Lemma 85.17.16. Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which has the cancellation property. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $g \circ f$ and $g$ satisfies the equivalent conditions of Lemma 85.17.3 then so does $f : X \to Y$.

**Proof.**
Choose a covering $\{ Z_ k \to Z\} $ as in Definition 85.7.1. For each $k$ choose a covering $\{ Y_{kj} \to Z_ k \times _ Z Y\} $ as in Definition 85.7.1. For each $k$ and $j$ choose a covering $\{ X_{kji} \to Y_{kj} \times _ Y X\} $ as in Definition 85.7.1. Let $X_{kji} \to Y_{jk}$ and $Y_{jk} \to Z_ k$ correspond to arrows $B_{kj} \to A_{kji}$ and $C_ k \to B_{kj}$ of $\text{WAdm}^{count}$. If $g \circ f$ and $g$ satisfies the equivalent conditions of Lemma 85.17.3 then $C_ k \to B_{kj}$ and $C_ k \to A_{kji}$ satisfy $P$. Hence $B_{kj} \to A_{kji}$ does too and we conclude.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)