The Stacks project

97.20 Strong formal effectiveness

In this section we demonstrate how a strong version of effectiveness of formal objects implies openness of versality. The proof of [Theorem 1.1, Bhatt-Algebraize] shows that quasi-compact and quasi-separated algebraic spaces satisfy the strong formal effectiveness discussed in Remark 97.20.2. In addition, the theory we develop is nonempty: we use it later to show openness of versality for the stack of coherent sheaves and for moduli of complexes, see Quot, Theorems 98.6.1 and 98.16.12.

Lemma 97.20.1. 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 97.19.1 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 97.20.2 (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 97.20.2 is effective. An example is given in Examples, Section 109.72.

Lemma 97.20.3. 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 97.20.2 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 97.19.2 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 97.20.1. 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 93.10.11. 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 97.12.6 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 97.20.1 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 75.19.6). 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 97.20.1. This contradiction finishes the proof. $\square$

Remark 97.20.4. 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 97.20.2 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 97.20.3 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 (2)

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).

Comment #6082 by on

Thanks! I have added a remark to the introduction of the section. See here.


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