Soft sheaf
A sheaf of sets on a topological space
any section of which over some closed subset in
can be extended to a section of
over all of
. Examples of soft sheaves are: the sheaf of germs of discontinuous sections of an arbitrary sheaf of sets on
; any flabby sheaf
on a paracompact space
; and any fine sheaf
of Abelian groups on a paracompact space
. The property of softness of a sheaf
on a paracompact space
is local: A sheaf
is soft if and only if any
has an open neighbourhood
such that
is a soft sheaf on
. A soft sheaf on a paracompact space induces a soft sheaf on any closed (and, if
is metrizable, any locally closed) subspace. A sheaf of modules over a soft sheaf of rings is a soft sheaf.
If
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is an exact sequence of soft sheaves of Abelian groups on a paracompact space , then the corresponding sequence of groups of sections
![]() |
is also exact. The cohomology group of any soft sheaf
of Abelian groups on a paracompact space
is trivial for
.
References
[1] | R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) |
[2] | R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) |
Comments
References
[a1] | G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967) pp. §9 |
Soft sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Soft_sheaf&oldid=43509