Proposition 6

— Let K be a commutative ring, A the polynomial algebra $K[X_i]_{i \in I}$, and S the set of elements F of A such that $dF = 0$.

a) If K is a ring of characteristic 0, then S = K. b) If K is a ring of characteristic $p \neq 0$, then $S = K[X_i^p]_{i \in I}$; if moreover K is perfect, then $K = A^P$.

The mapping $F \mapsto dF$ of A into the module $\Omega_K(A)$ of K-differentials of A is K-linear and satisfies the relation

(III, p. 569). Therefore S is a subalgebra.

When K is of characteristic $p \neq 0$, put $T = K[X_i^p]_{i \in I}$; we thus have $T = A^P$ if K is perfect ($V$, p. 4, Prop. 2); moreover, we have $d(X_i^p) = pX_i^{p-1} \cdot dX_i = 0$ for all $i \in I$, hence the subalgebra S of A contains T. If K is of characteristic 0, we put $T = K$, and still find that $T \subset S$. It remains to show that S is contained in T.

For every finite subset J of I let $A_J$ be the subalgebra of A generated by the family $(X_j)_{j \in J}$. We have $A_\emptyset = K$ and $A = \bigcup_{J \in I} A_J$; so it suffices to prove the relation $S \cap A_J \subset T$, which we shall accomplish by induction on the cardinal of J. Thus let J be a finite subset of I such that $S \cap A_J \subset T$, let i be an element of $I - J$ and $J' = J \cup \{ i \}$. Every element F of A may be written in just one way in the form

with $F_n \in A_J$ for all $n \geq 0$, and then

Suppose that F belongs to S; the family $(dX_r)_{r \in J}$ is a basis of the A-module $\Omega_K(A)$ (III, p. 570) and $dF_n$ is a linear combination of the differentials $dX_j$ for $j \in J$ because $F_n \in A_J = K[X_j]_{j \in J}$. By (6) we then have $dF_n = 0$ and $nF_n = 0$ for each integer $n \geq 0$. By the induction hypothesis $F_n \in T$ for all $n$, since $dF_n = 0$.

a) If K is of characteristic 0, we have $nF_n = 0$ for all $n \geq 1$, whence $F_n = 0$ by Prop. 1 ($V$, p. 2); so we have $F = F_0$, whence $F \in T$.

b) If K is of characteristic $p \neq 0$, then A is an algebra over the field $\mathbf{F}_p$ and the relation $nF_n = 0$ implies $F_n = 0$ for every integer $n$ not divisible by $p$. So we have $F = \sum_{m=0}^m F_{mp} X_i^{mp}$, whence $F \in T$