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Transactions of the American Mathematical Society

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Functorial structure of units
in a tensor product


Author: David B. Jaffe
Journal: Trans. Amer. Math. Soc. 348 (1996), 4339-4353
MSC (1991): Primary 14C22, 18F20
DOI: https://doi.org/10.1090/S0002-9947-96-01680-7
MathSciNet review: 1361641
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Abstract: 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

\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].


References [Enhancements On Off] (What's this?)

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Additional Information

David B. Jaffe
Affiliation: Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323
Email: jaffe@cpthree.unl.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01680-7
Received by editor(s): March 6, 1995
Additional Notes: Partially supported by the National Science Foundation
Article copyright: © Copyright 1996 American Mathematical Society

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