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Functorial structure of units in a tensor product

Authors:	David B Jaffe

Institution:	Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323

Abstract:	The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let be an algebraically closed field. Let and be reduced rings containing , 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 be a field, and let $\varphi :R\to S$ be a homomorphism of finitely generated -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 -algebras. For a -algebra, define to be $(S\otimes _k\,A)^/(R\otimes _k\,A)^$ . Then satisfies the following sheaf property: the sequence $\begin{displaymath}0\to Q(B)\to Q(C)\to Q(C\otimes _B\,C)\end{displaymath}$ is exact. This and another result are used to prove (5.2) of 7].

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