The Stacks project

87.22 Taut ring maps and representability by algebraic spaces

In this section we briefly show that morphisms between locally countably index formal algebraic spaces correspond étale locally to taut continuous ring homomorphisms between weakly admissible topological rings having countable fundamental systems of open ideals. In fact, this is rather clear from Lemma 87.19.8 and we encourage the reader to skip this section.

Lemma 87.22.1. Let $B \to A$ be an arrow of $\textit{WAdm}^{count}$. The following are equivalent

  1. $B \to A$ is taut (Definition 87.5.1),

  2. for $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots $ a fundamental system of weak ideals of definitions there exist a commutative diagram

    \[ \xymatrix{ A \ar[r] & \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 \\ B \ar[r] \ar[u] & \ldots \ar[r] & B/J_3 \ar[r] \ar[u] & B/J_2 \ar[r] \ar[u] & B/J_1 \ar[u] } \]

    such that $A_{n + 1}/J_ nA_{n + 1} = A_ n$ and $A = \mathop{\mathrm{lim}}\nolimits A_ n$ as topological ring.

Moreover, these equivalent conditions define a local property, i.e., they satisfy axioms (1), (2), (3).

Proof. The equivalence of (a) and (b) is immediate. Below we will give an algebraic proof of the axioms, but it turns out we've already proven them. Namely, using Lemma 87.19.8 the equivalent conditions (a) and (b) translate to saying the corresponding morphism of affine formal algebraic spaces is representable by algebraic spaces. Since this condition is “étale local on the source and target” by Lemma 87.19.4 we immediately get axioms (1), (2), and (3).

Direct algebraic proof of (1), (2), (3). Let a diagram (87.21.2.1) as in Situation 87.21.2 be given. By Example 87.19.11 the maps $A \to (A')^\wedge $ and $B \to (B')^\wedge $ satisfy (a) and (b).

Assume (a) and (b) hold for $\varphi $. Let $J \subset B$ be a weak ideal of definition. Then the closure of $JA$, resp. $J(B')^\wedge $ is a weak ideal of definition $I \subset A$, resp. $J' \subset (B')^\wedge $. Then the closure of $I(A')^\wedge $ is a weak ideal of definition $I' \subset (A')^\wedge $. A topological argument shows that $I'$ is also the closure of $J(A')^\wedge $ and of $J'(A')^\wedge $. Finally, as $J$ runs over a fundamental system of weak ideals of definition of $B$ so do the ideals $I$ and $I'$ in $A$ and $(A')^\wedge $. It follows that (a) holds for $\varphi '$. This proves (1).

Assume $A \to A'$ is faithfully flat and that (a) and (b) hold for $\varphi '$. Let $J \subset B$ be a weak ideal of definition. Using (a) and (b) for the maps $B \to (B')^\wedge \to (A')^\wedge $ we find that the closure $I'$ of $J(A')^\wedge $ is a weak ideal of definition. In particular, $I'$ is open and hence the inverse image of $I'$ in $A$ is open. Now we have (explanation below)

\begin{align*} A \cap I' & = A \cap \bigcap (J(A')^\wedge + \mathop{\mathrm{Ker}}((A')^\wedge \to A'/I_0A')) \\ & = A \cap \bigcap \mathop{\mathrm{Ker}}((A')^\wedge \to A'/JA' + I_0 A') \\ & = \bigcap (JA + I_0) \end{align*}

which is the closure of $JA$ by Lemma 87.4.2. The intersections are over weak ideals of definition $I_0 \subset A$. The first equality because a fundamental system of neighbourhoods of $0$ in $(A')^\wedge $ are the kernels of the maps $(A')^\wedge \to A'/I_0A'$. The second equality is trivial. The third equality because $A \to A'$ is faithfully flat, see Algebra, Lemma 10.82.11. Thus the closure of $JA$ is open. By Lemma 87.4.10 the closure of $JA$ is a weak ideal of definition of $A$. Finally, given a weak ideal of definition $I \subset A$ we can find $J$ such that $J(A')^\wedge $ is contained in the closure of $I(A')^\wedge $ by property (a) for $B \to (B')^\wedge $ and $\varphi '$. Thus we see that (a) holds for $\varphi $. This proves (2).

We omit the proof of (3). $\square$

Lemma 87.22.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent

  1. for every commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to a taut map $B \to A$ of $\textit{WAdm}^{count}$,

  2. there exists a covering $\{ Y_ j \to Y\} $ as in Definition 87.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 87.11.1 such that each $X_{ji} \to Y_ j$ corresponds to a taut ring map in $\textit{WAdm}^{count}$,

  3. there exist a covering $\{ X_ i \to X\} $ as in Definition 87.11.1 and for each $i$ a factorization $X_ i \to Y_ i \to Y$ where $Y_ i$ is an affine formal algebraic space, $Y_ i \to Y$ is representable by algebraic spaces and étale, and $X_ i \to Y_ i$ corresponds to a taut ring map in $\textit{WAdm}^{count}$, and

  4. $f$ is representable by algebraic spaces.

Proof. The property of a map in $\textit{WAdm}^{count}$ being “taut” is a local property by Lemma 87.22.1. Thus Lemma 87.21.3 exactly tells us that (1), (2), and (3) are equivalent. On the other hand, by Lemma 87.19.8 being “taut” on maps in $\textit{WAdm}^{count}$ corresponds exactly to being “representable by algebraic spaces” for the corresponding morphisms of countably indexed affine formal algebraic spaces. Thus the implication (1) $\Rightarrow $ (2) of Lemma 87.19.4 shows that (4) implies (1) of the current lemma. Similarly, the implication (4) $\Rightarrow $ (1) of Lemma 87.19.4 shows that (2) implies (4) of the current lemma. $\square$


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