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.