Let
A
A
be a commutative noetherian ring. We investigate a class of functors from
\lBrack commutative
A
A
-algebras
\rBrack to
\lBrack sets
\rBrack , which we call coherent. When such a functor
F
F
in fact takes its values in
\lBrack abelian groups
\rBrack , we show that there are only finitely many prime numbers
p
p
such that
p
F
(
A
)
{}_pF(A)
is infinite, and that none of these primes are invertible in
A
A
. This (and related statements) yield information about torsion in
Pic
(
A
)
\operatorname {Pic}(A)
. For example, if
A
A
is of finite type over
Z
\mathbb {Z}
, we prove that the torsion in
Pic
(
A
)
\operatorname {Pic}(A)
is supported at a finite set of primes, and if
p
Pic
(
A
)
{}_p\operatorname {Pic}(A)
is infinite, then the prime
p
p
is not invertible in
A
A
. These results use the (already known) fact that if such an
A
A
is normal, then
Pic
(
A
)
\operatorname {Pic}(A)
is finitely generated. We obtain a parallel result for a reduced scheme
X
X
of finite type over
Z
\mathbb {Z}
. We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.