The Stacks project

Lemma 69.23.4. In Situation 69.23.1. Let $f : X \to Y$ be a morphism of algebraic spaces quasi-separated and of finite type over $B$. Let

\[ \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_{i_1} } } \quad \text{and}\quad \vcenter { \xymatrix{ Y \ar[r] \ar[d] & V \ar[d] \\ B \ar[r] & B_{i_2} } } \]

be diagrams as in ( Let $X = \mathop{\mathrm{lim}}\nolimits _{i \geq i_1} X_ i$ and $Y = \mathop{\mathrm{lim}}\nolimits _{i \geq i_2} Y_ i$ be the corresponding limit descriptions as in Lemma 69.23.3. Then there exists an $i_0 \geq \max (i_1, i_2)$ and a morphism

\[ (f_ i)_{i \geq i_0} : (X_ i)_{i \geq i_0} \to (Y_ i)_{i \geq i_0} \]

of inverse systems over $(B_ i)_{i \geq i_0}$ such that such that $f = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} f_ i$. If $(g_ i)_{i \geq i_0} : (X_ i)_{i \geq i_0} \to (Y_ i)_{i \geq i_0}$ is a second morphism of inverse systems over $(B_ i)_{i \geq i_0}$ such that such that $f = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} g_ i$ then $f_ i = g_ i$ for all $i \gg i_0$.

Proof. Since $V \to B_{i_2}$ is of finite presentation and $X = \mathop{\mathrm{lim}}\nolimits _{i \geq i_1} X_ i$ we can appeal to Proposition 69.3.10 as improved by Lemma 69.4.5 to find an $i_0 \geq \max (i_1, i_2)$ and a morphism $h : X_{i_0} \to V$ over $B_{i_2}$ such that $X \to X_{i_0} \to V$ is equal to $X \to Y \to V$. For $i \geq i_0$ we get a commutative solid diagram

\[ \xymatrix{ X \ar[d] \ar[r] & X_ i \ar[r] \ar@{..>}[d] \ar@/_2pc/[dd] |!{[d];[ld]}\hole & X_{i_0} \ar[d]^ h \\ Y \ar[r] \ar[d] & Y_ i \ar[r] \ar[d] & V \ar[d] \\ B \ar[r] & B_ i \ar[r] & B_{i_0} } \]

Since $X \to X_ i$ has scheme theoretically dense image and since $Y_ i$ is the scheme theoretic image of $Y \to B_ i \times _{B_{i_2}} V$ we find that the morphism $X_ i \to B_ i \times _{B_{i_2}} V$ induced by the diagram factors through $Y_ i$ (Morphisms of Spaces, Lemma 66.16.6). This proves existence.

Uniqueness. Let $E_ i \to X_ i$ be the equalizer of $f_ i$ and $g_ i$ for $i \geq i_0$. We have $E_ i = Y_ i \times _{\Delta , Y_ i \times _{B_ i} Y_ i, (f_ i, g_ i)} X_ i$. Hence $E_ i \to X_ i$ is a monomorphism of finite presentation as a base change of the diagonal of $Y_ i$ over $B_ i$, see Morphisms of Spaces, Lemmas 66.4.1 and 66.28.10. Since $X_ i$ is a closed subspace of $B_ i \times _{B_{i_0}} X_{i_0}$ and similarly for $Y_ i$ we see that

\[ E_ i = X_ i \times _{(B_ i \times _{B_{i_0}} X_{i_0})} (B_ i \times _{B_{i_0}} E_{i_0}) = X_ i \times _{X_{i_0}} E_{i_0} \]

Similarly, we have $X = X \times _{X_{i_0}} E_{i_0}$. Hence we conclude that $E_ i = X_ i$ for $i$ large enough by Lemma 69.6.10. $\square$

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 0CPA. Beware of the difference between the letter 'O' and the digit '0'.