View source for Flabby sheaf

A sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405401.png" /> of sets over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405402.png" /> such that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405403.png" /> open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405404.png" /> the restriction mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405405.png" /> is surjective. Examples of such sheaves include the sheaf of germs of all (not necessarily continuous) sections of a [[Fibre space|fibre space]] with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405406.png" />, the sheaf of germs of divisors (cf. [[Divisor|Divisor]]), and a prime sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405407.png" /> over an irreducible algebraic variety. Flabbiness of a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405408.png" /> is a local property (i.e. a flabby sheaf induces a flabby sheaf on any open set). A quotient sheaf of a flabby sheaf by a flabby sheaf is itself a flabby sheaf. The image of a flabby sheaf under a continuous mapping is a flabby sheaf. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405409.png" /> is paracompact, a flabby sheaf is a [[Soft sheaf|soft sheaf]], i.e. any section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054010.png" /> over a closed set can be extended to the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054011.png" />.

Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054012.png" /></td> </tr></table>

be an exact sequence of flabby sheaves of Abelian groups. Then, for any family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054013.png" /> of supports, the corresponding sequence of sections (the supports of which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054014.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054015.png" /></td> </tr></table>

is exact, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054016.png" /> is a left-exact functor.



====Comments====
Flabby sheaves are used for resolutions in the construction of sheaf cohomology (i.e. [[Cohomology|cohomology]] with values in a sheaf) in algebraic geometry and topology, [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) {{MR|0559531}} {{ZBL|0433.14012}} </TD></TR></table>