Exercise 4, § 7

Let $k$ be a noetherian ring, $A$ a finitely generated $k$-algebra. We say that $A$ is an étale (resp. net) $k$-algebra if it is formally étale (resp. formally net) when provided with the discrete topology.

a) Proving that the following conditions are equivalent:

(i) the $k$-algebra $A$ is net ; (ii) $\Omega_k(A) = 0$ ; (iii) the $A \otimes_k A$-module $A$ is projective.

(Let $I$ be the kernel of the canonical homomorphism $A \otimes_k A \to A$ ; now prove that (ii) and (iii) are both equivalent to

(iv) $I_q = 0$ for every prime ideal $q$ of $A \otimes_k A$ containing $I$.)

b) Let $f \in k[T]$ be a polynomial such that $f$ and $f'$ generate the unit ideal of $k[T]$. Prove that the $k$-algebra $k[T]/(f)$ is étale (\emph{cf.} No. 3, example 3).

c) If $k$ is a field, the net $k$-algebras are the étale $k$-algebras, and this notion coincides with that introduced in A, V, p. 28 (the case of algebras of finite degree over $k$ results from \emph{loc. cit.}, p. 32, th. 3 ; if $A$ is a net $k$-algebra, consider a maximal ideal $m$ of $A$ and apply the preceding case to $A/m^n$). The net $k$-algebras are therefore the finite products of separable extensions of finite degree of $k$ (A, V, p. 34, th. 4).

d) For the $k$-algebra $A$ to be net, it is necessary and sufficient that for every prime ideal $p$ of $k$ the $\kappa(p)$-algebra $\kappa(p) \otimes_k A$ be étale.