Does anyone know any interesting examples of flasque sheaves? Ideally, I would like to see one that both arises naturally and is geometric in some sense. On the other hand, I know so few examples other than direct products of stalks that I would be happy to see anything new.

$\begingroup$ The sheaf of rational functions on a variety. $\endgroup$– user2035Feb 17, 2011 at 13:42

1$\begingroup$ Skyscraper sheaves (and their direct products) are flasque. $\endgroup$– Charles StaatsFeb 17, 2011 at 13:44

$\begingroup$ Another very common way you get flasque sheaves is as pushforwards of other flasques. Sometimes via other functors as well. $\endgroup$– Karl SchwedeFeb 17, 2011 at 13:48

$\begingroup$ I probably should have been less vague in my statement. Other than direct products of stalks or constant sheaves on irreducible spaces, I don't know of any other examples of flasque sheaves. So I was hoping for something other than these. $\endgroup$– user332Feb 17, 2011 at 17:01

$\begingroup$ I am having a hard time thinking of unusual examples of flasque sheaves, but an easier time thinking of reasons why the usual examples are useful, hence perhaps "interesting". E.g. Karl's example reminds that this fact, plus Serre's vanishing theorem, implies affine morphisms do not change cohomology, i.e. the affine pushforward of any sheaf still has the same cohomology. $\endgroup$– roy smithFeb 18, 2011 at 17:48
4 Answers
the field of rational functions $\mathcal K_X$ on an integral scheme $X$ ( for example an algebraic variety) is flasque and so is the sheaf of its invertible elements $\mathcal K^\ast_X$. This has as a nice consequence that the divisor class group $CaCl(X)$ of Cartier divisors on $X$ is isomorphic to the Picard group $Pic(X)$ of isomorphism classes of line bundles on $X$. Indeed we have an exact sequence of sheaves of abelian groups on $X$:
$$0\to \mathcal O^\ast_X \to \mathcal K^\ast_X \to \mathcal K^\ast_X/ \mathcal O^\ast_X \to0 $$
Taking the associated long exact sequence of cohomology we get the portion
$$\Gamma (X,\mathcal K^\ast_X) \to \Gamma (X, \mathcal K^\ast_X/ \mathcal O^\ast_X ) \to H^1(X,\mathcal O^\ast_X) \to H^1(X,\mathcal K^\ast_X)$$
The cokernel of the first arrow is precisely $CaCl(X)$, whereas the cohomology group $H^1(X,\mathcal O^\ast_X)$ is the Picard group $Pic(X)$. And now for the sting: $H^1(X,\mathcal K^\ast_X)=0$ because $\mathcal K^\ast_X$ is flasque, hence acyclic ! And we have our isomorphism $CaCl(X) \simeq Pic(X)$, the paraphrase of which being that every line bundle on $X$ comes from a Cartier divisor, unique up to linear equivalence..

$\begingroup$ Dear roy, I'm very happy to read you here, but your interesting example deserves a better fate than being just a comment to my answer! Why not post it as a new answer , so that I and others can show our appreciation by upvoting it? $\endgroup$ Feb 17, 2011 at 22:47

$\begingroup$ Thank you Georges, I have moved it. I want to emphasize however this is just a brief account of the beautiful discussion in George Kempf's terrific book. Unfortunately this book has no major publisher so many have not seen it. $\endgroup$ Feb 18, 2011 at 2:43

$\begingroup$ Dear Georges, I came to this answer of yours googling for some references. Just a notational suggestion: in my opinion, it is better to write $\textrm{DivCl}(X)$ instead of $\textrm{Cl}(X)$ for the divisor class group of Cartier divisors. In fact, $\textrm{Cl}(X)$ usually denotes the divisors class group of Weil divisors, and the two groups are in general different, unless our scheme $X$ is locally factorial (for example, smooth). By the way: what is the standard reference where I can find the proof that $\mathcal{K}_X$ is flasque for a noetherian integral scheme? $\endgroup$ Jul 8, 2013 at 10:34

$\begingroup$ Dear @Francesco, I agree that there is some ambiguity with the notation $Cl(X)$. I have edited my answer and adopted Hartshorne's a little clumsy but unambiguous notation $CaCl(X)$.The sheaf $\mathcal K_X$ is flasque on any integral scheme $X$ because it is constant, with value the generic stalk $\mathcal O_{X,\eta}$.Cf. Qing Liu's book, Chapter 7, Definition 1.13, page 255 (No noetherian condition is necessary). $\endgroup$ Jul 8, 2013 at 12:17

1$\begingroup$ E' un piacere, Francesco. [Sono appena tornato da Firenze e vorrei esprimermi in italiano un' ultima volta:)] $\endgroup$ Jul 8, 2013 at 12:23
In the spirit of Georges' beautiful example, note that the usual computation of cohomology of an invertible sheaf on a complete curve proceeds largely by means of the simplest types of flasque sheaves. Recall given L, the natural map from rational sections of L to the "principal parts" of those sections, is a flasque resolution of L, so computes H(L). I.e. the induced map on global sections is a linear map of infinite dimensional spaces with finite dimensional kernel and cokernel: H^0(L) and H^1(L). In particular, since this is a 2 step complex, H^r(L) = 0 if r > 1.
To compute more one typically approximates this resolution by a smaller one. Restricting to a fixed divisor D, we get a subresolution L(D)>L(D)D, with finite diml global sections, hence more useful for computing H(L). Since L(D) restricted to D is also flasque, we get the formula chi(L(D))  chi(L) = deg(D), for all L,D. Using the result in Georges' answer above, this includes weak Riemann Roch: chi(L)  chi(O) = deg(L).
This is as far as we can go with only flasque sheaves since L(D) is not flasque. But if we choose D with H^1(L(D)) = 0, we have an acyclic subresolution L(D)>L(D)D, of the original flasque resolution of L. This gives a map of fairly explicit finite dimensional spaces with kernel and cokernel isomorphic to H^0(L), H^1(L).
Thus most of the standard theory of invertible sheaves on curves arises from these concrete examples of the simplest flasque sheaves, i.e. constant sheaves of rational sections as in Georges' answer, and direct sums of stalks, illustrating further how useful those apparently trivial cases can be. (see Kempf, Abelian Integrals.)
I don't think that they could be very geometric unless you regard spaces with Zariskilike topology as geometric. For example, sheaves naturally arising on smooth manifolds are rather soft, not flasque.
But on an irreducible topological space (e.g. an algebraic variety), there are examples. For example, any locally constant sheaf is flasque.
A useful example is that of injective modules. Assume your space $X$ is endowed with a sheaf of local rings $\mathcal{O}_X$. Then any injective $\mathcal{O}_X$module is flasque. I don't think that this is geometric though, because injective modules are rather artificial monsters used to define derived functors than naturally arising objects.
Edit. Concerning the title of your question, I think of a flasque sheaf as a synonym for a very very uninteresting sheaf (i.e. for which most statements become trivial).
An exemple of flasque sheaf is the sheaf of hyperfunction. It has important application in the theory of Dmodules.

2$\begingroup$ This sounds interesting. Would you mind saying more? $\endgroup$– user332Feb 18, 2011 at 12:13