The Stacks project

Lemma 70.12.2. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $\mathbf{Z}$ with $Y$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_ i : X_ i \to Y_ i)$ of morphisms of algebraic spaces over $I$, such that the transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$ are affine, such that $f_ i$ is proper and of finite presentation, such that $Y_ i$ is of finite presentation over $\mathbf{Z}$, and such that $(X \to Y) = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$.

Proof. By Lemma 70.12.1 we can write $X = \mathop{\mathrm{lim}}\nolimits _{k \in K} X_ k$ with $X_ k \to Y$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition 70.8.1) we can write $Y = \mathop{\mathrm{lim}}\nolimits _{j \in J} Y_ j$ with $Y_ j$ of finite presentation over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to Y_ j$ of finite presentation with $X_ k \cong Y \times _{Y_ j} X_{k, j}$ as algebraic spaces over $Y$, see Lemma 70.7.1. After increasing $j$ we may assume $X_{k, j} \to Y_ j$ is proper, see Lemma 70.6.13. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to Y_ j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $Y_{j'} \to Y_ j$ whose base change to $Y$ gives the morphism $X_{k'} \to X_ k$ (follows again from Lemma 70.7.1). These morphisms form the transition morphisms of the system. Some details omitted. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A0X. Beware of the difference between the letter 'O' and the digit '0'.