  For a homomorphism f: A --> B of commutative rings, let D(A,B) denote
Ker[Pic(A) --> Pic(B)]. Let k be a field and assume that A is a f.g. k-algebra.
  We prove a number of finiteness results for D(A,B). Here are four of them.
  1: Suppose B is a f.g. and faithfully flat A-algebra which is geometrically
integral over k. If k is perfect, we find that D(A,B) is f.g. (In positive
characteristic, we need resolution of singularities to prove this.) For an
arbitrary field k of positive characteristic p, we find that modulo p-power
torsion, D(A,B) is f.g.
  2: Suppose B = A tensor k^sep. We find that D(A,B) is finite.
  3: Suppose B = A tensor L, where L is a finite, purely inseparable extension.
We give examples to show that D(A,B) may be infinite.
  4: Assuming resolution of singularities, we show that if K/k is any algebraic
extension, there is a finite extension E/k contained in K/k such that D(A
tensor E,A tensor K) is trivial.
  The remaining results are absolute finiteness results for Pic(A).
  5: For every n prime to char(k), Pic(A) has only finitely many elements of
order n.
  6: Structure theorems are given for Pic(A), in the case where k is absolutely
f.g.
  All of these results are proved in a more general form, valid for schemes.
Hard copy is available from the authors.
