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I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started learning Scheme theory (Hartshorne Ch II). My question is :

Shall I read coherent sheaves and the cohomology from the translation of Serre's FAC or from the Hartshorne, this is going to be my first encounter with Coherent sheaves. Do you recommend FAC as an introduction to Coherent Sheaves.

My prime interest at present is the derived categories of coherent sheaves on projective varieties, specially the work of Tom Bridgeland and the Mirror Symmetry.

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Your question appears to be quite borderline and subjective to me. Is there any way you can make your question more specific? –  Ryan Budney Nov 14 '10 at 6:43
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Moreover, I imagine this question would be more appropriate to pose to someone who knows you and your sensibilities. From your profile it looks like you're a grad or undergraduate student at Kansas State University. There should be several profs in the department willing to help you with this query. –  Ryan Budney Nov 14 '10 at 6:56
    
I voted to close, per Ryan Budney's comments. This is the definition of "too localized". –  Harry Gindi Nov 14 '10 at 7:21
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Dear J, I think that FAC is well worth reading. It contains a large bulk of the results from Hartshorne Chapter III, in a somewhat streamlined expositional setting. I recommend at the same time that you read Zariski's article from the Bulletin of the AMS on Sheaf Theory. (It is Zariski's survey of FAC, with a discussion of how the results there relate to various important geomeric questions.) –  Emerton Nov 14 '10 at 14:29
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Wait just a minute, Harry: the full embedding of sheaves into presheaves preserves limits, and so to compute a limit of a diagram of sheaves, you can just compute the limit as presheaves: the limit $L$ there is guaranteed to be a sheaf already, and so sheafification there is superfluous (i.e., the unit component $L \to iaL$ is already an isomorphism, where sheafification $a$ is left adjoint to $i$). On the other hand, to take a colimit of a diagram of sheaves, the prescription is to first embed in presheaves, take the colimit there, and then sheafify (there it is not superfluous). –  Todd Trimble Nov 14 '10 at 16:26
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3 Answers

up vote 15 down vote accepted

As always, the source you use may be related to what your goals are. To give some perspective, recall there are several ways to define sheaf cohomology, and Serre and Hartshorne feature different methods. Serre used Cech cohomology, and there the important long exact sequence property does not always hold. He was able to prove however that it does hold for "coherent" sheaves. One big advantage of Cech theory is its easier computability in specific cases, such as on projective space. Hartshorne presents first Grothendieck's theory of derived functor cohomology, but then proves it agrees with Cech cohomology before using that theory to compute the cohomology of coherent sheaves on projective space. But if you want to learn the derived functor theory you must choose Hartshorne over Serre.

The distinction made above between schemes and varieties is also relevant. Serre teaches Cech cohomology on varieties,and Hartshorne presents derived functor cohomology on schemes. If you are only interested in varieties, or prefer learning cohomology in the easier setting of varieties, then you may prefer Serre's FAC. Another good source is the book Algebraic Varieties by George Kempf, where the derived functor theory is presented on varieties and used for basic computations, including coherent cohomology of projective space and even the full Riemann Roch theorem, before being linked with Cech theory. So if you want to learn to make computations with the abstract derived functor theory you might prefer George's treatment, although some details are missing there, and some misprints exist.

Finally there are slight differences in Serre's and Hartshorne's results which can be relevant in some settings. E.g. in Beauville's book on surfaces, he uses Cech theory to relate rank two vector bundles on curves with ruled surfaces. To prove that all ruled surfaces arise from vector bundles he then uses Serre's result that Cech H^2 vanishes on a curve with coefficients in any sheaf coherent or not. (He also gives a second argument.) But this vanishing theorem for Cech cohomology does not follow from Hartshorne's treatment, since he proves vanishing for derived functor cohomology but does not relate derived functor and Cech cohomology on non coherent sheaves above degree one.

There is a sentence in Hartshorne, at the end of chapter III, section 2, page 212 in the 1977 edition, which says that Serre proved vanishing for "coherent sheaves on algebraic curves and projective algebraic varieties", whereas the correct statement would be that he proved it "for curves, and for coherent sheaves on projective algebraic varieties". Since Robin is very careful, one wonders whether some well intentioned copy editor did not change this sentence's meaning unwittingly to make it flow better.

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In order to do this stuff with derived categories, you might consider learning a little bit about derived functors :) –  Daniel Pomerleano Nov 15 '10 at 5:24
    
A source for sheaves in the context of derived categories, is Methods of homological algebra, by Gelfand and Manin. I recommend studying the more elementary sources above first though. –  roy smith Nov 16 '10 at 20:09
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Dear J, here is a little technical warning which might be relevant to your question.

If you open Hartshorne and read the definition of "coherent" (Chapter II, §5, page 111) you might get the impression that the structural sheaf $ \mathcal O $ of a scheme is coherent: after all the scheme can be covered by affine open subsets $U_i= Spec (A_i)$ on which $\mathcal O |U_i =\tilde A_i$ and $A_i$ is certainly finitely generated over itself . But this is false: there exist affine schemes whose structural scheme is not coherent! So where did Hartshorne make a mistake? Actually he didn't make any : in the next paragraph he writes that he will only mention coherent sheaves on noetherian scheme "because the notion of coherence is not at all well behaved on a nonnoetherian scheme" (loc.cit.) .

Serre on the other hand gives the general definition, applicable to all schemes without any noetherian hypotheses and even to complex spaces (and even to ringed spaces) : this is not surprising since the notion of coherence comes from complex geometry ! He also proves the main general result on coherent sheaves: if in a short exact sequence involving three sheaves of $\mathcal O$-modules two of them are coherent, then the third is also coherent. Serre's definition is the one used in E.G.A. and in De Jong and collaborators' Stack Project ( in those references you will find statements starting with "Suppose $\mathcal O _X$ is coherent...").

So that might be another reason, beside Peter's excellent ones, to read FAC !

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When I opened EGA 0, I found that a lot of the basic facts about coherent sheaves (over general locally ringed spaces) weren't actually proved but referred to FAC. –  Akhil Mathew Nov 15 '10 at 3:41
    
A very relevant observation, Akhil. –  Georges Elencwajg Nov 15 '10 at 13:02
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The question is indeed a little bit "localized", so in the MO tradition of casting an answer in more general terms, I claim that the following is a reasonable answer to:

Question: I wish to learn about subject X. I have my hands on references Y and Z, but don't know which one is better. What should I do?

Answer: Start by reading a little bit of both: e.g. put in a few hours with both Y and Z. Probably at some point you'll find one to be more appealing than the other -- without loss of generality, let's say it's Y -- so go ahead and read Y more thoroughly, including taking some notes and working through some exercises (whether or not exercises are actually included!). Then, if and when you're still interested, go back and take a look at Z with your more experienced and critical eye: look both for similarities to and differences from Y. With regard to the former, are you able to translate from Z to the language of Y? (If you are finding this difficult, it may not be worthwhile to continue reading Z, depending on how satisfied you were with Y and what your goals are.) With regard to the latter: what does Z do that Y does not? Take note of this material. Do you see how it could have been done in the language of Y, or is the setup of Z somehow essential?

Let me add that this really is nonlocalized advice in that I have never given Serre's FAC more than the briefest flip through, but nevertheless I imagine that what I say would be applicable and useful in this particular situation (among others).

Added: Okay, let me say something about the specific situation of this question. Serre's FAC predates scheme theory so is not written in scheme theoretic language but rather a somewhat modernized version of Weil-style foundations. Therefore you will not be able to get away with only reading FAC: you will need to make contact with scheme-theoretic ideas and constructions so for this you will have to read Hartshorne or some other similar material (e.g. EGA) later on anyway. I think it is fair to say that most students of algebraic and arithmetic geometry in the last 30 years have put in significant time with Hartshorne's book at some stage of their career.

So the shorter and more standard modern path is just to read Hartshorne (that's what I did, for instance). However, Serre is such a wonderful mathematician and expositor that I think it is worthwhile to read through all of his writings at some point. So I would say that reading through FAC first -- maybe not in complete detail, but reading for the main ideas -- would give you a good perspective for your future studies and work. I'll read it myself when I get the chance!

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Pete: Sorry to be picky, but I would argue that Serre's viewpoint is much closer in spirit to schemes than than to Weil's foundations. Of course I agree that any serious student has to learn schemes eventually, but learning FAC might be a reasonable bridge to it. In Mumford's Red Book, he starts off with FAC-style varieties, and switches to schemes a 3rd of the way into it. –  Donu Arapura Nov 14 '10 at 12:35
    
@Donu: not at all, your comment seems reasonable and helpful. –  Pete L. Clark Nov 14 '10 at 21:11
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