In this post, I will prove the Mayer-Vietoris Sequence for sheaf cohomology.
In what follows, is a topological space and are sheaves of abelian groups on . Let be a closed subset of . We let denote the global sections of with support in . The functor is a left-exact additive functor from sheaves on to abelian groups, and its right derived functors, denoted , is the -th cohomology of with support in . If is a sheaf, the presheaf is also a sheaf on , denoted and called the “subsheaf of with support in “.
The Mayer-Vietoris sequence, for a sheaf and for a pair of closed subsets , is the long exact sequence of cohomology with supports
We will prove the existence of this sequence in several steps.
Lemma 1 Let be a flasque sheaf, a closed subset of , and . Then the sequence
Lemma 2 Let be a flasque sheaf, and let be closed subsets of . Then the sequence
is exact (where the first map is the diagonal embedding, and the second map is ).
Proof: Exactness is clear except possibly on the right. Let , and Let be the short exact sequences
where the maps are defined similarily as in the statement of the Lemma. There is an obvious morphism of short exact sequences . Since is flasque, this morphism is surjective onto each term of . By the snake lemma, and using Lemma 1, we get the desired short exact sequence.
Now we are ready to prove the existence of the Mayer-Vietoris sequence for . Let
be a flasque resolution of . By the lemma, we have a short exact sequence of complexes
The long exact sequence of cohomology associated to this short exact sequences of complexes is precisely the Mayer-Vietoris sequence.