Exercise 2, § 3

Let A be a topological ring, B a topological A-algebra. One says that B is a formally smooth A-algebra¹ if, for every discrete topological A-algebra C, every ideal j of C, of square zero, and every continuous A-homomorphism $u : B \to C/j$, there exists a continuous A-homomorphism $v : B \to C$ which, by passing to the quotient, induces $u$.

Let A be a ring, B an A-algebra. One says that B is a smooth A-algebra if B is a formally smooth A-algebra when A and B are endowed with the discrete topologies. a) Let A be a topological ring and B a formally smooth A-algebra. Let C be a ring and j an ideal of C. Endow C with the j-adic topology, and suppose that C is separated and complete for this topology. Then, for every continuous A-homomorphism $u : B \to C/j$, there exists a continuous A-homomorphism $v : B \to C$ which, by passing to the quotient, induces $u$. b) Let A be a ring. Every polynomial algebra over A is a smooth A-algebra. c) Let A be a topological ring. If B is a formally smooth A-algebra and C a formally smooth B-algebra, then C is a formally smooth A-algebra. d) Let A be a topological ring, B a formally smooth A-algebra, A’ a topological A-algebra. Then the topological A’-algebra $B \otimes_A A'$ (III, § 2, exerc. 28) is a formally smooth A’-algebra. e) Let A be a topological ring, B a formally smooth A-algebra. Let S (resp. T) be a multiplicative subset of A (resp. B) such that the image of S in B is contained in T. Then $T^{-1}B$ is a formally smooth $S^{-1}A$-algebra (see III, § 2, exerc. 27 for the topology of $T^{-1}B$ and $S^{-1}A$). f) Let $n$ be an integer $\geqslant 1$, A a topological ring, $(B_i)_{1 \leqslant i \leqslant n}$ a family of topological A-algebras. In order that $\prod_{i=1}^n B_i$ be a formally smooth A-algebra, it is necessary and sufficient that $B_i$ be a formally smooth A-algebra for $1 \leqslant i \leqslant n$. g) Let A be a topological ring, B a topological A-algebra, $\hat{A}$ and $\hat{B}$ the respective separated completions of A and B. Then the following three conditions are equivalent: (i) B is a formally smooth A-algebra; (ii) $\hat{B}$ is a formally smooth A-algebra; (iii) $\hat{B}$ is a formally smooth $\hat{A}$-algebra. h) Let us take up again the notations and hypotheses of e). Then $B\{T^{-1}\}$ is a formally smooth $A\{S^{-1}\}$-algebra.