Lemma 10.48.2. Let k be a field. Let R be a k-algebra. The following are equivalent
for every field extension k'/k the spectrum of R \otimes _ k k' is connected, and
for every finite separable field extension k'/k the spectrum of R \otimes _ k k' is connected.
Proof. For any extension of fields k'/k the connectivity of the spectrum of R \otimes _ k k' is equivalent to R \otimes _ k k' having no nontrivial idempotents, see Lemma 10.21.4. Assume (2). Let k \subset \overline{k} be a separable algebraic closure of k. Using Lemma 10.43.4 we see that (2) is equivalent to R \otimes _ k \overline{k} having no nontrivial idempotents. For any field extension k'/k, there exists a field extension \overline{k}'/\overline{k} with k' \subset \overline{k}'. By Lemma 10.48.1 we see that R \otimes _ k \overline{k}' has no nontrivial idempotents. If R \otimes _ k k' has a nontrivial idempotent, then also R \otimes _ k \overline{k}', contradiction. \square
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