In order to give people a place and time to meet their fellow students working on algebraic geometry, Tuesday afternoons in the tea room. There is a schedule problem: based on class polling, most people become free after 4 PM, I will almost always need to leave by 5, and 4-5 is the colloquium. I will come to the tea room between 4 and 5 unless I am going to the colloquium, but I hope that you will also be able to help each other.


Textbooks: Algebraic Geometry, by Robin Hartshorne. I also strongly recommend Foundations of Algebraic Geometry by Ravi Vakil.

The Update: A day by day summary of the course, written by the students and edited by me.

Other valuable online sources: Mel Hochster's commutative algebra notes. The algebraic-geometry tag at mathoverflow and math.stackexchange (but see the homework policy below). Mark Haiman's synopses of EGA, Ravi's notes and blog, the stacks project, the algebraic geometry tag at nLab.

Intended Level: This is not a first course in algebraic geometry. Many of you took Math 631 with me last term, and I am glad to see you back. Whether through my course or another, you should already have some geometric intuition for varieties before you work with schemes.

The commutative algebra prerequisite will also be much more serious this term. Last term, I made an effort to avoid significant use of homological algebra, localization, completion or tensor products, and it was often possible to do so by some clever planning. That is no longer possible. You should have taken Math 614 or the equivalent before taking this class.

Anticipated topics The primary goals of this construction are schemes and sheaf cohomology — that is to say, Chapters 2 and 3 of Hartshorne. I hope to have time to say a bit about K0-theory, which is one of my favorite subjects, and about algebraic de Rham theory. Ultimately, though, the two main technical subjects are large enough that there will not be time for another large topic.

Problem Sets: I will assign weekly problem sets, due on Wednesdays. See below for the homework policy. Unlike last term, most of my problem sets will be taken from Hartshorne's book.

Reading:I think Hartshorne's book is best used as a reference after understanding the big picture. I will once again assign reading, but you should focus on reading the statements of theorems and definitions, and only on reading those theorems that you want to read the proofs of. It is very easy to get lost in the details with Hartshorne.

The obvious assertion quizzes: The foundations of scheme theory involve an extraordinary number of plausible assertions, each of which takes only a few minutes to check. I will maintain a file of such assertions and, on each Wednesday, I will challenge you to prove your choice of one of three of them. My tentative plan is that assertions from k weeks ago will appear with probability proportional to 2-k; I may change this formula if it produces absurd results.

Update Like Karen Smith's fantastic 632 classes, this class will have a Daily Update covering what has happened in class each day. Unlike Karen, I will insist that you write it! I will have you take turns writing a one or two paragraph summary of the key definitions and theorems of the day. (I will be glad to help you with this.) These will be concatenated into a single document, which I hope will serve as a valuable record.

Schedule and reading

By popular request, I have added references to where you can find the same material in Ravi Vakil's book. (Scroll down to the bottom to download the newest version. All references here are to the December 30, 2014 version, but the table of contents has been stable for a while, so they should work for other versions.)

January 7-16 We will discuss sheaves. Sheaves are a formalism for discusing constructions which are local. Allen Knutson observed to me, and I agree, that there is no significant "theory of sheaves", just as there is no significant "theory of topological spaces". Rather, there are various theories — algebraic geometry, algebraic topology, certain aspects of complex analysis or PDEs — in which the language of sheaves is valuable.

Please read Chapter II.1 by Monday, January 12.

In Vakil, this is Chapter 2. Chapter 2.7 gives a more detailed discussion of gluing sheaves from a basis of open sets than I was able to do.

January 21-23 We will introduce our main objects of study, schemes. Schemes are what you get when you want to use the methods of algebraic geometry to talk about all commutative rings, not just finitely generated algebras over an algebraically closed field. The last pages of this section discuss the category of varieties, which are the quasi-projective varieties of last term. You can look back to Section I.3 if you want to see exactly how Harsthorne sets up this category.

Please read Chapter II.2 through the end of Example 2.3.6 by Wednesday, January 21, and finish Chapter II.2 by Friday, January 23.

In Vakil, this is Chapter 3, Chapter 4.1-4.4 and 6.1-6.3. Vakil's proof that regular functions form a sheaf (Theorem 4.1.2) is much closer to mine than Hartshorne's is. I really like Vakil's discussion of visualizing schemes in Sections 3.3 and 4.2. The claim that morphisms of affine schemes are exactly the maps which come from maps of rings is Exercise 6.3.D, and the corresponding statement for general schemes is Exercise 6.3.C.

January 26-28 We will discuss the Proj construction, which turns a graded ring into a scheme.

In Vakil, this is Chapter 4.5. Both sources leave a lot of assertions to the exercises, but they leave different assertions, so it may be helpful to see if one checks the proof that the other omits.

January 30-February 2 Chapter II.3 introduces the crucial product construction. I'll talk about products on January 30 and February 2.

Products appear as Chapter 9.1-9.3 of Vakil. This material is really good! I highly endorse it.

February 4 Chapter II.3 also features a lot of definitions. Talking about all of them would be absurd. Therefore, I will poll the class in order to find out which of them would be most useful to discuss. In order for me to have time to write these lectures, please skim Chapter II.3 and respond to the poll I will send out by Monday, January 26.

The various adjectives in Chapter II.3 of Hartshorne are scattered throughout chapters 6, 7 and 8.

February 6-February 9 We will discuss separated and proper morphisms. In my experience, the easy direction of the valuative criterion (if a scheme is separated/proper then...) is much more useful than the converse (if ... then the scheme is separated/proper). I only intend to talk about the easy direction. Please read Chapter II.4 by Monday, February 2. Vakil discusses these notions in Chapter 10. Like me, he omits the hard direction of the valuative criteria.

February 11 At this point, I will start going a bit out of order. Hartshorne puts all the big theorems about sheaves of modules in Chapter II.5, before you see any applications of them in Chapters II.6 and II.7. I don't like this, and I don't fully like Hartshorne's choice of applications, so I am going to skip ahead a bit and come back. Please look through the main definitions and concepts of Chapter II.5 by Monday, February 9, but don't worry about proofs yet.

February 13-16 I'll talk about vector bundles and the relation ship between vector bundles and locally free modules. Hartshorne relegates this material to Exercise II.5.18; I want to discuss it in class.

This is Chapter 13 in Vakil.

February 18-23 or so I'll talk about the many different ways to think about line bundles: As line bundles, as invertible sheaves, as divisors and as maps to projective space. Please read Chapter II.6 to learn about divisors by Monday, February 16. I won't have time to talk about all of II.7, but please read through the end of Example II.7.8.6 by Friday, February 20.

This is Chapter 14 in Vakil.

In the week of March 9-13, we'll introduce sheaf cohomology, using injective resolutions. Hartshorne introduces the general formalism of derived functors in Chapter III.1 and then discusses the particular case of sheaf cohomology in Chpater III.2. Derived functors are very worth understanding eventually, but I'll move directly to the special case. If I have time, I'll talk a bit about hypercohomology on Friday, March 13.

Please read Chapter III.1 and III.2 by Friday, March 13.

Other optional reading: Both Hartshorne and I will skip over a lot of routine verifications in the name of getting to the good stuff. If you'd like to see them, I recommend chapters 1 and 2 of Weibel's Homological Algebra. The corresponding chapter in Vakil's book is Chapter 18.

In March 16-20, we'll talk about Cech cohomology, including proving some of the results about quasi-coherent sheaves that we missed the first time around. Please read Chapter III.3 and III.4 by Friday, March 20.

Vakil takes Cech cohomology as his fundamental definition, so the relevant material in Vakil is chapter 18 again.

On Monday, March 23 I'll compute the cohomology groups of line bundles on projective space. This computation can be found in Chapter III.5 of Hartshorne; please read it by Monday, March 23. (Actually, this wound up delayed until Wednesday.)

On March 25 and 27, I'll talk about two cohomology vanishing results due to Grothendieck and Serre respectively (Theorems III.2.7 and III.5.2 in Hartshorne.)

In the week of March 30-April 3, I'll talk about Hilbert series. Hartshorne puts this material in a series of exercises: III.5.1-4, I think it deserves class discussion. So no reading assignments for this week, but this would be a great week to return to earlier material.

In the final two weeks of class, April 6-20, I'll talk about flatness and the Hilbert scheme. Please read Chapter III.9 by Wednesday April 8.

The corresponding chapter in Vakil is Chapter 24.

Problem Sets

Problem Set 1, due Wednesday, January 21 Chapter II.1, problems 2, 5, 7, 8, 17, 21, 22 and this problem. Also remember to take a look at the obvious assertions file.

Problem Set 2, due Wednesday, January 28 Chapter II.2, problems 3, 7, 8, 12, 17 and this problem. Also remember to take a look at the obvious assertions file.

Problem Set 3, due Wednesday, February 4 Chapter II.3, problems 1, 8, 12 and these problems. Also remember to take a look at the obvious assertions file.

Problem Set 4, due Wednesday, February 11 Friday, February 13 Chapter II.3 problems 4, 5, 10. Chapter II.4 problems 2, 3 and these problems.

Problem Set 5, due Wednesday, February 18 Chapter II.4 problems 4, 7, Chapter II.5 problems 1, 7, 8 and these problems.

Problem Set 6, due Wednesday, February 25 Chapter II.5 problems 12, 13, 14, 16, 17, this problem, and II.6.9. (I recommend doing the extra problem before 6.9.)

Problem Set 7, due Wednesday, March 18 I didn't like any of the Hartshorne problems this week. Instead, here are two problems, on partitions of unity and homological algebra.

Problem Set 8, due Friday, March 27 (delayed due date due to 115 and 116 exams) Chapter III.4 problems 1, 3, 5, 7 and these problems.

Problem Set 9, due Wednesday, April 3 Chapter III.4.6, 8 and 9, III.5.5, 7, 8 and 9.

Homework Policy: You are welcome to consult each other provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. If you seek help from mathematicians/math students outside the course, you should be seeking general advice, not specific solutions, and must disclose this help. I am, of course, glad to provide help!

I don't intend for you to need to consult books and papers outside your notes. If you do consult such, you should be looking for better/other understanding of the definitions and concepts, not solutions to the problems.

You MAY NOT post homework problems to internet fora seeking solutions. Although I participate in some such fora, I feel that they have a major tendency to be too explicit in their help; you can read further thoughts of mine here. You may post questions asking for clarifications and alternate perspectives on concepts and results we have covered.

The daily update

When it is your turn to write the Update (TeX), download the TeX template and write a 1/2 – 1 page summary of what happened that day. The goal is to get the key definitions and theorems; you certainly do not need to record everything. Then e-mail it back to me. The deadline for editing the update is 24 hours after the lecture. I will, in turn, proofread and edit your entries in the next 24 hours and post them back to this webpage, so that the class always has a good record of what we have covered.

Note: Last term, we had some trouble with conflicting edits to the Update file. Therefore, this term, start with the blank Template.tex, and I'll merge your LaTeX into Update.tex.

If you have forgetten when you are scheduled to write the update, you can check here.