Separation theorems in the commutative algebra of C∞-rings and applications

Abstract

In this paper we state and prove ad hoc “Separation Theorems” of the so-called Smooth Commutative Algebra, the Commutative Algebra of ${\mathcal{C}}^{\infty }$-rings. These results are formally similar to the ones we find in (ordinary) Commutative Algebra. However, their proofs are not so straightforward, since they depend on the introduction of the key concept of “smooth saturation.” As an application of these theorems we present an interesting result that sheds light on the natural connection between the smooth Zariski spectrum and smooth real spectrum of a ${\mathcal{C}}^{\infty }$-ring, the ${\mathcal{C}}^{\infty }$-analog of the real spectrum of a commutative unital ring.

Keywords:

• ${\mathcal{C}}^{\infty }$-rings
• real smooth spectrum
• separation theorems
• smooth commutative algebra
• smooth spectrum

2020 MATHEMATICS SUBJECT CLASSIFICATION::

• 13B30

Notes

1 Here considered simply as syntactic symbols rather than functions.