Reducedness of formally unramified algebras over fields (Q2662203)

If \(A\) is a commutative ring with identity, then an \(A\)-algebra \(S\) is said to be {\em formally unramified} over \(A\) if the corresponding module of Kähler differential \(\Omega_{S/A}\) is zero. The paper under review establishes conditions under which a formally unramified algebra is reduced. The authors show that if \(A\) is a local algebra separated in its \(m\)-adic topology and formally unramified over a perfect field, then \(A\) is reduced. As a consequence of this result, it is shown that any Noetherian \(k\)-algebra formally unramified over a perfect field is reduced. Considering the graduate case, the authors prove that if \(R\) is an \(\mathbb{N}\)-graded formally unramified algebra over a perfect field such that the degree zero graded piece of \(R\) is Noetherian, then \(R\) is reduced. The paper also contains an analysis of several examples including a construction of a non-reduced formally unramified algebra over an arbitrary field of characteristic zero which is due to O. Gabber.