# Smooth manifolds as locally ringed spaces #

This file equips a smooth manifold-with-corners with the structure of a locally ringed space.

## Main results #

`smoothSheafCommRing.isUnit_stalk_iff`

: The units of the stalk at`x`

of the sheaf of smooth functions from a smooth manifold`M`

to its scalar field`𝕜`

, considered as a sheaf of commutative rings, are the functions whose values at`x`

are nonzero.

## Main definitions #

`SmoothManifoldWithCorners.locallyRingedSpace`

: A smooth manifold-with-corners can be considered as a locally ringed space.

## TODO #

Characterize morphisms-of-locally-ringed-spaces (`AlgebraicGeometry.LocallyRingedSpace.Hom`

) between
smooth manifolds.

The units of the stalk at `x`

of the sheaf of smooth functions from `M`

to `𝕜`

, considered as a
sheaf of commutative rings, are the functions whose values at `x`

are nonzero.

The non-units of the stalk at `x`

of the sheaf of smooth functions from `M`

to `𝕜`

, considered
as a sheaf of commutative rings, are the functions whose values at `x`

are zero.

The stalks of the structure sheaf of a smooth manifold-with-corners are local rings.

## Equations

- One or more equations did not get rendered due to their size.

A smooth manifold-with-corners can be considered as a locally ringed space.

## Equations

- One or more equations did not get rendered due to their size.