The first post is the August 26 version here.

I hope to post approximately 25 pages of reading for learners (with more optional material for experts) every fortnight.

**For learners.**

If you are reading to learn, the first reading is, I’m afraid, quite heavy if you’ve never seen category before. You should just skim the starred subsections and skip the spectral sequence section 2.7: you should read (slowly) 2.1-2.6, and get a sneak peak at 3.1. It is essential to be familiar with the exercises, and which ones you should do depends on what you know. If you force me to name twelve, I won’t be happy, but I’ll try: 2.3.F, 2.3.K, 2.3.O, 2.3.T, 2.3.Y, 2.4.A, 2.4.C, 2.5.C, 2.5.E, 2.6.A, 2.6.B, 2.6.F (from the Aug 26 version).

I don’t want to give you much guidance, because I want the notes to speak for themselves as much as possible.

**For experts.**

I hope to get across not just the category theory people need, but also a sense of how to think categorically. I always find it a good question when people in class start asking wise categorically-minded questions, and thinking clearly. There are also a few constructions that are essential. But I don’t want to tell people anything they don’t need to know, and I try maintain a balance between informality and precision. I’ve noticed that most of the following topics is easy once you understand them (with some exceptions!), but that they are quite unmotivated when you first see them. Developing fluency with these ideas requires working through as many trivial examples as possible. I hope many of the exercises are (after you figure them out) trivial, although some are hard.

In 2.1, I try to give them motivation for why we think in this way.

In 2.2, I introduce categories and functors and related notions, such as natural transformations. I didn’t understand the notion of equivalent categories in my gut until Johan de Jong gave me some baby examples.

Section 2.3 is devoted to universal properties, again through key examples. Yoneda’s lemma sneaks in at the end, and I hope readers end up finding it reasonable (even if they don’t yet appreciate it).

(Question: after 3.3.D: “0” is a zero-divisor. Is that so horrible? Question: I originally denoted the contravariant functor representing X by h^X, and the covariant functor representing X by h_X. Arthur Ogus and Jason Ferguson made strong cases that I’m violating accepting convention, so I’ve attempted to reverse myself. Does anyone strongly disagree with Arthur and Jason?)

(Co)limits are discussed at length in 2.4, with examples, and constructions of when they exist in cases that will actually be used.

In 2.5, adjoints are discussed. I find this tricky to explain. It is a pain in the butt to show that two things are adjoint.

Abelian categories are introduced in 2.6. We need them, but they are a tarpit in which many people have been irretrievably lost. We need some things, but remarkably little. It’s really true that if you understand modules over a ring, you can get by in almost any case in algebraic geometry (of the sort most people do) — even without Freyd-Mitchell. I am the most worried about this section. If someone hasn’t seen long exact sequences before, Theorem 2.6.5 will be a bolt from the blue.

Section 2.6.7, two useful facts in homological algebra, contains facts that I really wish I’d collected in my head earlier than I did (on the relationship between homology, (right/left)-exact functors, adjoints, and colimits — I would often be confused as to when various things would commute, and only later realized that there are some general principles at work). I hope I have collected them correctly. People seeing homological algebra for the first time shouldn’t read this, but people seeing it for the second time might benefit from it.

Spectral sequences are necessary later, so I included a brief introduction in 2.7. I really mean it when I say that I hope that people don’t peek until they actually need it. I do something slightly unusual (that may upset some people). For me, spectral sequences are useful because of how they can be used with little thought. Too many people just tune out when they hear the word, but when they turn up, they should make you happy, not sad. The best way to understand how to use something is to see it used to do something you already know how to do. So I deliberately concentrate on how to use them, in the special case of double complexes, which is the only case we’ll use later. I give a proof in this case which I believe is complete (and which hasn’t yielded many complaints), but which I hope most people don’t read. My only goal with this section is that people should leave it unafraid of spectral sequences in this setting, ready to use them on a moment’s notice, and with a sharp sense of how to use them. (Questions for experts: (1) I also introduce a convention of having an arrow in the spectral sequence of a double complex to indicate which is the first differential you use, the horizontal or the vertical — both in the names of the pages and in the names of the differentials on that page. Is anyone really offended? (2)

Should the indices of spectral sequences be ordered “(row, column)” or “(x,y)”? I don’t think there is consistency in the literature, so I’ve gone with the first. (3) Is there some classical important fact that would be useful practice for people learning spectral sequences, that doesn’t require much additional background?)

August 23, 2010 at 11:27 pm

In my wordpress setup there is a way to take a post offline. Namely, in dashboard you click on trash. This does not delete the post and you can edit it at your leisure and then repost it… I am not sure your setup is exactly the same.

[Thanks! I’ll do this in the future. -R.]August 26, 2010 at 7:17 pm

A response from Tyler Lawson I wanted to include:

Ravi,

I know you explicitly stated that your blog post on “first notes”

should be ignored, but some things are hard to wait on. So far as the

spectral sequence questions go:

(2)

(p,q) = (row,column) is standard indexing almost universally in the

alg top community with two exceptions:

– when running the “horizontal first” spectral sequence of a double complex, and

– for the people who work with these things on a daily basis, who use

a completely different convention called Adams indexing that puts

“output degree” in a column rather than a diagonal.

(3)

Numerous short examples exist coming from double complexes, run in

both the horizontal and vertical directions, to compute the same thing

in two ways. Favorites (you may already have these handy):

– View a map of short exact sequences as a double complex: proves the

snake lemma.

– View a short exact sequence of chain complexes as a double complex:

proves the existence of the long exact sequence of homology.

– Given a composite f:A->B and g:B->C, construct the double complex

(vertical arrows suppressed)

A -> B -> 0

0 -> B -> C

and this proves a six-term exact sequence relating the kernels and

cokernels of f, g, and gf.

– One that actually came up during the reading course we did from

Fulton-Harris: given a ring A with non-zero-divisors a,b in A,

construct a double complex

A -> A

A -> A

with the horizontal maps multiplication by a, and the vertical maps

multiplication by b. This shows that the kernel of multiplication by

a on A/(b) is the same as the kernel of multiplication by b on A/(a),

and this was related to commutativity of intersection pairings.

There are obviously a lot more, and from a practical standpoint some

of the handiest ones for AG probably relate computations over a ring

to computations over a quotient using the associated graded of an

ideal, but they likely don’t fit into this introductory material.

—

Tyler.

August 26, 2010 at 7:19 pm

Thanks Tyler! Responses:

(2) Great, I’ll leave things as is.

(3) Very useful. The snake lemma and long exact sequence are already there. The f/g/fg thing is nice, and I’ll keep this as a potential one to add. The A –> A I’ll keep in mind in particular, as it will be useful in the as-yet-unwritten Cohen-Macaulay section.

August 27, 2010 at 2:06 am

Embarrassingly, I misinterpreted and made a mistake in the email I sent to you. The entry (p,q) is normally in column p and row q. Sorry about that. I wouldn’t go so far as to regard the other convention as offensive, especially if the main examples are double complexes.

September 8, 2010 at 5:32 am

Tyler, sorry then for quickly posting your response publicly! My tentative intent is to eventually switch to the convention (column, row), unless someone argues convincingly otherwise. (Matt Emerson agrees with you, see below in comment ~20.)

Later update April 24 2011: Chuck Weibel, who wrote what I take as the bible in the area, agrees. So I consider the issue settled.

Later update May 4 2011: The change is made. I’m sure I forgot or messed up something. A change such as this is no fun to do.

August 29, 2010 at 8:08 pm

Looks great! I can’t wait to start using this for my course this semester.

Comments: (a) 0 is a zero divisor, (b) h_X should denote the contravariant functor Y |—> Mor(Y, X), (c) I am happy that you are using the terminology “limits” and “colimits”, (d) the stacks project uses the terminology “directed partially ordered set” and “filtered index category”, (e) in your definition 2.4.6 you also have to assume the partially ordered set and index category to be nonempty — a (co)limit over an empty category is not considered filtered, (f) in the table of adjoints there is an “\otimes” missing, (g) I am not that happy with introducing the notion (in the setting of additive categories) of a “monic” morphism and even less with “epic” morphisms… is it standard? Especially the fact that you are using 0 to define them makes me a bit unhappy… I think you should at least mention “monomorphisms” and “epimorphisms” somewhere nearby, in a footnote maybe? (h) what you call an abelian category is also what the stacks project calls an abelian category, but students should be aware that in other texts additional hypotheses are imposed, (i) I like the characterization of an abelian category in terms of Coim = Im, (j) Maybe formulate the snake lemma as an exercise?

More later!

September 2, 2010 at 6:06 pm

Great: (a) (b) (c) (i)

Thanks, changed (in version to be posted text): (e) (f) (h)

(d) I now mention this alternate terminology. I assume the two things you write down correspond to the two things I wrote (filtered (po)set and filtered category), but let me know if I’m misunderstanding.

(g) (I hope I’m not saying something silly; someone should stop me if I am.) You’re right; I’ve just realized that the notions I wrote are equivalent to monomorphism and epimorphism (introduced earlier). Now fixed. (The terminology “monic” and “epi” are indeed used elsewhere, e.g. see Weibel. But no longer used here.)

(j) I wanted to have one “basic” fact as a worked example in the spectral sequence section (where it is). (Or perhaps you were suggesting *also* having it as an exercise in the abelian category section because it is so central? If so, let me know.)

September 3, 2010 at 6:50 pm

Sorry, if what I wrote at first sounded a bit brisk! My remarks will probably always be of the form “you should add more here” which is not always the right thing to do for your text. So feel free to ignore them! (I’m trying to apologize here, but I’m not used to it.)

OK, great. Yes, I was thinking of having the snake lemma in the section on abelian categories… before theorem 2.6.5… I did not even realize the snake lemma does exist in the manuscript until you just mentioned it and I searched for it which I should have done before submitting my comment. One very weak argument in favor of adding a separate snake lemma exercise is that it is a tiny bit more general than what you is in 2.7.5 or in 2.6.5…

Still very excited! Thanks Ravi.

September 3, 2010 at 9:41 pm

To Johan’s later (Sept 3 2010) comment: I’ve now added links forward from this section to the snake lemma (2.7.5, and in stronger form Ex 2.7.B), and the five lemma (2.7.6 and in stronger form Ex. 2.7.C). Are there still stronger forms that are very useful? These 2 stronger versions get used later in the notes.

September 6, 2010 at 4:01 pm

Hey, here is an idea: If any of the students tex up answers to the exercises, then feel free to submit these to stacks.project@gmail.com for inclusion in the stacks project. For example 2.7.5, 2.7.B, 2.7.6, and 2.7.C…

[Sounds great — I think you’ve already implemented this idea by posting that! – R.][Later follow-up: people *should* do this, as the stacks project is rapidly becoming the canonical modern version of EGA. – R.]August 29, 2010 at 10:34 pm

This looks great! I’m a topologist who often has a need for

some algebraic geometry, but who never learned the subject properly (though I’ve picked quite a bit up along the way). I plan on reading things as you post them, though as the semester picks

up I might fall a little behind…

Anyway, I have a few comments/errata.

2.3.A. You use the notion of an initial object which isn’t defined until 2.3.3.

2.3.B. I think the definition of the localization of a module is

scrambled; as it is, $S^{-1} M$ doesn’t seem to depend on $M$. Also, isn’t a large chunk of Exercise 2.5.G coming up with a suitable defn of $S^{-1} M$ via universal properties?

2.3.N. Yoneda’s lemma has not been stated. In 2.3.Z, you state this exercise again and this time state Yoneda’s lemma.

2.3.4. A “Z” needs to be capitalized in the 1st sentence.

2.3.Y. Since fibered coproducts have not been introduced, it might be useful to make the goal of 2.3.X to define both coproducts and fibered coproducts.

2.5.B. An “A” needs to be capitalized in the 2nd line.

[Thanks! All fixed (for next version). As always, changes made just before posting were botched. – R.]August 30, 2010 at 10:40 pm

Regarding the table of adjoint pairs, there are other examples you could mention which arise in the notes or solutions to the exercises. (I’m not sure if you intended the table to be a more-or-less complete “reference guide” so that people can check there to remember, for example, which is the left adjoint and which is the right adjoint.) The examples I have in mind are stalks/skyscraper sheaves and the adjunction related to graded modules and projective schemes developed in 16.4.F,G.

[I’ve added a little related to this to the table, and elsewhere (in the next version). The stalk/skyscraper is a special case of f_*, f^{-1}, so I don’t mention it here, but because it is (as you say) particularly useful, I flag it as such later. The graded module projective scheme example is currently not done very convincingly later in the notes and doesn’t get used, so I’m not emphasizing it here. -R.]August 31, 2010 at 3:39 am

In 3.1.1, after the first paragraph you write “Notice also that if $p\in U$, you get a map $\mathcal{O}(U)\rightarrow O_x$. Experts may already see

that we are talking about germs as colimits.”

Instead of $x$ do you mean $p$?

[Yes, thanks — it’s now fixed (in the version to be posted next). -R.]August 31, 2010 at 2:00 pm

I have really been enjoying reading these notes. Here are a few typos I saw.

1. On page 20, 2.2.16 there is a missing = after $ latex f(m1\circ m2)$

2. 2.3.A “the the same” should be “are the same”

3. I got a little confused in 2.3. You are using initial objects but they haven’t been defined yet?

I look forward to the next set of notes.

[Thanks! All fixed (in the next version to appear). -R.]September 2, 2010 at 7:21 pm

Fun side comment: I only realized when working on these notes that there was some asymmetry in the description of the definition of homology of a complex as kernel mod image. It can just as well be defined as the kernel of (this definition is “dual”), see Exercise 2.6.A. This formulation can be more useful in some circumstances.

September 2, 2010 at 11:59 pm

I’m a little confused with the definition of the universal property for localization of a module (between 2.3B and 2.3C) – it seems like setting A = Z, S = { 1 }, M = Z would be a counter-example, since there are many maps from Z, to say, Z x Z, but I might be missing something.

[You’re right to be confused, because the text is messed up there. I’ve fixed it in the next version. -R.]September 3, 2010 at 8:06 pm

There seems to be a problem with your definition of the localization of the module; the obvious red flag is that the universal property doesn’t mention M at all, only S. Is this a typo?

(On an unrelated note, you’ve used initial objects in one or two of the exercises before you define them.)

[Thanks! Both are now fixed in the next version (to be posted with the next batch of notes). The patch of the localization definition is pre-posted at http://math.stanford.edu/~vakil/216blog/p23-24.pdf

If it would be really helpful to have edits posted in between the “main posts”, just let me know. But because edits happen on most days, and there are advantages to having everyone comment on the same version, right now I’m still planning on posting the entire version every fortnight. -R.]

September 3, 2010 at 8:28 pm

I think your definition of the universal property for the localization of a module is not quite correct. (Or at least, I’m having trouble seeing how to use it for exercise 2.3.C since it seems like this map from M to S^(-1) M has to materialize out of nowhere.)

The universal property should be: for any A-module N with the property that multiplication by s, for any s in S, is a bijection on N, and where pi: M -> S^(-1) M is the natural map and psi : M -> N is any morphism, then there is a unique f : S^(-1) M -> N with (f)(pi) = psi.

However, I am used to thinking of universal properties in terms of being able to draw an arrow in a commutative diagram, as with your treatment of the universal property of the tensor product, rather than in terms of a particular object being initial. (But I would be interested to know if there is some useful intuition to be gained by thinking in terms of initialness.)

Also tangentially, the definition of an initial object isn’t given until 2.3.3, two pages after this definition.

[Thanks! See the patch on the localization error in a response above (posted at the same time as your comment). The (correct) definition of localization you give is exactly of the form you like: the existence of a unique arrow. You ask if there is useful intuition to be gained by thinking in terms of initialness. I find that what I get is the opposite of intuition: intuition for me comes from the construction or a picture (others are different!), but sometimes a characterization by initial property leads to devastatingly short (and surprising and non-intuitive) proofs. If you’d like an example: it is possible to do part of the right-exactness of tensor product using the fact that it satisfies a universal property. The idea that comes up is precisely the one that is generalized in the section on how (co)limits commute with adjoints and such. After the first time you come up with such an argument as a solution to an exercise, your life will never again be the same. – R.]September 3, 2010 at 11:44 pm

Here is a list of ‘bugs’ I’ve found during the first reading (without 2.7.) and which are not mentioned in the comments above:

-2.2.20. there should be g\circ f: B_{1} \to B_{2} \to A instead of g\circ f: B_{2}\to B_{1}\to A

-page 23 – there is one redundant “\otimes” in the second line from the bottom of page

-2.5.1. – there should be: “B\in\mathcal{B}” instead of “B\in\mathcal{G}” and “G(\epsilon_{B}):GFG(B)\to G(B)” instead of “G(\eta_{B}):GFG(B)\to G(B)”

-2.5.F – formally groupification of group is only the isomorphism, not identity (or should it be: “the identity is the groupification” instead of “groupification is the identity”?)

-2.6.4. – You haven’t defined a quotient of two objects in arbitrary abelian category, but You define homology of complex as a quotient ker g/im f

-2.6.H – the link to 2.6.H from 2.6.H itself looks redundant

-2.6.H(b) – in the second line there should be “H^{\bullet}F\to FH^{\bullet}” instead of “FH^{\bullet}\to H^{\bullet}F”

-last sentence before 2.6.K – it seems that F,G are assumed to be adjoints, but it isn’t mentioned in the text

Now, I have to say that I’m very grateful for Your idea of posting Your lectures here. I’m a student representing ‘wanna-be algebraic geometer’ species with some background from commutative algebra, and I’ll try to work hard with them.

I hope to be useful at least in finding typos.

[Thanks! You have sharp eyes. I’ve fixed/clarified everything (for the next posting). The self-reference in 2.6.H was intended to be a joke (no one else has called it the fernbahnhopf theorem, but I find the name handy to remember which way the arrows go), but you’re not the first person to be perplexed by the self-reference, so I’ve removed it, while keeping the new strange name “fernbahnhopf theorem”. – R]September 4, 2010 at 5:32 pm

Prof. Vakil:

I’m another student who is going to try to read these notes as they’re posted. I took a Hartshorne-based first course in AG last year and found previous incarnations of your course notes very helpful.

Here are some typos that I don’t think others in this thread have caught:

In section 2.5 on adjoints, in your explanation of “natural”, you use L for the left adjoint in one spot instead of F. In exercise 2.5.B, g: F(a) –> B should be g: F(A) –> B, and the map \epsilon_B, when first introduced in that exercise, is going the wrong way.

Right above exercise 2.2.C, “make can that” should be “can make that”.

In 2.3.2 on tensor products, right before “more formally”, you have “n_2, n_2” where you should have “n_1, n_2”, and below that, “genreated” should be “generated”. On the next page, in the definition of A-bilinear, f(m_1 + n_2, n) should be f(m_1 + m_2, n).

Preceding the two “important exercises” on page 25 is a confusing paragraph: you say that “the first exercise deals with localization” when it does not, and then say that “here is a brief introduction” (to localization, presumably) “in case you haven’t seen it before”, but such an introduction isn’t there (and localization was defined two pages earlier).

Finally, in 2.6 and 2.7 which I haven’t thoroughly read yet, a roman I in Proposition 2.6.11 should be a script I, and a subscript r is missing (I think…) in the Approximate Definition 2.7.2 of the differential of a spectral sequence.

Thanks again for these notes, and for this yearlong project. I learned a lot from your notes that I couldn’t find clearly explained anywhere else.

Regards,

Nick

[Dear Nick,Thanks for your sharp-eyed comments (and your thanks)! I think I’ve fixed everything, and the result will be posted by Sat. Sept. 11 (by when I hope to post the next batch of notes). The localization error was the result of a last-minute cut-and-paste (always dangerous!). In Proposition 2.6.11, the odd “I” seems to be what latex does. In 2.7.2, I caught two missing r’s. — Ravi]

September 6, 2010 at 1:19 pm

Dear Ravi Vakil,

I’m also a student trying to follow this online reading course and I am very excited and thankful to have this opportunity. I am halfway through the notes of Aug. 26 and I think I have found two typos that haven’t been mentioned yet:

2.3.2. in the last line there is one tensor symbol too much

2.3.M. the second “are” should not be there

Much more importantly though, I already have two questions! I am not sure if this is the right place to discuss them, please ignore them if this out of place. But I hope by posting them here other people can learn from my problems.

Here they are:

2.2.14 in this example you define a functor h^A that sends an object B to Mor(A,B), the collection of morphisms from A to B. By stating that h^A is a functor from C to Sets you imply that Mor(A,B) is always a set, why is this so? Couldn’t it possibly be bigger?

2.3.S (and 2.3.T) here the fibered product of X with X over Z is mentioned (I hope this is the right way to name it, or that you at least understand what I mean), however, as you have mentioned when introducing the fibered product, it also depends on the morphisms in this case from X to Z. Now in there exercise the only given morphisms are from X to Y and from Y to Z. Is it the composition of the two which is meant for the exercise? Does this also hold true for 2.3.T, where morphisms X_1 to Z and X_2 to Z are needed?

I would assume that my guess for 2.3.S and T is right, but I am confused that you write “natural” morphism and not “unique” morphism – which the morphism would be, if I then interpreted the exercise correctly.

Your “hot fix” on the universal property of localization helped very much, but I think edits in between the main posts are in general not necessary, but rather overwhelming. I like the idea two get something new every two weeks, and only then.

I am very excited to be part of this and to see how it will go, thank you so much for the opportunity!

Best Regards,

Felix

[Dear Felix,Many thanks — I’ve fixed 2.3.2 and 2.3.M (in the version to be posted by Saturday). You’ve also convinced me that I should post “hot fixes” only in case of emergency. About 2.2.14: you are absolutely right. I wanted to avoid such issues (i.e. let people with sufficient expertise patch them themselves, and not distract others). I’ve now included text to this effect, referring back the conventions section. About 2.3.S and 2.3.T, you are indeed right. If almost all readers will come to the correct conclusion (as you did), I’d prefer not to add more text to the exercises. But perhaps people will strongly disagree (or agree) with this. – R.]

September 6, 2010 at 10:05 pm

I will hopefully write up a long list of suggestions later, but here’s one thing:

In Example 2.4.5, the isomorphism is of abelian groups, right? It might then be confusing to say: “The ring . . . is a colimit”. It’s sort of obvious because 2^{-i}Z isn’t a ring, but I thought I would bring it up. I am hopefully not making a fool of myself.

Thank you for what you’re doing.

[That’s a good point, thanks! Now fixed. (I also realized that one $5^{\infty} Z$ should have been $5^{-\infty} \Z$.) – R.]September 7, 2010 at 2:27 pm

Set theoretic comment; feel free to ignore. In exercise 2.2.D of course the functor is essentially surjective and fully faithful. But for me, it is not an equivalence. The reason is that I work with ZFC set theory (virtually all mathematics is based on ZFC), and ZFC does not have choice for classes. Of course this is never a problem as most arguments never use more than a set worth of vector spaces anyway, and any full subcategory of f.d.Vect_k whose collection of objects is a set, and which contains at least one object of any given dimension _is_ equivalent to \cV.

September 10, 2010 at 3:29 pm

Hi Johan,

You’re of course right, one has a problem if one wants to live in a ZFC world (which reasonable people do). I’ve now added a more explicit fudge here. I don’t want people to worry about this issue, which means that they may need to know enough to know not to worry. (I wanted to include this example because it was actually when I learned this example from you that a lightbulb went off in my head that equivalences of categories were really simpler than I thought.) For a more correct response to the set-theoretic issue, Andrew Blumberg sent me the following enlightening email.

“So here’s what I think the deal is. You want to prove that having a pair of functors inducing an equivalence is the same as having a single functor which is full, faithful, and essentially surjective. The proof of course involves constructing an adjoint to such a functor, and you do this by choosing inverse images, essentially. This choice process is where the set theoretic assumptions enter; one has to have some form of the axiom of choice. If you have a class of objects, one might have to use Godel-Bernays set theory and assume choice for classes rather than ZFC (which has choice for sets). If you only work with small categories (with a set of objects), then you’re fine in ZFC, and most of the time small categories suffice. Hope this helps.”

September 10, 2010 at 4:12 pm

We had a bit of discussion on the stacks project blog about ZFC and universes, and Brian Conrad had a beautiful comment which I wholeheartedly agree with!

[Johan, I agree with what I think you mean! – R.]September 12, 2010 at 12:45 pm

A further later comment from Andrew Blumberg (in an email): “I guess one might want to say something about universes, but I don’t like that approach since when people use it, it always feels like they’re waving a wand to ignore problems.”

September 7, 2010 at 7:07 pm

Small typo: In 2.2.16, first paragraph, there is a missing “=”. The relevant parenthetical should read: (Thus F(m_2\circ m_1) = F(m_1)\circ …

[Thanks, fixed! – R.]

September 18, 2010 at 12:08 am

Hi Ravi,

Exercise 2.6.K asks to prove that filtered colimits are exact in an abelian category. Is that not equivalent to Grothendieck’s axiom AB5 (at least assuming the category is cocomplete) and thus not true in an arbitrary abelian category (for example the category Ab^op.) Maybe I am misinterpreting something.

These notes look great, thank you for making them available.

Best,

George

[Hi George,

I don’t have AB5 handy, but I realize I don’t need it to answer your question. You’re certainly right that what I intend there is something milder. I’ve changed this (in the version to be posted next) to be just for modules over a ring, and I state that the same argument will work equally well for categories whose objects can be interpreted as “sets with additional structure”. This isn’t a precise statement of course. But I think it’s better to understand the proof and then see that it applies without change to some other situation you care about rather than carefully making dry-sounding axioms and then proving things from those axioms. I’m fully aware that this point of view (of understanding examples well enough that the theory is clear) lies to one end of a spectrum. – R.]

September 25, 2010 at 6:38 am

Following Bjorn Poonen’s advice, for an index category to be filtered, it must also be nonempty, to avoid the dangers of the empty set. (This change is in the version ~Sept. 25.)

September 28, 2010 at 11:49 am

Dear Ravi,

Spectral sequences should be labelled (column,row),

which I think is the same as what you mean by (x,y).

I’ve never seen any other convention (other than the fancy Adams convention that Tyler Lawson mentions, which however only occurs deep in the interior of algebraic topology); of course I could have just missed something, but I am comfortable claiming that using the other convention will set up a lot of notational conflict with other standard references that people might look at.

Best wishes,

Matt

[I’ll do that (at some point), thanks! As mentioned above, this is now firmly on the to-do list. — R.] [Later comment: now done! — R.]September 29, 2010 at 5:26 am

Dear Prof. Vakil,

I am a student who is trying to keep up with your reading course. I thank you very much for doing this.

I found your category theory section very helpful. Even though this is probably the third or fourth time seeing most of the material, you by far present the material more clearly than any other source I have read.

There is one thing I have never seen clearly presented anywhere. I cannot seem to find a text that clearly describes what a “universal property” is. Though I have now internalized this, it would have been very helpful to understand this more clearly earlier on. In addition, it would be helpful to know why certain universal objects should be initial or final. I can never seem to remember them and have to look back at their universal definitions.

Thanks again,

-Robbie

September 30, 2010 at 4:44 am

Hi Robbie,

I think seeing category theory three or four times in different ways is the best way to internalize it. That’s also true for scheme theory; it’s the rare person who gets a good sense of it the first time through. And about universal properties: I think to understand them well, you have to see examples, which can be of quite different flavors. Any comprehensive formal definition would be so formal as to be even less comprehensible.

Or perhaps something slightly informal might be helpful: if at some point after the first few examples, where you’d seen uniqueness up to unique isomorphism proved (always by “essentially the same argument”), they key idea was explicitly stated, along the following lines. (I haven’t tried to craft the words well yet.)

In each case you’re looking for an object M with some certain auxiliary construction (such as a map to some other things) which pulls back under maps (in other words given N’ –> N, and you have that auxiliary construciton on N, then that determines it on N’). Then the “universal” such M, if it exists, is unique up to unique isomorphism. By universal, we mean that there is a “best” M with that construction, so that any other N with that construction arises via a

uniquemap N –> M.And there is a “dual” version involving properties that “push forward” under maps.

Those two statements informally abstract the notion of universal property. But I don’t know if they are helpful. (You and others should feel free to voice an opinion.)

And about initial and final objects: I think here the best thing is to understand what the initial and final objects are in your favorite simplest meaningful categories (sets, groups, vector spaces, etc.), and in those cases to know why they are initial and final, and to realize in those cases once you’ve thought it through once or twice it’s clear why they are initial or final.

September 30, 2010 at 2:43 am

Dear Ravi,

In your section on spectral sequences you give a proof of the snake lemma. Please correct me if I’m but I think this argument is circular since you need the snake lemma in order to define the spectral sequence in the first place?

Kind Regards,

Ciaran

September 30, 2010 at 4:34 am

Dear Ciaran,

I

thinkthe proof/definition of spectral sequences I give doesn’t use the snake lemma, although I can well imagine that other (perhaps faster or better) proofs do. But please let me know if I’m wrong. I wouldn’t be really bothered if it was circular, as working through how spectral sequence machinery hands you the snake lemma will still give practice with spectral sequences, and keep fear at bay. But if that was the case, I’d want to clearly point out the circularity. best,Ravi

p.s. On a related point, to everyone: If there is a fantastically short or enlightening proof(/definition) of spectral sequences in the literature, I’d be happy to either use that exposition instead, or else give a pointer to the reader.

October 1, 2010 at 3:23 am

Dear Ravi,

I think you hint at the circularity in the middle of p.47 of the sep version; the bit in parentheses. Each time you turn the page, you take (co)homology. In particular, you generate a long exact sequence whose connecting homomorphism requires the snake lemma. I might be wrong which is why I thought I would ask. I agree that it is a nice demonstration but the way it is set out makes it look like you get the snake lemma for free whereas I don’t think you do.

Kind Regards,

Ciaran

January 4, 2011 at 1:22 pm

(My apologies for taking so long to respond!) I still think it is okay — we take cohomology, but nowhere do we use a long exact sequence (coming from a short exact sequence of complexes). (If anyone else can weigh in one way or another, please do!)

January 9, 2011 at 7:03 am

I was under the impression that spectral sequences were a way of computing the cohomology of a double/total complex; taking cohomology along one of the differentials induces a differential in the other direction along which you can take cohomology again. Surely, the snake lemma is implicitly used when we induce the new differentials?

March 14, 2011 at 10:38 am

I’m pretty sure that the argument I give proving that spectral sequences work doesn’t use the snake lemma. (There could be other arguments that do, of course.) If I’m wrong, please let me know. (I should admit that I want most readers not to read my proof of the spectral sequences, and just feel happy that the proof is there and available. But for this to work, it needs to be correct! Darij Greenberg has given me a number of comments in the last month which have helped me clean it up.)

March 14, 2011 at 12:10 pm

There are definitely proofs that spectral sequences work which don’t require one to use the snake lemma; however, they all naturally require some snake-esque chasing. The one which is probably simplest to prove and doesn’t lead to indexing nightmares is the exact couple formalism. However, it takes some doing to “unwind” what is going on in an exact couple to show that, if you have a (nicely) filtered chain complex, you get a spectral sequence whose target is the homology of the complex.

Regarding the comment just after 2.7.2.2, one of the easiest ways I know to remember which is the quotient and which is the subobject is in terms of the differentials: differentials are maps, and maps naturally come out of quotients and into subobjects.

[Thanks Tyler! — R.]December 29, 2010 at 9:25 am

Dear Prof. Vakil,

I am a bit confused in the example where you define localizations of modules using universal properties. You say that the “..map phi … is initial among all A-module maps…”. What is the category in which it is initial – are we looking at a category of maps ? If so, what are the morphisms ? Is it related to the functor you define in 2.2.14 ?

I think I get an idea of what you mean because of the example with localization of rings – but there we were looking at the category of A-algebras – so it made sense to me. I am not sure what the category in the module case is.

You will have to pardon me if my questions above are stupid. This is my first brush with category theory – my background is Computer Science and my main interest in AG is because of Mulmuley’s approach for P vs NP.

January 5, 2011 at 11:40 am

Hi Amitabh,

That’s not a stupid question. Here’s what the objects and morphisms are in this category. The objects are maps M –> N, where elements of S are invertible in N. The morphisms in this category are as follows. A morphism from a_1: M –> N_1 to a_2 M –> N_2 is a map b: N_1 –> N_2 such that b \circ a_1 = a_2. It isn’t necessary to understand this to read on. (I’ve added a parenthetical comment to this effect in this notes.)

Mulmuley spoke in our joint algebraic geometry colloquium with Berkeley and Davis last fall. Despite being an organizer, I unexpectedly had to miss the talk, but by all accounts it was terrific.

January 5, 2011 at 1:38 pm

Dear Prof. Vakil,

Thanks for the response – I had managed to figure this out on my own and because of that I must say I now appreciate the concept of universal property a lot more.

I was able to attend Mulmuley’s seminar at the colloquium that you refer to (I had earlier heard him speak at Carnegie Mellon when I was a student). Though the approach still seems far from a resolution of P/NP, it is fun to learn AG ! Specially with great notes like yours ! Thanks again for what you are doing.

March 16, 2011 at 1:18 pm

Dear Ravi,

You remark in 2.5.1 that checking the fact that the right adjoint of a functor F is determined up to canonical natural isomorphism can be done by a universal property argument. Could you say a bit more about what property you had in mind, and how it is used?

I checked this claim by using the functorial bijections given by the adjunctions together with Yoneda’s lemma, but I didn’t notice any point where I was invoking a universal property.

Also, would it not simplify the exposition a bit if you restricted your attention to locally small categories (i.e. the Homs are sets) rather than an arbitrary category? For instance, you would not need to add that caveat about the Yoneda embedding. I had been under the impression that all categories in algebraic geometry are locally small. Indeed, I don’t think I’ve ever seen any categories in nature which are not locally small.

March 16, 2011 at 1:34 pm

Actually, after going back and reviewing the definitions, it appears that you DO, in fact, define categories so that the Homs are sets. In this case, I think there may be an inconsistency in the notes.

In 2.2.14 you state some concerns “for experts” about Mor(A,B) being a set. Of course, if you required in the definition of categories that Homs are sets, then Mor(A,B) should be a set by definition. So it seems these remarks are unnecessary. This passage was what gave me the impression that you did not require Homs to be sets.

March 17, 2011 at 9:22 am

Thanks for the sharp-eyed comments Rex!

About 2.5.1: I’m not sure what I was thinking at the time, although I have a hard time disentangling Yoneda’s lemma from universal property arguments in my head. I’ve now removed that comment in the notes, because it detracts from the discussion.

About the categorical issue: I must admit that I didn’t even know the correct definition of category until you made me look it up — I’ve been wrong all along! I’ve now stated at the outset that we work with locally small categories. (Chris Davis had caught the issue in the caveat about the Yoneda embedding just a few days earlier, coincidentally enough.)

March 17, 2011 at 9:43 am

Dear Ravi,

I wouldn’t say that you were wrong all along. I myself had believed all along that categories are defined so that the Homs are sets (I think many people define them this way). At some point I heard the term “locally small” and it didn’t click in my head until now that the existence of this term must mean that some people define categories without requiring Homs to be sets.

I still don’t see how one can get much mileage out of category theory for categories that are not locally small. For instance, it is not clear to me what it would mean for two functors to be adjoint. We typically define adjunction to be a natural bijection

Hom(F(a),b) = Hom(a,G(b))

But if one of these sides is not a set, then we have to articulate what it means to be a bijection of proper classes.

Doing category theory without being locally small seems like a huge hassle.

March 17, 2011 at 10:07 am

I agree! But then again, I realize some people need to work in such dangerous places where categories are not locally small. Fortunately, as you say, for the most part the categories one meets in nature *are* locally small.

March 19, 2011 at 3:52 pm

Dear Prof. Vakil,

I was just reading your notes on category theory when I had this question.

Is there a definition of equality of two morphisms in a category when they have the same domain and codomain?

The notion of equality of morphisms is used in category theory, for e.g. in the definition of a monomorphism or epimorphism. But, when are two morphisms equal? I could not find this in your notes, and I get a feeling that this question is really silly. But, I have not been able to come up with a satisfactory definition.

March 20, 2011 at 12:39 pm

Dear Rankeya,

By “domain” and “codomain”, I take it you mean “source” and “target”. If you mean something else, then feel free to follow up. The morphisms from one object to another are a set. So a morphism is an element of this set. Two morphisms are the same if they are the same elements of this set.

best,

Ravi

March 20, 2011 at 8:38 pm

Ahhhhh! Thank you Prof. Vakil. That was indeed simple. Yes, I meant source and target.

March 22, 2011 at 7:12 am

Dear Prof. Vakil,

I just saw some of the earlier comments by Rex. In the definition of equality of morphisms that you have given, the morphisms from one object to another form a set. But, in general (and I know you mention explicitly in your notes that you did not want to get into set theoretic issues) if the morphisms do not form a set, is there some additional complexity that is introduced in the definition of equality of two morphisms?

You do not have to answer this question if you do not want. But, I guess there must be some definition of equality of objects in a collection that is not a set. Also, when I asked this question, I had the general case in mind too, but it did not strike me when I read your comment earlier, that you said that the morphisms between objects form a set.

Then again, as bith Rex and you agreed, most categories we work with are locally small.

March 24, 2011 at 8:34 am

In the more general notion of classes, there is still the notion of equality of elements, so nothing changes. We still have a notion of when two objects are “the same” (which is different from “isomorphic”).

I understand where you are coming from though: there are many natural places where you don’t want to say the word “the same” — for example, in the notion of product in a category in general, it is only defined up to unique isomorphism. So to help reassure yourself, think through morphisms in any concrete case you find interesting, and you will realize that there is no question about what “same” means.

December 30, 2011 at 3:51 pm

The following is from Aaron Mazel-Gee at Berkeley. An earlier version looked good to me, but I suspect Aaron and I are missing something — does anyone else buy this?

It seems to me that 2.6.G can be strengthened, although I must be missing something because Eisenbud has the same statement and my proof is the first obvious thing to try…

I claim that if is a finitely generated -module (not necessarily finitely presented), then . We have an obvious rightward map

(which is fairly obviously well-defined).

First, this is surjective. Given a map of -modules, we get a map of -modules. We can bring the images in of the generators of to a common denominator since there are only finitely many of them, so our map is the image of some .

On the other hand, this is also injective. If is the zero map , then it is zero when we precompose with . This means that for each , , so there is some so that . Set . Then annihilates the images of all the generators of , hence it annihilates the images of all of its elements. So , meaning that in the first place.

December 31, 2011 at 8:59 am

In the argument for surjectivity, it is not clear to me if is well-defined. The following counterexample might be helpful: http://math.stackexchange.com/a/81671.

January 1, 2012 at 6:13 am

I fell into this trap too (1-line comment on 2.6.G buried in one of the lists of my comments placed at end of “Eighteenth Post”).

Yifei Zhu has pointed out where the error lies, and where there is a nice counterexample. The explanation could be a little simpler and is basically this: When you try to lift in that example to get a morphism M –>N, there are many relations in M and it is impossible to have all of them satisfied in the image.

January 6, 2012 at 1:14 am

Ah, that is subtle. Thanks for taking a look!

January 14, 2012 at 10:50 am

Yes, thanks Yifei! That was very helpful — as is Qing Liu’s enlightening example in that math.stackexchange reference.

January 29, 2012 at 5:09 pm

Dear Prof. Vakil,

I think the backward implication of exercise 2.3.V. needs the additional hypothesis that the projections of X \times_{Y} X —-> X are equal. Otherwise here is a counter example:

Let X = {1,2}, Y = {1}, and f:X—->Y be the constant map. Then certainly the pullback exists, but f is not a monomorphism.

Regards,

Rankeya Datta.

January 29, 2012 at 5:13 pm

Actually never mind. I over looked the comment that X —-> X \times_{Y} X is an isomorphism…

[Great! — R.]July 18, 2012 at 6:12 pm

I’d mentioned a book of Kashiwara-Schapira for some background in case people want to do some further reading. I don’t feel strongly about this, as I don’t want people to do background reading, or to feel there is some big machinery they have to make friends with first, but I don’t mind having good references. Peter Johnson recommended McLarty’s “Elementary Categories, Elementary Toposes”. If anyone has any opinions on that book (or any others), feel free to mention them here!

February 19, 2013 at 12:19 pm

I had written in 1.6.12: In an abelian category, colimits over filtered index categories are exact.

Yang Zhou, an undergraduate at Zhejiang University, wrote to me:

“There is an explicit counter-example to your statement. Since the axioms for abelian categories are self-dual, it suffices to find an example in which filtered limits fail to be exact. The category is the category of abelian groups. Let be a fixed prime. Then for any possitive interger , is clearly exact. This is an exact sequence of inverse systems in an obvious way. But if we take the limits, we get , which is no longer exact. The p-adic integers is much larger a group than the integers.”

He is absolutely right, and this is now fixed, in the version likely to go public in the next week. (The statement actually proved in the notes is for the category of modules over a given ring, which is what we actually used.) Andrew Blumberg recommended that I actually give this canonical example in the notes, because it is so short, so I am doing so.

Thank you Yang Zhou!

October 18, 2013 at 1:35 pm

Zev Rosengarten points out that this isn’t exactly what we want; the dual version we seek is an example in which

cofilteredlimits fail to be exact. Back to square one…October 19, 2013 at 9:27 am

Dear Ravi, Zev Rosengarten isn’t correct on this one. When dualizing the exactness condition for filtered colimits, we won’t change the index category, so it’ll remain a filtered category. This is explicitly stated in several places, including Stacks (Tag 079A) and Weibel (p.427). Just as a side remark, the index category in Yang Zhou’s example is just the natural numbers with opposite order, which is both filtered and cofiltered.

[I will respond to this soon. – R, Oct. 1, 2015]