This chapter brings us to the essential core of geometry, as expressed in the language of sheaf theory. We study spaces equipped with a sheaf of rings, and particularly the geometric spaces, where the stalks are all local rings: we show that there is some justification for this name, since morphisms between geometric spaces specialise to the appropriate kinds of maps between several types of manifolds (differentiate, analytic, and so on).

We construct a universal geometric space associated with each commutative ring, and this leads us to the definition of schemes, which are central in modern algebraic geometry.

We then consider sheaves of modules over a ringed space, which generalise the idea of vector bundles, and globalise the idea of a module over a ring. The module constructions of direct sum and product, tensor product and module of homomorphisms also globalise to these sheaves, with appropriate universal properties. Similarly, change of base space by a morphism of ringed spaces gives rise to direct and inverse image functors. Finally, we define the picard group of a ringed space; we shall see later that this can be interpreted as a cohomology group.

Throughout this Chapter the word ‘ring’ will mean commutative ring with a one, and ring morphisms are required to preserve ones.