The Stacks project

96.20 Strong formal effectiveness

In this section we demonstrate how a strong version of effectiveness of formal objects implies openness of versality.

Lemma 96.20.1. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ having (RS*). Let $x$ be an object of $\mathcal{X}$ over an affine scheme $U$ of finite type over $S$. Let $u_ n \in U$, $n \geq 1$ be finite type points such that (a) there are no specializations $u_ n \leadsto u_ m$ for $n \not= m$, and (b) $x$ is not versal at $u_ n$ for all $n$. Then there exist morphisms

\[ x \to x_1 \to x_2 \to \ldots \quad \text{in }\mathcal{X}\text{ lying over }\quad U \to U_1 \to U_2 \to \ldots \]

over $S$ such that

  1. for each $n$ the morphism $U \to U_ n$ is a first order thickening,

  2. for each $n$ we have a short exact sequence

    \[ 0 \to \kappa (u_ n) \to \mathcal{O}_{U_ n} \to \mathcal{O}_{U_{n - 1}} \to 0 \]

    with $U_0 = U$ for $n = 1$,

  3. for each $n$ there does not exist a pair $(W, \alpha )$ consisting of an open neighbourhood $W \subset U_ n$ of $u_ n$ and a morphism $\alpha : x_ n|_ W \to x$ such that the composition

    \[ x|_{U \cap W} \xrightarrow {\text{restriction of }x \to x_ n} x_ n|_ W \xrightarrow {\alpha } x \]

    is the canonical morphism $x|_{U \cap W} \to x$.

Proof. Since there are no specializations among the points $u_ n$ (and in particular the $u_ n$ are pairwise distinct), for every $n$ we can find an open $U' \subset U$ such that $u_ n \in U'$ and $u_ i \not\in U'$ for $i = 1, \ldots , n - 1$. By Lemma 96.19.1 for each $n \geq 1$ we can find

\[ x \to y_ n \quad \text{in }\mathcal{X}\text{ lying over}\quad U \to T_ n \]

such that

  1. the morphism $U \to T_ n$ is a first order thickening,

  2. we have a short exact sequence

    \[ 0 \to \kappa (u_ n) \to \mathcal{O}_{T_ n} \to \mathcal{O}_ U \to 0 \]
  3. there does not exist a pair $(W, \alpha )$ consisting of an open neighbourhood $W \subset T_ n$ of $u_ n$ and a morphism $\beta : y_ n|_ W \to x$ such that the composition

    \[ x|_{U \cap W} \xrightarrow {\text{restriction of }x \to y_ n} y_ n|_ W \xrightarrow {\beta } x \]

    is the canonical morphism $x|_{U \cap W} \to x$.

Thus we can define inductively

\[ U_1 = T_1, \quad U_{n + 1} = U_ n \amalg _ U T_{n + 1} \]

Setting $x_1 = y_1$ and using (RS*) we find inductively $x_{n + 1}$ over $U_{n + 1}$ restricting to $x_ n$ over $U_ n$ and $y_{n + 1}$ over $T_{n + 1}$. Property (1) for $U \to U_ n$ follows from the construction of the pushout in More on Morphisms, Lemma 37.14.3. Property (2) for $U_ n$ similarly follows from property (2) for $T_ n$ by the construction of the pushout. After shrinking to an open neighbourhood $U'$ of $u_ n$ as discussed above, property (3) for $(U_ n, x_ n)$ follows from property (3) for $(T_ n, y_ n)$ simply because the corresponding open subschemes of $T_ n$ and $U_ n$ are isomorphic. Some details omitted. $\square$

Remark 96.20.2 (Strong effectiveness). Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume we have

  1. an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$,

  2. an inverse system $(R_ n)$ of $\Lambda $-algebras with surjective transition maps whose kernels are locally nilpotent,

  3. a system $(\xi _ n)$ of objects of $\mathcal{X}$ lying over the system $(\mathop{\mathrm{Spec}}(R_ n))$.

In this situation, set $R = \mathop{\mathrm{lim}}\nolimits R_ n$. We say that $(\xi _ n)$ is effective if there exists an object $\xi $ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(R)$ whose restriction to $\mathop{\mathrm{Spec}}(R_ n)$ gives the system $(\xi _ n)$.

It is not the case that every algebraic stack $\mathcal{X}$ over $S$ satisfies a strong effectiveness axiom of the form: every system $(\xi _ n)$ as in Remark 96.20.2 is effective. An example is given in Examples, Section 108.68.

Lemma 96.20.3. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume

  1. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,

  2. $\mathcal{X}$ has (RS*),

  3. $\mathcal{X}$ is limit preserving,

  4. systems $(\xi _ n)$ as in Remark 96.20.2 where $\mathop{\mathrm{Ker}}(R_ m \to R_ n)$ is an ideal of square zero for all $m \geq n$ are effective.

Then $\mathcal{X}$ satisfies openness of versality.

Proof. Choose a scheme $U$ locally of finite type over $S$, a finite type point $u_0$ of $U$, and an object $x$ of $\mathcal{X}$ over $U$ such that $x$ is versal at $u_0$. After shrinking $U$ we may assume $U$ is affine and $U$ maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$ of $S$. Let $E \subset U$ be the set of finite type points $u$ such that $x$ is not versal at $u$. By Lemma 96.19.2 if $u \in E$ then $u_0$ is not a specialization of $u$. If openness of versality does not hold, then $u_0$ is in the closure $\overline{E}$ of $E$. By Properties, Lemma 28.5.13 we may choose a countable subset $E' \subset E$ with the same closure as $E$. By Properties, Lemma 28.5.12 we may assume there are no specializations among the points of $E'$. Observe that $E'$ has to be (countably) infinite as $u_0$ isn't the specialization of any point of $E'$ as pointed out above. Thus we can write $E' = \{ u_1, u_2, u_3, \ldots \} $, there are no specializations among the $u_ i$, and $u_0$ is in the closure of $E'$.

Choose $x \to x_1 \to x_2 \to \ldots $ lying over $U \to U_1 \to U_2 \to \ldots $ as in Lemma 96.20.1. Write $U_ n = \mathop{\mathrm{Spec}}(R_ n)$ and $U = \mathop{\mathrm{Spec}}(R_0)$. Set $R = \mathop{\mathrm{lim}}\nolimits R_ n$. Observe that $R \to R_0$ is surjective with kernel an ideal of square zero. By assumption (4) we get $\xi $ over $\mathop{\mathrm{Spec}}(R)$ whose base change to $R_ n$ is $x_ n$. By assumption (3) we get that $\xi $ comes from an object $\xi '$ over $U' = \mathop{\mathrm{Spec}}(R')$ for some finite type $\Lambda $-subalgebra $R' \subset R$. After increasing $R'$ we may and do assume that $R' \to R_0$ is surjective, so that $U \subset U'$ is a first order thickening. Thus we now have

\[ x \to x_1 \to x_2 \to \ldots \to \xi ' \text{ lying over } U \to U_1 \to U_2 \to \ldots \to U' \]

By assumption (1) there is an algebraic space $Z$ over $S$ representing

\[ (\mathit{Sch}/U)_{fppf} \times _{x, \mathcal{X}, \xi '} (\mathit{Sch}/U')_{fppf} \]

see Algebraic Stacks, Lemma 92.10.11. By construction of $2$-fibre products, a $T$-valued point of $Z$ corresponds to a triple $(a, a', \alpha )$ consisting of morphisms $a : T \to U$, $a' : T \to U'$ and a morphism $\alpha : a^*x \to (a')^*\xi '$. We obtain a commutative diagram

\[ \xymatrix{ U \ar[rd] \ar[rdd] \ar[rrd] \\ & Z \ar[r]_{p'} \ar[d]^ p & U' \ar[d] \\ & U \ar[r] & S } \]

The morphism $i : U \to Z$ comes the isomorphism $x \to \xi '|_ U$. Let $z_0 = i(u_0) \in Z$. By Lemma 96.12.6 we see that $Z \to U'$ is smooth at $z_0$. After replacing $U$ by an affine open neighbourhood of $u_0$, replacing $U'$ by the corresponding open, and replacing $Z$ by the intersection of the inverse images of these opens by $p$ and $p'$, we reach the situation where $Z \to U'$ is smooth along $i(U)$. Note that this also involves replacing $u_ n$ by a subsequence, namely by those indices such that $u_ n$ is in the open. Moreover, condition (3) of Lemma 96.20.1 is clearly preserved by shrinking $U$ (all of the schemes $U$, $U_ n$, $U'$ have the same underlying topological space). Since $U \to U'$ is a first order thickening of affine schemes, we can choose a morphism $i' : U' \to Z$ such that $p' \circ i' = \text{id}_{U'}$ and whose restriction to $U$ is $i$ (More on Morphisms of Spaces, Lemma 74.19.6). Pulling back the universal morphism $p^*x \to (p')^*\xi '$ by $i'$ we obtain a morphism

\[ \xi ' \to x \]

lying over $p \circ i' : U' \to U$ such that the composition

\[ x \to \xi ' \to x \]

is the identity. Recall that we have $x_1 \to \xi '$ lying over the morphism $U_1 \to U'$. Composing we get a morphism $x_1 \to x$ whose existence contradicts condition (3) of Lemma 96.20.1. This contradiction finishes the proof. $\square$

Remark 96.20.4. There is a way to deduce openness of versality of the diagonal of an category fibred in groupoids from a strong formal effectiveness axiom. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume

  1. $\Delta _\Delta : \mathcal{X} \to \mathcal{X} \times _{\mathcal{X} \times \mathcal{X}} \mathcal{X}$ is representable by algebraic spaces,

  2. $\mathcal{X}$ has (RS*),

  3. $\mathcal{X}$ is limit preserving,

  4. given an inverse system $(R_ n)$ of $S$-algebras as in Remark 96.20.2 where $\mathop{\mathrm{Ker}}(R_ m \to R_ n)$ is an ideal of square zero for all $m \geq n$ the functor

    \[ \mathcal{X}_{\mathop{\mathrm{Spec}}(\mathop{\mathrm{lim}}\nolimits R_ n)} \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n \mathcal{X}_{\mathop{\mathrm{Spec}}(R_ n)} \]

    is fully faithful.

Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ satisfies openness of versality. This follows by applying Lemma 96.20.3 to fibre products of the form $\mathcal{X} \times _{\Delta , \mathcal{X} \times \mathcal{X}, y} (\mathit{Sch}/V)_{fppf}$ for any affine scheme $V$ locally of finite presentation over $S$ and object $y$ of $\mathcal{X} \times \mathcal{X}$ over $V$. If we ever need this, we will change this remark into a lemma and provide a detailed proof.


Comments (1)

Comment #5870 by MCO on

It may be worth pointing out (unless it is already in here somewhere) that strong formal effectiveness holds for qcqs algebraic spaces (Theorem 1.1 in Bhatt-Algebraize).


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