Functorial structure of units in a tensor product

Abstract

We study the units in a tensor product of rings. For example, let k be an algebraically closed field. Let A and B be reduced rings containing k, having connected spectra. Let u \in A tensor_k B be a unit. Then u = a tensor_k b for some units a \in A and b \in B. Here is a deeper result, stated for simplicity in the affine case only. Let k be a field, and let f: R --> S be a homomorphism of f.g. k-algebras such that Spec(f) is dominant. Assume that every irreducible component of Spec(R_red) or Spec(S_red) is geometrically integral and has a rational point. Let B --> C be a faithfully flat homomorphism of reduced k-algebras. For A a k-algebra, define Q(A) to be (S tensor_k A)^*/(R tensor_k A)^*. Then Q satisfies the following sheaf property: the sequence 0 --> Q(B) --> Q(C) --> Q(C tensor_B C) is exact. This and another result are used in the proof of the following statement from "The kernel of the map on Picard groups induced by a faithfully flat homomorphism" by R. Guralnick, D. Jaffe, W. Raskind, R. Wiegand: Let K/k be an algebraic field extension and let A be a f.g. k-algebra. Assume resolution of singularities. Then there is a finite extension E/k contained in K/k such that Pic(A tensor_k E) --> Pic(A tensor_k K) is injective.