Lemma 1

Suppose that the $A$-algebra $B$ admits a generating family $t = (t_i)_{i \in I}$, with defining relations $f_\lambda \in A[T]$ ($\lambda \in A$). The $B$-linear homomorphism

defined by $\psi(\alpha, (b_i)) = \Omega_0(\rho)(\alpha) + \sum_{i \in I} b_i dt_i$, is surjective; its kernel is generated by the elements $r_\lambda = \left( d^A f_\lambda(t), \left( \frac{\partial f_\lambda}{\partial T_i}(t) \right)_{i \in I} \right)$ for $\lambda \in A$.

Consider the sequence of B-modules and B-linear mappings

where $\varphi$ is the homomorphism such that $\varphi(e_\lambda) = n_\lambda$; it is a matter of proving that this sequence is exact. By A, II, p. 36, th. 1, it is enough to prove that, for every B-module M, the sequence

is exact. In view of the universal property of the module of differentials (A, III, p. 134), this sequence is identified with

where $\psi'(D) = (D \circ \rho, (D(t_i)))$ and $\varphi'(\Delta, (m_i)) = (f^\Delta_\lambda(t) + \sum_i \frac{\partial f_\lambda}{\partial T_i}(t) m_i)_{\lambda \in A}$ (in accordance with A, V, p. 121, for every polynomial $f = \sum_{\alpha \in \mathbf{N}^{(I)}} c_\alpha T^\alpha$ of $A[T]$, $f^\Delta(t)$ denotes the element $\sum_\alpha t^\alpha \Delta(c_\alpha)$). But the exactness of this sequence follows from loc. cit., prop. 1, in view of the fact that a derivation $D : B \to M$ is $k$-linear if and only if this is so for $D \circ \rho$.

Let A be a ring. There exists a unique structure of $\mathbf{Z}$-algebra on A; one writes simply $\Omega(A)$ for the A-module $\Omega_{\mathbf{Z}}(A)$. If $\rho : k \to A$ is a homomorphism of rings, there is a canonical exact sequence of A-modules (A, III, p. 136, prop. 21)

Suppose that A contains a subfield, and let P be the prime subfield of A; then $\Omega(P)$ is zero and the canonical homomorphism of A-modules $\Omega(A) \to \Omega_P(A)$ is bijective. If moreover A is of characteristic $p \neq 0$ (which means by definition that $p$ is a prime number, that $p1_A = 0$ and $1_A \neq 0$), then P is identified with $\mathbf{F}_p$. Moreover, every derivation of A vanishes on the subring $A^p$; for every subring $k$ of A contained in $A^p$ (and, in particular, for every perfect subfield $k$ of A), the canonical mapping $\Omega(A) \to \Omega_k(A)$ is bijective.

Let A be a ring of characteristic $p \neq 0$ and $(f_i)_{1 \leq i \leq n}$ a finite sequence of elements of A. Let $A_n$ denote the quotient ring of the polynomial ring $A[T_1, \ldots, T_n]$ by the ideal generated by the polynomials $T_i^p - f_i$, for $1 \leq i \leq n$.