On the approximation property of excellent rings (Q1820212)

Let A be a semilocal noetherian ring, with Jacobson radical I, whose generic formal fibres are geometrically regular. Suppose either that \({\mathbb{Q}}\subseteq A\) or that, for each minimal \(prime\quad g_ i\) of A, the fraction field of \(A/g_ i\) has a separating transcendence basis over a perfect subfield \(P_ i\) such that \(\prod P_ i\subset A.\quad U\sin g\) a theorem of \textit{M. Artin} [Contemp. Math. 13, 223-227 (1982; Zbl 0528.13021)], it is shown that \(\hat A,\) the I-adic completion of A, is a direct limit of generically smooth A-algebras. It is deduced that, for A an excellent henselian ring containing \({\mathbb{Q}}\), if a finite system of polynomials over A has a zero in \(\hat A\) then it has a zero A.