Let \((A_i,\varphi_{ji})\) be a direct system of noetherian rings, \(A=\varinjlim A_i\) being noetherian and \(\varphi_{ji}\) being flat for \(i\leq j\) if \(A\) is a Nagata (or an excellent) ring for all \(i\), is the same true for \(A\)? -The answer is “yes” in the local case, with an hypothesis of separability about the residue fields: more precisely, let \((A_i,\varphi_{ji})\) be a direct system of noetherian local rings such that
(i) \((A=\varinjlim A_i\) is a noetherian ring,
(ii) for \(i\leq j, \varphi_{ij}\) is local and flat, (iii) for \(i\leq j, k_j\) is a separable extension of \(k_i\) by \(\varphi_{ji} (k_i\) is the residual field of \(A_i\) for all \(i\)); then, if \(A_i\) is a Nagata (or an excellent) ring for all \(i\), the same is true for \(A\).- It is necessary to assume that \(k_j\) is a separable extension of \(k_i\).
The answer is “no” in the global case: more precisely, let \(k\) be an arbitrary field; then there exists a direct system of noetherian semi-local \(k\)-algebras \((A_i,\varphi_{ji})\) such that:
(i) \(A =\varinjlim A_i\) is a noetherian domain of dimension 2,
(ii) for all \(1\leq j, \varphi_{ji}\) is a flat \(k\)-homomorphism,
(iii) for all \(i\), \(A_i\) is an excellent ring,
(iv) \(A\) is a Nagata non excellent ring, (v) for all maximal ideal \(\mathfrak m\) of \(A\), \(A_{\mathfrak m} \) is an excellent ring.