Welcome to MathOverflow

Question

2

1

Let's open R.O.Wells "Differential Analysis on Complex Manifolds" p. 53 and have a look at the Proposition 3.5 stating that all fine sheaves are soft (over a paracompact Hausdorff $X$). In the proof we consider the covering of a closed $S \subset X$ by open $U_i$. Why can't we just take a single $U_1$ covering $S$? A section $s$ over $S$ by definition is an element of a direct limit, so it should have a representative in some neighborhood of $S$ and we could just set $U_1$ to be that neighborhood. Or couldn't we? But the proof is more complicated than that and I'm confused...

flag	asked May 12 2010 at 13:49 Kestutis Cesnavicius 1,198●5●12

3

Direct limit presheaf is not a sheaf. Must sheafify it. (Direct sum is a special case of this.) – BCnrd May 12 2010 at 14:18

Sure, but I don't think we need that. – Kestutis Cesnavicius May 12 2010 at 15:38

3

@Kestutis: the issue is the same as for sheaf pullbacks: the initial construction is just a presheaf, not a sheaf, so need to sheafify. So a section over $S$ is not generally an element of a direct limit; a sheafification intervenes. That's the error. The example of direct limit sheaf is just a toy version of the same issue. – BCnrd May 12 2010 at 15:42

4

@Kestutis: It is proved in Godement's book on sheaf theory that on such nice spaces as you consider, a section over a closed set does actually arise from a section over an open around the closed set, but that is not the definition (for Godement) and its proof requires some real work. I don't know offhand what foundations Wells is using, but if he takes the theorem in Godement's book as a "definition" then maybe it shifts the burden of work to proving that the definition has good properties (e.g., behaves like a sheaf), so possibly Wells is running into that issue in the proof you ask about? – BCnrd May 12 2010 at 15:46

Oh, it looks like this is indeed a reason. Wells defines sections over a closed $S$ to be elements of the direct limit, and he doesn't really care if that is a sheaf or not (just an abelian group or whatever for each closed $S$). And in this later theorem he suddenly starts using this other viewpoint of a sheaf associated to a presheaf that you're pointing out. Thanks a lot! I'll have a look at Godement. – Kestutis Cesnavicius May 12 2010 at 18:15

show 1 more comment

A question on a proof that fine sheaves are soft

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.