As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter.

Question 1. What is sheaf cohomology? I have a vague idea that it has something to do with right derived functors, but this seems rather far removed from the (admittedly very little) cohomology of (co)chain complexes I do know. I would also like to know why sheaf cohomology appears to be so much more fundamental in algebraic geometry than algebraic topology—for instance, I will be taking second courses in algebraic geometry and topology this coming autumn, but sheaf cohomology only appears in the former, suggesting that perhaps sheaf cohomology is not as relevant in basic algebraic topology. (For example, is there an ‘intuitive’ reason why de Rham cohomology cannot be made to work for algebraic varieties?)

Question 2. Are there any good introductions to sheaf cohomology in a general context? I have tried reading Chapter III of Hartshorne, but very little is getting through, perhaps because I'm not yet comfortable with schemes. A different take—perhaps with an emphasis on manifolds, say—may prove more accessible to me, but since I also need to learn it in the context of algebraic geometry, it would be nice if there were a single text which introduces the theory with applications in both subjects.

Zhen Lin
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    This [MO-thread](http://mathoverflow.net/questions/38966/what-is-sheaf-cohomology-intuitively) may be of interest. Also, two standard references are [Godement](http://books.google.com/books?id=CUZAAQAAIAAJ) and [Bredon](http://books.google.com/books?id=zGdqWepiT1QC). The topological context is treated well in [Iversen](http://books.google.com/books?id=rwzDQgAACAAJ). For a very abstract approach there are the books by Kashiwara and Schapira: [Sheaves on manifolds](http://books.google.com/books?id=qfWcUSQRsX4C) and [Categories and Sheaves](http://books.google.com/books?id=K-SjOw_2gXwC). – t.b. Jul 31 '11 at 13:49
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    (I'm only mentioning the books of KS for the sake of completeness - also, because I know about your interest in category theory in general, and the newer Categories and Sheaves may have many constructions of interest to you. I would definitely *not* recommend them as an intro to sheaf theory). You are certainly aware of [Mac Lane-Moerdijk](http://books.google.com/books?id=SGwwDerbEowC) but IIRC it doesn't have much sheaf cohomology -- if any at all. Also for the algebraic geometry context, you might want to look at [Shafarevich](http://books.google.com/books?id=jZW2-UW4DkMC). – t.b. Jul 31 '11 at 13:55
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    @Theo: Thank you for the suggestions. The absence of sheaf cohomology in Mac Lane and Moerdijk is precisely why I am asking this question—the authors regret its omission yet I do not see its relevance, so I am just very puzzled. I don't read French (yet), unfortunately, and the MR review for Bredon's text seems rather critical. (But perhaps it wasn't a fair review?) [continued] – Zhen Lin Jul 31 '11 at 14:24
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    The accepted answer on the MO thread seems to be applicable to cohomology in all its forms—in the sense of measuring ‘obstructions’—and I can imagine that the cohomology of, say, the $\Omega^1$ sheaf might be connected with de Rham cohomology, but how do the right derived functors magically give the same answers, without ‘knowing’ anything about $\Omega^n$ for $n > 1$? – Zhen Lin Jul 31 '11 at 14:25
  • Does Akhil's answer address your second comment satisfactorily for you? Concerning the first comment: I think the review of Bredon is not entirely fair, as it definitely has a bias towards the then very fashionable triples (= monads = standard constructions). Although I agree with many points made there. Bredon's book is definitely not a book for me, but many people do like it, that's why I mentioned it. I'm not aware of an English translation of Godement. – t.b. Jul 31 '11 at 16:42
  • Can anyone comment on Strooker's "Introduction to Categories, Homological Algebra and Sheaf Cohomology"? – Matthew Towers Jul 31 '11 at 17:13
  • @mt_: I never had this book in my hands. I don't know what to make of the [review on MathSciNet](http://www.ams.org/mathscinet-getitem?mr=498767) which reads in its entirety(!): *"This is a well-written textbook on category theory with an emphasis on homological algebra and cohomology of sheaves."* Doesn't seem to have excited the reviewer very much... – t.b. Jul 31 '11 at 17:34
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    @mt_: Here is a [much more in-depth review](http://projecteuclid.org/euclid.bams/1183544904) - it looks like it's a nice book. – Zev Chonoles Jul 31 '11 at 19:49
  • @Zev and Theo: many thanks for the reviews. – Matthew Towers Jul 31 '11 at 19:51
  • @Zev: This is a brilliant review (what do you want: it's by the great [Lambek](http://en.wikipedia.org/wiki/Joachim_Lambek) of all people!). I don't like the given proof of the snake lemma very much though - I prefer the one I give in Cor. 8.13 and Ex. 8.15 [here](http://projecteuclid.org/euclid.bams/1183544904), as Hilton's squares don't generalize well beyond abelian categories. Two asides: 1. The next book review is on the book *Convexity in the theory of lattice gases* by fellow MSE user Robert Israel. 2. It would be better to use the permanent link (displayed at *links and identifiers) :) – t.b. Jul 31 '11 at 20:38

2 Answers2


Sheaf cohomology is the right derived functor of the global section functor, regarded as a left-exact functor from abelian sheaves on a topological space (more generally, on a site) to the category of abelian groups. In fact, one can regard this functor as $\mathcal{F} \mapsto \hom_{\mathrm{sheaves}}(\ast, \mathcal{F})$ where $\ast$ is the constant sheaf with one element (the terminal object in the category of all -- not necessarily abelian -- sheaves, so sheaf cohomology can be recovered from the full category of sheaves, or the "topos:" it is a fairly natural functor.

de Rham cohomology can be made to work for arbitrary algebraic varieties: there is something called algebraic de Rham cohomology (which is the hyper-sheaf cohomology of the analog of the usual de Rham complex with algebraic coefficients) and it is a theorem of Grothendieck that this gives the usual singular cohomology over the complex numbers. Incidentally, sheaf cohomology provides a very simple proof that de Rham cohomology agrees with ordinary cohomology (at least when you agree that ordinary cohomology is cohomology of the constant sheaf, here $\mathbb{R}$) because the de Rham resolution is a soft resolution of the constant sheaf $\mathbb{R}$, and you can thus use it to compute cohomology.

Sheaf cohomology is quite natural if you want to consider questions like the following: say you have a surjection of vector bundles $M_1 \to M_2$: then when does a global section of $M_2$ lift to one of $M_1$? The obstruction is in $H^1$ of the kernel. So, for instance, this means that on an affine, there is no obstruction. On a projective scheme, there is no obstruction after you make a large Serre twist (because it is a theorem that twisting a lot gets rid of cohomology). Sheaf cohomology arises when you want to show that something that can be done locally (i.e., lifting a section under a surjection of sheaves) can be done globally.

$H^1$ is also particularly useful because it classifies torsors over a group: for instance, $H^1$ of a Lie group on a manifold $G$ classifies principal $G$-bundles, $H^1$ of $GL_n$ classifies principal $GL_n$-bundles (which are the same thing as $n$-dimensional vector bundles), etc.

Also, sheaf cohomology does show up in algebraic topology. In fact, the singular cohomology of a space with coefficients in a fixed group is just sheaf cohomology with coefficients in the appropriate constant sheaf (for nice spaces, anyway, say locally contractible ones; this includes the CW complexes algebraic topologists tend to care about). For instance, Poincare duality in algebraic topology can be phrased in terms of sheaves. Recall that this gives an isomorphism $H^p(X; k) \simeq H^{n-p}(X; k)$ for a field $k$ and an oriented $n$-dimensional manifold $X$, say compact. This does not look very sheaf-ish, but in fact, since these cohomologies are really $\mathrm{Ext}$ groups of sheaves (sheaf cohomology is a special case of $\mathrm{Ext}$), so we get a perfect pairing $$ \mathrm{Ext}^p(k, k) \times \mathrm{Ext}^{n-p}(k, k)\to \mathrm{Ext}^n(k,k)$$ where the $\mathrm{Ext}$ groups are in the category of $k$-sheaves. This can be generalized to singular spaces, but to do so requires sheaf cohomology (and derived categories): the reason, I think, that for manifolds those ideas don't enter is that the "dualizing complex" that arises in this theory is very simple for a manifold. You might find useful these notes on Verdier duality, which explains the connection (and which mostly follow the book by Iversen).

Akhil Mathew
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    Plus there is Serre's GAGA and/or the Artin comparison theorem (for the etale site) that shows we get the same answer in lots of other situations as well. – Matt Jul 31 '11 at 20:27
  • @Akhil: I know that $\Gamma \cong \mathrm{Hom}(1, -)$. (This isn't unintuitive at all, at least in the presheaf topos $\mathbf{C}^{\mathrm{op}} \to \mathbf{Set}$!) What I am not so familiar with is this business with derived functors. Assume I don't know any homological algebra. Could you explain, informally, why derived functors are relevant? I still find it mysterious how the cohomology of the injective resolution should magically match up with the cohomology of a specific cochain complex. I'm also curious as to how ‘constant’ sheaves manage to retain enough topological data to be useful. – Zhen Lin Aug 01 '11 at 01:18
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    @Zhen: There is a general fact about derived functors: given $F$ (say, right-exact, and on nice abelian categories), if you want to compute the derived functors $R^i F(X)$ for an object $X$, you don't *have* to use an injective resolution. Rather, if you have a resolution $X \to J^\bullet $ where $J^\bullet $ is a complex of $F$-acyclics, then you can compute $R^i F(X)$ via the cohomology of $F(J^\bullet)$. The point is that the sheaves of $i$forms are acyclic with respect to the global section functor, being soft. This is the reason de Rham cohomology coincides with sheaf cohomology, ... – Akhil Mathew Aug 01 '11 at 01:53
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    ...even though the de Rham resolution is generally not a resolution by injectives. I would also point out that the cohomology depends very much on the space, so the data in the cohomology comes not from the coefficient group as from the space. For instance, suppose you have a cover $\mathfrak{A}$ of your space such that every intersection is either empty or contractible: then, just by encoding which intersections are empty (data that a constant sheaf can detect!), you can recover the (weak) homotopy type of your space. – Akhil Mathew Aug 01 '11 at 01:56
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    The fact I mentioned in my first comment, incidentally, is not that mysterious. Here is an exercise: given a quasi-isomorphism between bounded-below complexes of $F$-acyclics, applying $F$ to the map between both complexes gives a new quasi-isomorphism. (To do this, form the mapping cone, after which you reduce to showing that applying $F$ to a bounded-below, acyclic complex of $F$-acyclics gives an acyclic complex, which is a fun exercise in homological algebra.) Given this, it is easy to check that any $F$-acyclic resolution will do, not necessarily an injective one. – Akhil Mathew Aug 01 '11 at 01:57
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    @Akhil: I've just had a chance to think about your comments carefully. I can see that the fact that the cohomology of the constant sheaf agrees with de Rham cohomology follows from the fact that acyclic resolutions give the same answer as injective resolutions. (I was at first puzzled about how the de Rham complex could even be exact, let alone be an acyclic resolution, but then I realised I needed to look at stalks rather than sections.) What's still mysterious to me is why acyclic resolutions should give the same answer. Could you provide a reference or an informal explanation? – Zhen Lin Aug 02 '11 at 08:43
  • (A reference for the claim that soft sheaves are acyclic would also be much appreciated.) – Zhen Lin Aug 02 '11 at 08:48
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    @Zhen: Let's see, this should be in books on homological algebra; for instance, I think it's in the first of Lang's two chapters on the subject (in "Algebra"). This is a general fact about derived functors, but the short explanation is that you can get a quasi-isomorphism of such an acyclic resolution into an injective resolution. So we want to show that applying $F$ preserves the quasi-isomorphism property. To do this, it suffices to show that the mapping cone is acyclic. In other words, the key lemma is that $F$ of a bounded-below, acyclic complex of $F$-acyclics is exact. – Akhil Mathew Aug 02 '11 at 13:58
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    ...and this can be done directly (by induction). Iversen's book should contain the fact about soft sheaves. – Akhil Mathew Aug 02 '11 at 13:58
  • @Akhil: Thanks for the reference. To think that one textbook would go from simple groups, rings, and modules all the way up to homological algebra...! I will have a look at it, and Iversen's book, sometime when I can obtain it. – Zhen Lin Aug 03 '11 at 03:52

Are there any good introductions to sheaf cohomology in a general context?

A source is J.-P. Serre's Faisceaux Algébriques Cohérents. It is very readable. Although the language of algebraic geometry is a bit older, everything carries over to schemes without any problems. A translated version is also available.