Sheaves

Summary

In this chapter we construct the abelian category of sheaves on a topological space, and the usual associated functors, such as the inverse image f ⁻¹, the direct image f _*, the proper direct image f _!, the tensor product ⊗ and the inner hom Hom. Making use of the results of the first chapter, one then defines the derived category D ^b(X) of sheaves, and the derived functors of the preceding ones. In the course of the chapter, we also introduce the notions of injective sheaves, flat sheaves, flabby sheaves, c-soft sheaves, and give the tools of sheaf theory that we shall use later: non-characteristic deformation lemma and homotopy invariance of cohomology. Although we do not really need it, we (briefly) present Čech cohomology. We end this chapter by recalling some natural sheaves on real or complex manifolds.

Most of the results we explain here are classical, and we refer to Bredon [1], Godement [1], Iversen [1] for further developments.