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Let $k$ be a field, and $R,S$ be commutative $k$-algebras. I would like to know what is the structure of the group of units $$ (R \otimes_k S)^{\times}. $$

We have a map $$R^{\times} \times S^{\times} \to (R \otimes_k S)^{\times}, \qquad (a,b) \mapsto a \otimes b.$$ This is not an isomorphism in general, see $(\mathbb R \otimes_{\mathbb R} \mathbb C)^{\times} \cong \mathbb C^{\times}$ as groups. But for instance, I think that we have $(\mathbb C \otimes_{\mathbb R} \mathbb C)^{\times} \cong \mathbb C^{\times} \times \mathbb C^{\times}$ as groups, but I'm not sure how to prove it. In general, how does $(R \otimes_k S)^{\times}$ compare to $R^{\times}, S^{\times}, k^{\times}$, etc. ?

Alphonse
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    In general, the map $R^\times S^\to (R\otimes_k S)^$ may not be an isomorphism. For an example, let $R=S=K$, a finite field extension of $k=\mathbb{F}_p$, finite field of $p$ elements. If $[K:k]=n$, you can check that $(K\otimes_k K)^$ is a product of $n$ copies of $K^$, while of course $K^\times K^*$ is product of just two copies. – Mohan Oct 15 '17 at 16:12
  • @Mohan : yes, thank you. I've noticed that already with $(\mathbb R \otimes_{\mathbb R} \mathbb C)^{\times} \cong \mathbb C^{\times} \not\cong \mathbb R^{\times} \times \mathbb C^{\times}$. What would be a good description of the group of units of $R \otimes_k S$ (under some assumptions if these can help) ? – Alphonse Oct 15 '17 at 16:45
  • I know that if $S = k[X_1,...,X_n] / I$ is a finitely generated $k$-algebra, then $R \otimes_k S \cong R[X_1, ..., X_n] / I$ as $k$-algebras, in particular their unit groups are isomorphic. When $S = k[X] / (f)$ is a finite separable field extension of $K$, of degree $d$ and $R$ is any field extension of $k$ over which $f$ splits, then $R \otimes_k S \cong R^d$ as $k$-algebras, so $(R \otimes_k S)^{\times} \cong (R^{\times})^d$. – Alphonse Oct 28 '17 at 15:52
  • I also found this: https://math.stackexchange.com/questions/2387905/ – Alphonse Oct 28 '17 at 15:52
  • I found that: https://math.stackexchange.com/questions/181788/tensor-product-algebra-mathbbc-otimes-mathbbr-mathbbc/2265459#2265459 – Alphonse Oct 31 '17 at 13:13
  • I also found that: https://math.stackexchange.com/questions/263192, https://math.stackexchange.com/questions/2070412/ – Alphonse Nov 22 '17 at 15:18

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