Basic constructions over C∞-schemes.

C ∞ -Rings are R -algebras equipped with operations ϕ f for every f ∈ C ∞ (R n) and every n ∈ N. Therefore, a C ∞ -version of algebraic geometry can be developed using C ∞ -rings instead of ordinary rings and many classical constructions can be performed in this context. In particular, C ∞ -schemes are the C ∞ counterpart of classical schemes. Examples of schemes are often obtained by gluing schemes or using fiber products. Another useful way to give examples of schemes is looking for representable functors F : Schemes → Sets. In this work, we show that constructions such as gluing schemes and fiber products can be done in the context of C ∞ -algebraic geometry and they can be used to exhibit some examples of C ∞ -schemes such as projective spaces and Grassmannians as well as necessary and sufficient conditions for a functor F : C ∞ − Schemes → Sets to be representable. [ABSTRACT FROM AUTHOR]