Exercise 1, § 7

Let $k$ be a ring, $A$ a linearly topologized $k$-algebra. We say that $A$ is a formally étale (resp. formally neat) $k$-algebra if it satisfies the following condition: whatever the $k$-algebra $C$, endowed with the discrete topology, and the square zero ideal $N$ of $C$, every continuous homomorphism of $k$-algebras $A \to C/N$ admits a unique (resp. at most one) lifting $A \to C$. In order that $A$ be formally étale over $k$, it is necessary and sufficient that it be formally smooth and formally neat. Every quotient of $k$ is formally neat.

a) Prove the statements analogous to those of props. 3 and 4 for formally neat or formally étale algebras.

b) Let $J$ be an ideal of $A$ such that the topology of $A$ is the $J$-adic topology; denote by $\widehat{\Omega}_k(A)$ the separated completion of the $A$-module $\Omega_k(A)$ (for the $J$-adic topology). In order that the topological $k$-algebra $A$ be formally neat, it is necessary and sufficient that $\widehat{\Omega}_k(A)$ be zero (observe that each of these properties is equivalent to the fact that every $k$-derivation of $A$ into an $A$-module annihilated by a power of $J$ is zero).