nLab
family of supports

A family of supports in a topological space XX is a family ϕ\phi of closed subsets SXS\subset X such that

  1. if S 1,S 2ϕS_1,S_2\in \phi then S 1S 2ϕS_1\cup S_2\in \phi;
  2. if S 1ϕS_1\in \phi and S 2S 1S_2\subset S_1 is closed, then S 2ϕS_2\in \phi.

In other words, it is an ideal in the lattice of closed subsets.

Families of supports are used to introduce a variant of sheaf cohomology with supports in ϕ\phi and also for developing certain homology theories using sheaves (see the book by Bredon, Sheaf theory). Especially useful are the so-called paracompactifying families of supports on non-paracompact spaces.

Let FF be a sheaf of abelian groups over a topological space XX. Denote by Γ ϕ(X,F)\Gamma_\phi(X,F) the subset of the space of all sections fΓ(X,F)=F(X)f\in \Gamma(X,F) = F(X) for which suppfϕsupp\,f\in \phi. This gives rise to a covariant left exact functor FΓ ϕ(X,F)F\mapsto \Gamma_\phi(X,F). Its right-derived functors

H ϕ k(X,F):=R kΓ ϕ(X,F)H_\phi^k(X,F) := R^k\Gamma_\phi(X,F)

are called the cohomology groups of XX with coefficients in the sheaf FF and with supports in the family ϕ\phi of supports. Or sometimes one simply says sheaf cohomology with supports.

Revised on August 24, 2009 18:27:40 by Toby Bartels (71.104.230.172)