Separation Theorems in Smooth Commutative Algebra and Applications

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