Lukas Heger

There's an interesting relation between these metaproperties (is this a metametaproperty?) which says that any property that may be checked on a partition of unity may also be checked on stalks, but the converse does not a hold. Interestingly, the property of being Noetherian can be checked on a partition of unity, but cannot be checked on stalks.

These two metaproperties are absolutely fundamental and natural in algebraic geometry.

There are also corresponding metaproperties for module-properties over commutative rings. For example, it is true that a module $M$ over $R$ is flat if and on…

@rschwieb I would like to suggest two metaproperties for properties of (commutative) rings that are used frequently in commutative algebra in algebraic geometry. Both are about relating a ring $R$ having a property $P$ to certain localisations of $R$ having $P$.

(I think these properties are very unusual in the noncommutative setting though.)

The first one could be called "may be checked on stalks" (if you want a geometric name) or "may be checked at prime ideals" (for a more algebraic name). A commutative ring property $P$ has this metaproperty if for any commutative ring $R$, $R$ has $P…