Small embeddings of integral domains (Q2272779)

The commutative algebra \(A\) over a field \(k\) is geometrically integral over \(k\) if for every field extension \(K/k\) the ring \(K\otimes_kA\) is a domain. The main result of the paper under review states that if \(A\) is geometrically integral over \(k\) then it has the small embedding property. This means that if the \(k\)-algebra \(R\) is a finitely generated domain and there exists an extension \(K/k\) such that \(R\subseteq K\otimes_kA\) and \(R\not\subseteq K\), then there exists an extension \(L/k\) with \(R\subseteq L\otimes_kA\) \(\text{trdeg}_k(L)<\text{trdeg}_k(R)\). As a consequence the authors obtain that the small embedding property holds for \(k[x_1,\ldots,x_n]\) and \(k[x_1^{\pm 1},\ldots,x_n^{\pm 1}]\) which generalizes a result in [\textit{G. Freudenburg}, Kyoto J. Math. 55, No. 3, 663--672 (2015; Zbl 1339.14032)] for \(A=k[x]\) and answers a problem for \(A=k[x,x^{-1}]\) in the same paper. When \(k\) is algebraically closed, then every domain \(A\) over \(k\) is geometrically integral, and hence the small embedding property holds for all domains.