nLab
germ-determined C-infinity ring

Contents

Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 C^\infty-ideal)

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

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 C^\infty-ring)

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

References

Created on December 10, 2020 at 05:27:46. See the history of this page for a list of all contributions to it.