7 - Coherent sheaves

Summary

We have spent many chapters now worrying about ringed spaces. In Chapter 2 we used the example of C^k–manifolds to motivate the definition and convince the reader that ringed spaces, that is spaces with sheaves of rings on them, are natural objects that are worth studying. In Chapter 3 we constructed the ringed spaces that form the object of study of this book, namely the schemes (locally) of finite type over ℂ. Chapters 4, 5 and 6 told us that, if (X, O) is a scheme locally of finite type over ℂ, then there is a natural way to attach to it another ringed space (X^an, O^an). Intuitively we think of 0 as the sheaf of polynomial functions on X, and O^an is the sheaf of holomorphic functions. For every open set U ⊂ X one can speak of the polynomial functions on it, that is the ring Γ(U, O). For any open subset V ⊂ X^an (there are many more such open sets) one can talk of the holomorphic functions on V, that is the elements of Γ(V, O^an).

If the reader glances back to the introduction she will discover that the results we want to prove are not only about spaces and functions on them, but also about vector bundles. It is only natural that we should again carefully define what we mean by an algebraic vector bundle, and what we mean by an analytic vector bundle.