\(C^\infty\)-differentiable spaces (Q1415057)

Differential geometry is traditionally regarded as the study of smooth manifolds, but sometimes this framework is too restrictive since it does not admit certain basic geometric constructions. The study of differentiable spaces allows including these constructions. For example, elementary surfaces of classical geometry such as a quadratic cone or a doubly counted plane are not smooth manifolds, but they have a natural differentiable structure defined by means of the consideration of an appropriate algebra of differentiable functions. In this monograph, the authors develop the theory of differentiable spaces. For the sheaf \(C^\infty(\mathbb R^ n)\) of differentiable functions of class \(C^\infty\) and some closed ideal \(I\) with respect to the Fréchet topology of uniform convergence on compact sets of functions, the quotient \(C^\infty(\mathbb R^ n)/I\) can be regarded as an algebra of differentiable functions on the closed subset \((I)_ 0=\{x;\;f(x)=0,\;\forall f\in I\}\) of \(\mathbb R^ n\) and is called a differentiable algebra. If \(I\) is the ideal of all \(C^ \infty\)-functions vanishing on a given closed set \(X\subset \mathbb R^ n\), then the quotient algebra \(C^\infty(\mathbb R^ n)/I\) is identified with a ring of real valued functions on \(X\) and is called a reduced differentiable algebra. Each differential algebra \(A\) can be replaced by a ringed space \(\text{Spec}_ r(A)\), a topological space with a sheaf of rings, called the real spectrum of \(A\). A ringed space is said to be affine differentiable space if it is isomorphic to the real spectrum of some differentiable algebra. A ringed space in which each point has an open neighborhood that is an affine differentiable space is called a differentiable space. The manuscript comprises eleven chapters and two appendices. Chapter 1 presents the elementary theory of smooth manifolds in the spirit of differentiable spaces. In chapter 2, the authors study differential algebras to introduce the definition and properties of differentiable spaces in chapter 3. Chapter 4 is devoted to the study of some basic topological properties of differentiable spaces, including the existence of partitions of unity and the equivalence between locally free sheaves of bounded rank and finitely generated projective modules over the ring of global differentiable functions. Chapter 5 introduces differentiable subspaces and embeddings. The main result is an embedding theorem for separated differentiable spaces whose topology has a countable basis. In chapter 6, locally convex modules over a Fréchet algebra \(A\) are defined and topological products of these modules are studied. Tensor products provide the basic tool for the theorem of existence of finite direct products and fibred products in the category of differentiable spaces, which is the main result of chapter 7. In chapter 8, modules of fractions including the localization theorem for Fréchet modules are studied. Modules of fractions are used in chapter 9 to study finite morphisms. In chapter 10, the authors use topological modules to introduce the module of relative differentials for any morphism of differentiable algebras and study its properties. The main result is the characterization of smooth morphisms over a reduced space as open maps with a locally free sheaf of relative differentials. Also in this chapter, the authors introduce formally smooth spaces and prove that a differentiable space \(M\) is formally smooth if and only if it is locally isomorphic to the Whitney space of a closed set in \(\mathbb R^ n\). In the last chapter, chapter 11, quotients of smooth manifolds by compact Lie groups of transformations are studied. Finally, two appendices are presented. In the first one, sheaves of Fréchet modules are studied, and in the second one, the space of \(r\)-jets of special morphisms is introduced.