Basic constructions over C∞-schemes (2023)
C∞-Rings are R-algebras equipped with operations φf for every f ∈ C∞(Rn) 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.