Basic constructions over C∞-schemes

C-infinity-Rings are R-algebras equipped with operations phi(f) every f is an element of C-infinity (R-n) and every n is an element of N. Therefore, a C-infinity-version of algebraic geometry can be developed using C-infinity-rings instead of ordinary rings and many classical constructions can be performed in this context. In particular, C-infinity-schemes are the C-infinity 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-infinity-algebraic geometry and they can be used to exhibit some examples of C-infinity-schemes such as projective spaces and Grassmannians as well as necessary and sufficient conditions for a functor F: C-infinity - Schemes -> Sets to be representable. (c) 2023 The Author(s). Published by Elsevier B.V.

Automated summary

C-infinity-Rings are R-algebras with operations phi(f) for every f in C-infinity (R-n) and every n in N. This allows for the development of C-infinity algebraic geometry using C-infinity rings instead of ordinary rings. Classical constructions, such as gluing schemes and fiber products, can be performed in this context. The article demonstrates the use of these constructions to exhibit examples of C-infinity-schemes and provides necessary and sufficient conditions for a functor F: C-infinity - Schemes -> Sets to be representable.