A Universal Algebraic Survey of C∞-Rings

Jean Cerqueira Berni; Hugo Luiz Mariano

doi:10.14244/lajm.v1i01.5

Authors

Jean Cerqueira Berni Instituto de Matemática e Estatística da Universidade de São Paulo - IME-USP
Hugo Luiz Mariano Instituto de Matemática e Estatística da Universidade de São Paulo, IME-USP

DOI:

https://doi.org/10.14244/lajm.v1i01.5

Keywords:

C∞−rings, Algebraic constructions

Abstract

This survey brings a detailed and systematic exposition of some fundamental results regarding the Universal Algebra of C∞−rings. Some of these were nowhere to be found – stated or proved – in the current literature. Our main contribution is to bundle these results up in a single text, using the unifying language of Universal Algebra and referring the reader to detailed proofs. Such a presentation is inspired by the treatment given by D. Joyce in [1] to some concepts involving these rings. Thus, we provide a comprehensive material with many known “taken for granted” results and constructions used everywhere in the literature about C∞−rings and their applications, providing a “propaedeutic exposition” for the reader’s benefit.

References

Ieke Moerdijk and Gonzalo E Reyes. Models for smooth infinitesimal analysis. Springer Science & Business Media, 2013.

Dominic Joyce. Algebraic Geometry over C∞-rings. Number 1256. American Mathematical Society, 2019.

Eduardo J Dubuc. C∞-schemes. American Journal of Mathematics, 103(4):683– 690, 1981.

Jean Cerqueira Berni and Hugo Luiz Mariano. A geometria diferencial sintética e os mundos onde podemos interpretá-la: um convite ao estudo dos aneis C∞. Revista Matemática Universitária, 1:5–30, 2022.

Dominic Joyce. An introduction to C∞-schemes and C∞-algebraic geometry. arXiv:1104.4951, 2011.

Ieke Moerdijk and Gonzalo E Reyes. Rings of smooth functions and their localizations, i. Journal of Pure and Applied Algebra, (99):324–336, 1986.

Jean Cerqueira Berni and Hugo Luiz Mariano. Classifying toposes for some theories of C∞- rings. South American Journal of Logic, 4(2):313–350, 2018.

Jean Cerqueira Berni, Rodrigo Figueiredo and Hugo Luiz Mariano. On the order theory for C ∞ −reduced C ∞ −rings and applications. Journal of Applied Logics, 9(1):93–134, 2022.

Jean Cerqueira Berni and Hugo Luiz Mariano. A universal algebraic survey of C∞−rings. arXiv preprint arXiv:1904.02728, 2019.

Jean Cerqueira Berni and Hugo Luiz Mariano. Topics on smooth commutative al- gebra. arXiv preprint arXiv:1904.02725, 2019.

Jean Cerqueira Berni and Hugo Luiz Mariano. Von Neumann regular C∞−rings and applications. arXiv preprint arXiv:1905.09617, 2019.