On the Picard Group: Torsion and the Kernel Induced by a Faithfully Flat Map

For a homomorphism f : A → B of commutative rings, letD(A,B) denote the kernel of the map Pic(A) → Pic(B). Let k be a field and assume that A is a finitely generated k-algebra. We prove a number of finiteness results for D(A,B). Here are four of them. [1]: Suppose B is a finitely generated and faithfully flat A-algebra which is geometrically integral over k. If k is perfect, we find that D(A,B) is finitely generated. (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 finitely generated. [2]: Suppose B = A ⊗k k. We find that D(A,B) is finite. [3]: Suppose B = A ⊗k 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⊗k E,A⊗k K) is trivial. The remaining results are absolute finiteness results for Pic(A). [5]: For every n prime to the characteristic of 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 finitely generated. In the body of the paper, all of these results are stated in a more general form, valid for schemes.