Functorial structure of units in a tensor product

The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let $k$ be an algebraically closed field. Let $A$ and $B$ be reduced rings containing $k$, having connected spectra. Let $u\in A\otimes _k\,B$ be a unit. Then $u=a\otimes b$ for some units $a\in A$ and $b\in B$.

Here is a deeper consequence, stated for simplicity in the affine case only. Let $k$ be a field, and let $\varphi :R\to S$ be a homomorphism of finitely generated $k$-algebras such that $\operatorname {Spec}(\varphi )$ is dominant. Assume that every irreducible component of $\operatorname {Spec}(R_{\operatorname {red}})$ or $\operatorname {Spec}(S_{\operatorname {red}})$ is geometrically integral and has a rational point. Let $B\to C$ be a faithfully flat homomorphism of reduced $k$-algebras. For $A$ a $k$-algebra, define $Q(A)$ to be $(S\otimes _k\,A)^*/(R\otimes _k\,A)^*$. Then $Q$ satisfies the following sheaf property: the sequence

is exact. This and another result are used to prove (5.2) of [7].