commutative rings. Morphisms A morphism from to is a pair , where is a continuous map between the underlying topological spaces, and is a morphism from the structure sheaf of to the direct image of the structure sheaf of . In other words, a morphism from to is given by the following data: a continuous map a family of ring homomorphisms for every open set of which commute with the restriction maps. That is, if are two open subsets of , then the following diagram must commute (the vertical maps are the restriction homomorphisms): There is an additional requirement for morphisms between locally ringed spaces: the ring homomorphisms induced by between the stalks of and the stalks of must be local homomorphisms, i.e. for every the maximal ideal of the local ring (stalk) at is mapped into the maximal ideal of the local ring at . Two morphisms can be composed to form a new morphism, and we obtain the category of ringed spaces and the category of locally ringed spaces. Isomorphisms in these categories are defined as usual. Tangent spaces Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let be locally ringed space with structure sheaf ; we want to define the tangent space at the point . Take the local ring (stalk) at the point , with maximal ideal . Then is a field and is a vector space over that field (the cotangent space). The tangent space is defined as the dual of this vector space. The idea is the following: a tangent vector at should tell you how to "differentiate" "functions" at , i.e. the elements of . Now it is enough to know how to differentiate functions whose value at is zero, since all other functions differ from these only by a constant, and we know how to differentiate constants. So we only need to consider . Furthermore, if two functions are given with value zero at , then their product has derivative 0 at , by the product rule. So we only need to know how to assign "numbers" to the elements of , and this is what the dual space does. -modules Given a locally ringed space , certain sheaves of modules on occur in the applications, the -modules. To define them, consider a sheaf F of abelian groups on . If F(U) is a module