Definition 10.49.1. Let k be a field. Let S be a k-algebra. We say S is geometrically integral over k if for every field extension k'/k the ring of S \otimes _ k k' is a domain.
10.49 Geometrically integral algebras
Here is the definition.
Any question about geometrically integral algebras can be translated in a question about geometrically reduced and irreducible algebras.
Lemma 10.49.2. Let k be a field. Let S be a k-algebra. In this case S is geometrically integral over k if and only if S is geometrically irreducible as well as geometrically reduced over k.
Proof. Omitted. \square
Lemma 10.49.3. Let k be a field. Let S be a k-algebra. The following are equivalent
S is geometrically integral over k,
for every finite extension k'/k of fields the ring S \otimes _ k k' is a domain,
S \otimes _ k \overline{k} is a domain where \overline{k} is the algebraic closure of k.
Proof. Follows from Lemmas 10.49.2, 10.44.4, and 10.47.3. \square
Lemma 10.49.4. Let k be a field. Let S be a geometrically integral k-algebra. Let R be a k-algebra and an integral domain. Then R \otimes _ k S is an integral domain.
Proof. By Lemma 10.43.5 the ring R \otimes _ k S is reduced and by Lemma 10.47.7 the ring R \otimes _ k S is irreducible (the spectrum has just one irreducible component), so R \otimes _ k S is an integral domain. \square
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