Talk:Injective sheaf

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I think "flabby" is much more common than "flasque" (in functional analysis, at least - and of course, outside French speaking countries). — MFH: Talk 20:47, 13 May 2005 (UTC)

[edit] Question

What does "homomorphisms from the sheaf to itself" mean? Is this referring to sections? - Gauge 07:18, 8 February 2006 (UTC)

Yes, by definition a homomorphism of sheaves is a homomorphism of all groups of sections fitting together under restriction maps. In this case the homomorphisms constituting the partition of unity map a section s over some open U to f_i(x)·s, where the sum of the f_i is 1 at every point and f_i is zero outside some U_i. Jakob.scholbach 02:45, 18 April 2007 (UTC)

[edit] Calculations

I had been under the impression that the reason flasque sheaves were called upon is because flasque resolutions are easier to compute than injective resolutions. Are flasque sheaves really useless? —vivacissamamente 14:07, 30 December 2006 (UTC)

No, not at all. Your impression is absolutely correct. For example cohomology (in the sheaf-theoretic sense) of a manifold with ℝ-coefficients is defined by injective resolutions, which are hard to get your hands on. But there is a fine resolution, consisting of smooth differential forms, so the cohomology can be computed using global sections of the smooth de Rham complex. Similar examples exist for flasque sheaves: skyscraper sheaves are flasque and come in quite handy, for example when proving the Riemann-Roch theorem. Jakob.scholbach 02:42, 18 April 2007 (UTC)