# nLab germ-determined C-infinity ring

Contents

### Context

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

###### Definition

(germ-determined $C^\infty$-ideal)

For $M$ a smooth manifold, an ideal $I \,\subset\, C^\infty(M)$ in its C-infinity ring (smooth algebra) of smooth functions is called germ determined if for any $f \in I$ such that the germ of $f$ at any point $x$ in the zero locus of $I$ in $M$ coincides with the germ of some element of $I$ at $x$, then $f$ itself belongs to $I$.

Equivalently, germ-determined ideals are precisely those ideals that are closed under (possibly infinite) sums of locally finite families. One can also take sums with coefficients in a partition of unity.

###### Definition

(germ-determined $C^\infty$-ring)

A finitely generated C-infinity ring of the form $C^\infty(M)/I$ for some smooth manifold $M$ and ideal $I$ is germ-determined if the ideal $I$ is a germ-determined ideal (Def. ).