Local etale cohomology for sheaves of abelian groups.

Question

I have seen local cohomology (cohomology supported on a closed subspace) in different contexts, like for topological spaces and for quasi-coherent sheaves on a scheme. I was wondering whether the same naïve definition (i.e. taking the global sections of an injective resolution that are supported on the closed subspace) gives the correct local cohomology for let's say sheaves of abelian groups on the etale site of a scheme? (by the correct the definition I means fitting into a long exact sequence of cohomology groups of the space itself and the complement of the support.)