Schemes in algebraic geometry 3 : glued schemes and sheaves
André Weil was among the first ones to point out the importance of having a local description of varieties, especially projective spaces, which can always locally be described as an affine space with completion by a hyperplane at infinity, and projective varieties, which similarly look like varieties in affine space. The use of sheaves in local description of spaces was magnified by Cartan and Serre, in the context of complex analytic spaces, and generalised to the algebraic setting by Serre in Faisceaux algébriques cohérents.
The projective space is the simplest example of an algebro-geometric object which cannot be described by the prime spectrum or the functor of points of a ring. For example, there is no obvious ring whose ideals describe varieties in projective space, which come from homogeneous equations. We would like to give a correct definition of gluing affine lines (with coordinates and ) to define the projective line as the gluing of with given by . For functors of points, the latest article by Alain Connes and Caterina Consani, gives a definition. For prime spectra, one has to be aware that gluing only topological spaces do not give meaningful information on algebraic properties. This is illustrated by the case of differentiable manifolds, which are not the same as topological manifolds: gluing differentiable manifolds has to induce a correspondance between differentiable functions (this is equivalent to the requirement that gluing maps between charts be differentiable).
An elegant way of making the answers clear is to consider prime spectra to have not only a ring of functions (equations of subvarieties), but a sheaf of functions. For example, given an open set which is the complement of , the functions on this domain are defined to be fractions of the form . This ring of fractions is called the localised ring of along . The correct notion of gluing schemes is then the gluing of affine schemes (affine charts) along open subschemes with isomorphic sheaves of functions (this is Grothendieck’s definition of schemes in EGA1). This gives a consistent way of defining subschemes in a general scheme: this is the data of ideals of equations in each affine piece of the scheme, which define the same subscheme on intersections of charts. Another interesting property of affine schemes is the fact that a natural class of sheaves (quasi-coherent sheaves) has no cohomology for the Zariski topology: affine covers can be used to compute cohomology. The relevance of cohomology of coherent sheaves in the Zariski topology is not really trivial: it is especially interesting for projective varieties, where it coincides with cohomology of holomorphic sheaves, and allows to compute the dimension of linear systems. The Zariski topology also allows to define the algebraic de Rham cohomology, which is the same as ordinary de Rham cohomology for complex projective varieties.
This definition of prime spectra and gluing schemes is not restrictive, and there is no reason not to consider a scheme , or a scheme , which is topologically equivalent to . Such a non-reduced ring is useful to model finite Taylor expansions of functions, and has its own use when defining tangent spaces, multiplicities of intersections, or finite-order deformations. If I recall correctly, the main reason Grothendieck to look for a definition of schemes which would work with any ring was the will to prove Weil conjectures (but schemes were not enough, since étale cohomology was eventually needed). However, most of the schemes encountered in classical algebraic geometry are locally of finite type, which means they are obtained by gluing subschemes of affine spaces (with rational functions as gluing maps).
The availability of gluing gives the category of schemes nice properties: it has fibred products (given by taking tensor products of rings), and several coproducts (given by gluing schemes). These properties mirror standard properties of sets: that’s why working with schemes is usually as simple as working with sets (with several annoying complications though).