nLab
direct image with compact support

under construction

Contents

Idea

Let f:XYf:X\to Y be a morphism of locally compact topological spaces. Then there exist a unique subfunctor f !:Sh(X)Sh(Y)f_!: Sh(X)\to Sh(Y) of the direct image functor f *f_* such that for any abelian sheaf FF over XX the sections of f !(F)f_!(F) over U openYU^{open}\subset Y are those sections sf *(U)=Γ(f 1(U),F)s\in f_*(U)= \Gamma(f^{-1}(U),F) for which the restriction f| supp(s):supp(s)Uf|_{supp(s)} : supp(s)\to U is a proper map.

This is called the direct image with compact support.

It follows that f !f_! is left exact.

Let p:X*p:X\to {*} be the map into the one point space. Then for any FSh(X)F\in Sh(X) the abelian sheaf p !Fp_!F is the abelian group consisting of sections sΓ(X,F)s\in \Gamma(X,F) such that supp(s)supp(s) is compact. One writes Γ c(X,F):=p !F\Gamma_c(X,F):= p_! F and calls this group a group of sections of FF with compact support. If yYy\in Y, then the fiber (f !F) y(f_! F)_y is isomorphic to Γ c(f 1y,F| f 1(y))\Gamma_c(f^{-1}y,F|_{f^{-1}(y)}).

Last revised on May 22, 2018 at 08:00:25. See the history of this page for a list of all contributions to it.