Proposition 3

— Let $B$ be a perfect ring of characteristic $p$ and $A$ a subring of $B$. Write $\mathbf{A}^{p^{-\infty}} = \bigcup_{f \geq 0} \mathbf{A}^{p^{-f}}$ and denote by $j$ the canonical injection of $A$ in $\mathbf{A}^{p^{-\infty}}$. Then $\mathbf{A}^{p^{-\infty}}$ is the smallest perfect subring of $B$ containing $A$ and $(\mathbf{A}^{p^{-\infty}}, j)$ is a perfect closure of $A$.

For each integer $f \in \mathbf{Z}$ denote by $\pi_f$ the automorphism $b \mapsto b^{p^f}$ of $B$. The sequence of subrings $\pi_{-f}(A)$ of $B$ (for $f \geq 0$) is ascending and its union $\mathbf{A}^{p^{-\infty}}$ is thus a subring of $B$. We have $\pi_1(\mathbf{A}^{p^{-\infty}}) = \bigcup_{f \geq 0} \pi_{-(f-1)}(A) = \mathbf{A}^{p^{-m}}$, hence $\mathbf{A}^{p^{-\infty}}$ is a perfect subring of $B$. Finally let $B_0$ be a perfect subring of $B$ containing $A$; for every integer $f \geq 0$ we have $\pi_{-f}(A) \subset \pi_{-f}(B_0) = B_0$, whence $\mathbf{A}^{p^{-\infty}} \subset B_0$.

If $v$ is a homomorphism of $A$ into a perfect ring $B'$ of characteristic $p$, then for every integer $f \geq 0$ we can define a homomorphism $h_f$ of $\pi_{-f}(A)$ into $B'$ by $h_f(\pi_{-f}(a)) = v(a)^{p^{-f}}$ for all $a \in A$. We see at once that $h_{f+1}$ agrees with $h_f$ on $\pi_{-f}(A)$; thus there exists a homomorphism $h$ of $\mathbf{A}^{p^{-\infty}}$ into $B'$ which induces $h_f$ on $\pi_{-f}(A)$ for all $f \geq 0$ and in particular, $h$ extends $h_0 = v$. If $h'$ is another extension of $v$ to a homomorphism of $A^{p^{-\infty}}$ into B, the equality $h' = h$ may be established as in the proof of Th. 3.