More Sheaf Theory

Abstract

We introduced sheaves of functions in the previous chapter as a convenient language for defining manifolds and varieties. However, as we will see, there is much more to this story. In this chapter, we develop sheaf theory in a more systematic fashion. Presheaves and sheaves are somewhat more general notions than what we described earlier. We give the full definitions here, and then explore their formal properties. We define the notion of an exact sequence in the category of sheaves. Exact sequences and the associated cohomology sequences, given in the next chapter, form one of the basic tools used throughout the rest of the book.We also give a brief introduction to Grothendieck’s theory of schemes. A scheme is a massive generalization of an algebraic variety, and quite a bit of sheaf theory is required just to give the definition.