In this post, we consider the following question:
Question. Let be a Noetherian ring, and
and
-module. If
is a finite
-module for all primes
, is
finite?
That is, is finiteness a local property?
For the statement where local means the property is true on a cover by Zariski opens, see Tag 01XZ. Some properties (e.g. flatness) can also be checked at the level of local rings; however, we show that this is not true for finiteness.
Example 1. Let , and let
. Then
, because localisation commutes with direct sums and
if
is prime. Thus,
is finitely generated for all primes
. Finally,
, because
is torsion. But
is obviously not finitely generated.
Example 2. Again, let , and let
be the subgroup of fractions
with
such that
is squarefree. This is a subgroup because
can be written with denominator
, and that number is squarefree if
and
are. Clearly
is not finitely generated, because the denominators can be arbitrarily large. But
, which is finitely generated over
. If
is a prime, then
is the submodule
, which is finitely generated over
.
Another way to write is
.
Remark. The second example shows that over a PID, the property that is free of rank
can not be checked at the stalks. Of course it can be if
is finitely generated, for then
is finite projective [Tag 00NX] of rank
, hence free since
is a PID.