Primary spectrum of $\mathcal{C}^\infty(M)$ and jets theory

TL;DR

This paper extends the classical Lie theory of jets to a new class of primary ideals in the algebra of smooth functions on a manifold, incorporating algebraic and differentiable structures for enhanced functorial properties.

Contribution

It introduces a generalized jet theory based on primary ideals with maximal radicals, combining algebraic and differentiable approaches for a broader framework.

Findings

Defined the set of primary ideals with maximal radicals for smooth manifolds.

Extended jet theory to include these primary ideals with functorial properties.

Developed algebraic and differentiable structures on the new jets spaces.

Abstract

We consider, for each smooth manifold $M$, the set $\mathbb{M}$ comprised by all the primary ideals of $\mathcal{C}^\infty(M)$ which are closed and whose radical is maximal. The classical Lie theory of jets (jets of submanifolds) must be extended to $\mathbb{M}$ in order to have nice functorial properties. We will begin with the purely algebraic notions, referred always to the ring $\mathcal{C}^\infty(M)$. Subsequently it will be introduced the differentiable structures on each jets space of a given type. The theory of contact systems, which generalizes the classical one, has a part purely algebraic and another one which depends on the differentiable structures.