The Stacks project

91.14 Deformations of algebraic spaces

In this section we spell out what the results in Section 91.13 mean for deformations of algebraic spaces.

Lemma 91.14.1. Let $S$ be a scheme. Let $i : Z \to Z'$ be a morphism of algebraic spaces over $S$. The following are equivalent

  1. $i$ is a thickening of algebraic spaces as defined in More on Morphisms of Spaces, Section 76.9, and

  2. the associated morphism $i_{small} : (\mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}), \mathcal{O}_ Z) \to (\mathop{\mathit{Sh}}\nolimits (Z'_{\acute{e}tale}), \mathcal{O}_{Z'})$ of ringed topoi (Properties of Spaces, Lemma 66.21.3) is a thickening in the sense of Section 91.9.

Proof. We stress that this is not a triviality.

Assume (1). By More on Morphisms of Spaces, Lemma 76.9.6 the morphism $i$ induces an equivalence of small étale sites and in particular of topoi. Of course $i^\sharp $ is surjective with locally nilpotent kernel by definition of thickenings.

Assume (2). (This direction is less important and more of a curiosity.) For any étale morphism $Y' \to Z'$ we see that $Y = Z \times _{Z'} Y'$ has the same étale topos as $Y'$. In particular, $Y'$ is quasi-compact if and only if $Y$ is quasi-compact because being quasi-compact is a topos theoretic notion (Sites, Lemma 7.17.3). Having said this we see that $Y'$ is quasi-compact and quasi-separated if and only if $Y$ is quasi-compact and quasi-separated (because you can characterize $Y'$ being quasi-separated by saying that for all $Y'_1, Y'_2$ quasi-compact algebraic spaces étale over $Y'$ we have that $Y'_1 \times _{Y'} Y'_2$ is quasi-compact). Take $Y'$ affine. Then the algebraic space $Y$ is quasi-compact and quasi-separated. For any quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{F}$ we have $H^ q(Y, \mathcal{F}) = H^ q(Y', (Y \to Y')_*\mathcal{F})$ because the étale topoi are the same. Then $H^ q(Y', (Y \to Y')_*\mathcal{F}) = 0$ because the pushforward is quasi-coherent (Morphisms of Spaces, Lemma 67.11.2) and $Y$ is affine. It follows that $Y'$ is affine by Cohomology of Spaces, Proposition 69.16.7 (there surely is a proof of this direction of the lemma avoiding this reference). Hence $i$ is an affine morphism. In the affine case it follows easily from the conditions in Section 91.9 that $i$ is a thickening of algebraic spaces. $\square$

Lemma 91.14.2. Let $S$ be a scheme. Let $Y \subset Y'$ be a first order thickening of algebraic spaces over $S$. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. If there exists a flat morphism $f' : X' \to Y'$ of algebraic spaces over $S$ and an isomorphsm $a : X \to X' \times _{Y'} Y$ over $Y$, then

  1. the set of isomorphism classes of pairs $(f' : X' \to Y', a)$ is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/Y}, f^*\mathcal{C}_{Y/Y'})$, and

  2. the set of automorphisms of $\varphi : X' \to X'$ over $Y'$ which reduce to the identity on $X' \times _{Y'} Y$ is $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/Y}, f^*\mathcal{C}_{Y/Y'})$.

Proof. We will apply the material on deformations of ringed topoi to the small étale topoi of the algebraic spaces in the lemma. We may think of $X$ as a closed subspace of $X'$ so that $(f, f') : (X \subset X') \to (Y \subset Y')$ is a morphism of first order thickenings. By Lemma 91.14.1 this translates into a morphism of thickenings of ringed topoi. Then we see from More on Morphisms of Spaces, Lemma 76.18.1 (or from the more general Lemma 91.11.2) that the ideal sheaf of $X$ in $X'$ is equal to $f^*\mathcal{C}_{Y'/Y}$ and this is in fact equivalent to flatness of $X'$ over $Y'$. Hence we have a commutative diagram

\[ \xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{Y/Y'} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_ X \ar[r] & 0 \\ 0 \ar[r] & f_{small}^{-1}\mathcal{C}_{Y/Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_{Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_ Y \ar[u] \ar[r] & 0 } \]

Please compare with (91.13.0.1). Observe that automorphisms $\varphi $ as in (2) give automorphisms $\varphi ^\sharp : \mathcal{O}_{X'} \to \mathcal{O}_{X'}$ fitting in the diagram above. Conversely, an automorphism $\alpha : \mathcal{O}_{X'} \to \mathcal{O}_{X'}$ fitting into the diagram of sheaves above is equal to $\varphi ^\sharp $ for some automorphism $\varphi $ as in (2) by More on Morphisms of Spaces, Lemma 76.9.2. Finally, by More on Morphisms of Spaces, Lemma 76.9.7 if we find another sheaf of rings $\mathcal{A}$ on $X_{\acute{e}tale}$ fitting into the diagram

\[ \xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{Y/Y'} \ar[r] & \mathcal{A} \ar[r] & \mathcal{O}_ X \ar[r] & 0 \\ 0 \ar[r] & f_{small}^{-1}\mathcal{C}_{Y/Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_{Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_ Y \ar[u] \ar[r] & 0 } \]

then there exists a first order thickening $X \subset X''$ with $\mathcal{O}_{X''} = \mathcal{A}$ and applying More on Morphisms of Spaces, Lemma 76.9.2 once more, we obtain a morphism $(f, f'') : (X \subset X'') \to (Y \subset Y')$ with all the desired properties. Thus part (1) follows from Lemma 91.13.3 and part (2) from part (2) of Lemma 91.13.1. (Note that $\mathop{N\! L}\nolimits _{X/Y}$ as defined for a morphism of algebraic spaces in More on Morphisms of Spaces, Section 76.21 agrees with $\mathop{N\! L}\nolimits _{X/Y}$ as used in Section 91.13.) $\square$

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F} \to \mathcal{G}$ be a homomorphism of $\mathcal{O}_ X$-modules (not necessarily quasi-coherent). Consider the functor

\[ F : \left\{ \begin{matrix} \text{extensions of }f^{-1}\mathcal{O}_ B\text{ algebras} \\ 0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_ X \to 0 \\ \text{where }\mathcal{F}\text{ is an ideal of square zero} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{extensions of }f^{-1}\mathcal{O}_ B\text{ algebras} \\ 0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O}_ X \to 0 \\ \text{where }\mathcal{G}\text{ is an ideal of square zero} \end{matrix} \right\} \]

given by pushout.

Lemma 91.14.3. In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_ X(\mathcal{F}) \to DQ_ X(\mathcal{G})$ (Derived Categories of Spaces, Section 75.19) is an isomorphism. Then the functor $F$ is an equivalence of categories.

Proof. Recall that $\mathop{N\! L}\nolimits _{X/B}$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$, see More on Morphisms of Spaces, Lemma 76.21.4. Hence our assumption implies the maps

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathop{N\! L}\nolimits _{X/B}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathop{N\! L}\nolimits _{X/B}, \mathcal{G}) \]

are isomorphisms for all $i$. This implies our functor is fully faithful by Lemma 91.13.1. On the other hand, the functor is essentially surjective by Lemma 91.13.3 because we have the solutions $\mathcal{O}_ X \oplus \mathcal{F}$ and $\mathcal{O}_ X \oplus \mathcal{G}$ in both categories. $\square$

Let $S$ be a scheme. Let $B \subset B'$ be a first order thickening of algebraic spaces over $S$ with ideal sheaf $\mathcal{J}$ which we view either as a quasi-coherent $\mathcal{O}_ B$-module or as a quasi-coherent sheaf of ideals on $B'$, see More on Morphisms of Spaces, Section 76.9. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F} \to \mathcal{G}$ be a homomorphism of $\mathcal{O}_ X$-modules (not necessarily quasi-coherent). Let $c : f^{-1}\mathcal{J} \to \mathcal{F}$ be a map of $f^{-1}\mathcal{O}_ B$-modules and denote $c' : f^{-1}\mathcal{J} \to \mathcal{G}$ the composition. Consider the functor

\[ FT : \{ \text{solutions to }(08UF) \text{ for }\mathcal{F}\text{ and }c\} \longrightarrow \{ \text{solutions to }(08UF) \text{ for }\mathcal{G}\text{ and }c'\} \]

given by pushout.

Lemma 91.14.4. In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_ X(\mathcal{F}) \to DQ_ X(\mathcal{G})$ (Derived Categories of Spaces, Section 75.19) is an isomorphism. Then the functor $FT$ is an equivalence of categories.

Proof. A solution of (91.13.0.1) for $\mathcal{F}$ in particular gives an extension of $f^{-1}\mathcal{O}_{B'}$-algebras

\[ 0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_ X \to 0 \]

where $\mathcal{F}$ is an ideal of square zero. Similarly for $\mathcal{G}$. Moreover, given such an extension, we obtain a map $c_{\mathcal{O}'} : f^{-1}\mathcal{J} \to \mathcal{F}$. Thus we are looking at the full subcategory of such extensions of $f^{-1}\mathcal{O}_{B'}$-algebras with $c = c_{\mathcal{O}'}$. Clearly, if $\mathcal{O}'' = F(\mathcal{O}')$ where $F$ is the equivalence of Lemma 91.14.3 (applied to $X \to B'$ this time), then $c_{\mathcal{O}''}$ is the composition of $c_{\mathcal{O}'}$ and the map $\mathcal{F} \to \mathcal{G}$. This proves the 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 0D15. Beware of the difference between the letter 'O' and the digit '0'.