The Stacks project

91.10 Modules on first order thickenings of ringed topoi

In this section we discuss some preliminaries to the deformation theory of modules. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. We will freely use the notation introduced in Section 91.9, in particular we will identify the underlying topological topoi. In this section we consider short exact sequences

91.10.0.1
\begin{equation} \label{defos-equation-extension-ringed-topoi} 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \end{equation}

of $\mathcal{O}'$-modules, where $\mathcal{F}$, $\mathcal{K}$ are $\mathcal{O}$-modules and $\mathcal{F}'$ is an $\mathcal{O}'$-module. In this situation we have a canonical $\mathcal{O}$-module map

\[ c_{\mathcal{F}'} : \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \longrightarrow \mathcal{K} \]

where $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp )$. Namely, given local sections $f$ of $\mathcal{I}$ and $s$ of $\mathcal{F}$ we set $c_{\mathcal{F}'}(f \otimes s) = fs'$ where $s'$ is a local section of $\mathcal{F}'$ lifting $s$.

Lemma 91.10.1. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given extensions

\[ 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad \text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0 \]

as in (91.10.0.1) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$.

  1. If there exists an $\mathcal{O}'$-module map $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi $ and $\psi $, then the diagram

    \[ \xymatrix{ \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi } & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes _\mathcal {O} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} } \]

    is commutative.

  2. The set of $\mathcal{O}'$-module maps $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi $ and $\psi $ is, if nonempty, a principal homogeneous space under $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{L})$.

Proof. Part (1) is immediate from the description of the maps. For (2), if $\varphi '$ and $\varphi ''$ are two maps $\mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi $ and $\psi $, then $\varphi ' - \varphi ''$ factors as

\[ \mathcal{F}' \to \mathcal{F} \to \mathcal{L} \to \mathcal{G}' \]

The map in the middle comes from a unique element of $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{L})$ by Modules on Sites, Lemma 18.25.1. Conversely, given an element $\alpha $ of this group we can add the composition (as displayed above with $\alpha $ in the middle) to $\varphi '$. Some details omitted. $\square$

Lemma 91.10.2. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given extensions

\[ 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad \text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0 \]

as in (91.10.0.1) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$. Assume the diagram

\[ \xymatrix{ \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi } & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes _\mathcal {O} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} } \]

is commutative. Then there exists an element

\[ o(\varphi , \psi ) \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{L}) \]

whose vanishing is a necessary and sufficient condition for the existence of a map $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi $ and $\psi $.

Proof. We can construct explicitly an extension

\[ 0 \to \mathcal{L} \to \mathcal{H} \to \mathcal{F} \to 0 \]

by taking $\mathcal{H}$ to be the cohomology of the complex

\[ \mathcal{K} \xrightarrow {1, - \psi } \mathcal{F}' \oplus \mathcal{G}' \xrightarrow {\varphi , 1} \mathcal{G} \]

in the middle (with obvious notation). A calculation with local sections using the assumption that the diagram of the lemma commutes shows that $\mathcal{H}$ is annihilated by $\mathcal{I}$. Hence $\mathcal{H}$ defines a class in

\[ \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{L}) \subset \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}'}(\mathcal{F}, \mathcal{L}) \]

Finally, the class of $\mathcal{H}$ is the difference of the pushout of the extension $\mathcal{F}'$ via $\psi $ and the pullback of the extension $\mathcal{G}'$ via $\varphi $ (calculations omitted). Thus the vanishing of the class of $\mathcal{H}$ is equivalent to the existence of a commutative diagram

\[ \xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r] \ar[d]_{\psi } & \mathcal{F}' \ar[r] \ar[d]_{\varphi '} & \mathcal{F} \ar[r] \ar[d]_\varphi & 0\\ 0 \ar[r] & \mathcal{L} \ar[r] & \mathcal{G}' \ar[r] & \mathcal{G} \ar[r] & 0 } \]

as desired. $\square$

Lemma 91.10.3. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given $\mathcal{O}$-modules $\mathcal{F}$, $\mathcal{K}$ and an $\mathcal{O}$-linear map $c : \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}$. If there exists a sequence (91.10.0.1) with $c_{\mathcal{F}'} = c$ then the set of isomorphism classes of these extensions is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K})$.

Proof. Assume given extensions

\[ 0 \to \mathcal{K} \to \mathcal{F}'_1 \to \mathcal{F} \to 0 \quad \text{and}\quad 0 \to \mathcal{K} \to \mathcal{F}'_2 \to \mathcal{F} \to 0 \]

with $c_{\mathcal{F}'_1} = c_{\mathcal{F}'_2} = c$. Then the difference (in the extension group, see Homology, Section 12.6) is an extension

\[ 0 \to \mathcal{K} \to \mathcal{E} \to \mathcal{F} \to 0 \]

where $\mathcal{E}$ is annihilated by $\mathcal{I}$ (local computation omitted). Hence the sequence is an extension of $\mathcal{O}$-modules, see Modules on Sites, Lemma 18.25.1. Conversely, given such an extension $\mathcal{E}$ we can add the extension $\mathcal{E}$ to the $\mathcal{O}'$-extension $\mathcal{F}'$ without affecting the map $c_{\mathcal{F}'}$. Some details omitted. $\square$

Lemma 91.10.4. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. Assume given $\mathcal{O}$-modules $\mathcal{F}$, $\mathcal{K}$ and an $\mathcal{O}$-linear map $c : \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}$. Then there exists an element

\[ o(\mathcal{F}, \mathcal{K}, c) \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}) \]

whose vanishing is a necessary and sufficient condition for the existence of a sequence (91.10.0.1) with $c_{\mathcal{F}'} = c$.

Proof. We first show that if $\mathcal{K}$ is an injective $\mathcal{O}$-module, then there does exist a sequence (91.10.0.1) with $c_{\mathcal{F}'} = c$. To do this, choose a flat $\mathcal{O}'$-module $\mathcal{H}'$ and a surjection $\mathcal{H}' \to \mathcal{F}$ (Modules on Sites, Lemma 18.28.8). Let $\mathcal{J} \subset \mathcal{H}'$ be the kernel. Since $\mathcal{H}'$ is flat we have

\[ \mathcal{I} \otimes _{\mathcal{O}'} \mathcal{H}' = \mathcal{I}\mathcal{H}' \subset \mathcal{J} \subset \mathcal{H}' \]

Observe that the map

\[ \mathcal{I}\mathcal{H}' = \mathcal{I} \otimes _{\mathcal{O}'} \mathcal{H}' \longrightarrow \mathcal{I} \otimes _{\mathcal{O}'} \mathcal{F} = \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \]

annihilates $\mathcal{I}\mathcal{J}$. Namely, if $f$ is a local section of $\mathcal{I}$ and $s$ is a local section of $\mathcal{H}$, then $fs$ is mapped to $f \otimes \overline{s}$ where $\overline{s}$ is the image of $s$ in $\mathcal{F}$. Thus we obtain

\[ \xymatrix{ \mathcal{I}\mathcal{H}'/\mathcal{I}\mathcal{J} \ar@{^{(}->}[r] \ar[d] & \mathcal{J}/\mathcal{I}\mathcal{J} \ar@{..>}[d]_\gamma \\ \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \ar[r]^-c & \mathcal{K} } \]

a diagram of $\mathcal{O}$-modules. If $\mathcal{K}$ is injective as an $\mathcal{O}$-module, then we obtain the dotted arrow. Denote $\gamma ' : \mathcal{J} \to \mathcal{K}$ the composition of $\gamma $ with $\mathcal{J} \to \mathcal{J}/\mathcal{I}\mathcal{J}$. A local calculation shows the pushout

\[ \xymatrix{ 0 \ar[r] & \mathcal{J} \ar[r] \ar[d]_{\gamma '} & \mathcal{H}' \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{F}' \ar[r] & \mathcal{F} \ar[r] & 0 } \]

is a solution to the problem posed by the lemma.

General case. Choose an embedding $\mathcal{K} \subset \mathcal{K}'$ with $\mathcal{K}'$ an injective $\mathcal{O}$-module. Let $\mathcal{Q}$ be the quotient, so that we have an exact sequence

\[ 0 \to \mathcal{K} \to \mathcal{K}' \to \mathcal{Q} \to 0 \]

Denote $c' : \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}'$ be the composition. By the paragraph above there exists a sequence

\[ 0 \to \mathcal{K}' \to \mathcal{E}' \to \mathcal{F} \to 0 \]

as in (91.10.0.1) with $c_{\mathcal{E}'} = c'$. Note that $c'$ composed with the map $\mathcal{K}' \to \mathcal{Q}$ is zero, hence the pushout of $\mathcal{E}'$ by $\mathcal{K}' \to \mathcal{Q}$ is an extension

\[ 0 \to \mathcal{Q} \to \mathcal{D}' \to \mathcal{F} \to 0 \]

as in (91.10.0.1) with $c_{\mathcal{D}'} = 0$. This means exactly that $\mathcal{D}'$ is annihilated by $\mathcal{I}$, in other words, the $\mathcal{D}'$ is an extension of $\mathcal{O}$-modules, i.e., defines an element

\[ o(\mathcal{F}, \mathcal{K}, c) \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{Q}) = \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}) \]

(the equality holds by the long exact cohomology sequence associated to the exact sequence above and the vanishing of higher ext groups into the injective module $\mathcal{K}'$). If $o(\mathcal{F}, \mathcal{K}, c) = 0$, then we can choose a splitting $s : \mathcal{F} \to \mathcal{D}'$ and we can set

\[ \mathcal{F}' = \mathop{\mathrm{Ker}}(\mathcal{E}' \to \mathcal{D}'/s(\mathcal{F})) \]

so that we obtain the following diagram

\[ \xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r] \ar[d] & \mathcal{F}' \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] & \mathcal{E}' \ar[r] & \mathcal{F} \ar[r] & 0 } \]

with exact rows which shows that $c_{\mathcal{F}'} = c$. Conversely, if $\mathcal{F}'$ exists, then the pushout of $\mathcal{F}'$ by the map $\mathcal{K} \to \mathcal{K}'$ is isomorphic to $\mathcal{E}'$ by Lemma 91.10.3 and the vanishing of higher ext groups into the injective module $\mathcal{K}'$. This gives a diagram as above, which implies that $\mathcal{D}'$ is split as an extension, i.e., the class $o(\mathcal{F}, \mathcal{K}, c)$ is zero. $\square$

Remark 91.10.5. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. A first order thickening $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ is said to be trivial if there exists a morphism of ringed topoi $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ which is a left inverse to $i$. The choice of such a morphism $\pi $ is called a trivialization of the first order thickening. Given $\pi $ we obtain a splitting

91.10.5.1
\begin{equation} \label{defos-equation-splitting-ringed-topoi} \mathcal{O}' = \mathcal{O} \oplus \mathcal{I} \end{equation}

as sheaves of algebras on $\mathcal{C}$ by using $\pi ^\sharp $ to split the surjection $\mathcal{O}' \to \mathcal{O}$. Conversely, such a splitting determines a morphism $\pi $. The category of trivialized first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is equivalent to the category of $\mathcal{O}$-modules.

Remark 91.10.6. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a trivial first order thickening of ringed topoi and let $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a trivialization. Then given any triple $(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of $\mathcal{O}$-modules and a map $c : \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}$ we may set

\[ \mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K} \]

and use the splitting (91.10.5.1) associated to $\pi $ and the map $c$ to define the $\mathcal{O}'$-module structure and obtain an extension (91.10.0.1). We will call $\mathcal{F}'_{c, triv}$ the trivial extension of $\mathcal{F}$ by $\mathcal{K}$ corresponding to $c$ and the trivialization $\pi $. Given any extension $\mathcal{F}'$ as in (91.10.0.1) we can use $\pi ^\sharp : \mathcal{O} \to \mathcal{O}'$ to think of $\mathcal{F}'$ as an $\mathcal{O}$-module extension, hence a class $\xi _{\mathcal{F}'}$ in $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K})$. Lemma 91.10.3 assures that $\mathcal{F}' \mapsto \xi _{\mathcal{F}'}$ induces a bijection

\[ \left\{ \begin{matrix} \text{isomorphism classes of extensions} \\ \mathcal{F}'\text{ as in (08MB) with } c = c_{\mathcal{F}'} \end{matrix} \right\} \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}) \]

Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.

Remark 91.10.7. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i)$, $i = 1, 2$ be first order thickenings with ideal sheaves $\mathcal{I}_ i$. Let $h : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2)$ be a morphism of first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$. Picture

\[ \xymatrix{ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[ld] \ar[rd] & \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \ar[rr]^ h & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2) } \]

Observe that $h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1$ in particular induces an $\mathcal{O}$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}_ i$. Assume furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

\[ \xymatrix{ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } \]

is commutative. Then there is a canonical functoriality

\[ \left\{ \begin{matrix} \mathcal{F}'_2\text{ as in (08MB) with } \\ c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2 \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \mathcal{F}'_1\text{ as in (08MB) with } \\ c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1 \end{matrix} \right\} \]

Namely, thinking of all sheaves $\mathcal{O}$, $\mathcal{O}'_ i$, $\mathcal{F}$, $\mathcal{K}_ i$, etc as sheaves on $\mathcal{C}$, we set given $\mathcal{F}'_2$ the sheaf $\mathcal{F}'_1$ equal to the pushout, i.e., fitting into the following diagram of extensions

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{K}_1 \ar[r] & \mathcal{F}'_1 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

We omit the construction of the $\mathcal{O}'_1$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$).

Remark 91.10.8. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i)$, $\mathcal{I}_ i$, and $h : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2)$ be as in Remark 91.10.7. Assume that we are given trivializations $\pi _ i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ such that $\pi _1 = h \circ \pi _2$. In other words, assume $h$ is a morphism of trivialized first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}_ i$. Assume furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

\[ \xymatrix{ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } \]

is commutative. In this situation the construction of Remark 91.10.6 induces a commutative diagram

\[ \xymatrix{ \{ \mathcal{F}'_2\text{ as in (08MB) with } c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\} \ar[d] \ar[rr] & & \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\ \{ \mathcal{F}'_1\text{ as in (08MB) with } c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\} \ar[rr] & & \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}_1) } \]

where the vertical map on the right is given by functoriality of $\mathop{\mathrm{Ext}}\nolimits $ and the map $\mathcal{K}_2 \to \mathcal{K}_1$ and the vertical map on the left is the one from Remark 91.10.7.

Remark 91.10.9. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i)$, $\mathcal{I}_ i$, and $h : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2)$ be as in Remark 91.10.7. Observe that $h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1$ in particular induces an $\mathcal{O}$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}_ i$. Assume furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

\[ \xymatrix{ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } \]

is commutative. Then we claim the map

\[ \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_2) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_1) \]

sends $o(\mathcal{F}, \mathcal{K}_2, c_2)$ to $o(\mathcal{F}, \mathcal{K}_1, c_1)$.

To prove this claim choose an embedding $j_2 : \mathcal{K}_2 \to \mathcal{K}_2'$ where $\mathcal{K}_2'$ is an injective $\mathcal{O}$-module. As in the proof of Lemma 91.10.4 we can choose an extension of $\mathcal{O}_2$-modules

\[ 0 \to \mathcal{K}_2' \to \mathcal{E}_2 \to \mathcal{F} \to 0 \]

such that $c_{\mathcal{E}_2} = j_2 \circ c_2$. The proof of Lemma 91.10.4 constructs $o(\mathcal{F}, \mathcal{K}_2, c_2)$ as the Yoneda extension class (in the sense of Derived Categories, Section 13.27) of the exact sequence of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}_2 \to \mathcal{K}_2' \to \mathcal{E}_2/\mathcal{K}_2 \to \mathcal{F} \to 0 \]

Let $\mathcal{K}_1'$ be the cokernel of $\mathcal{K}_2 \to \mathcal{K}_1 \oplus \mathcal{K}_2'$. There is an injection $j_1 : \mathcal{K}_1 \to \mathcal{K}_1'$ and a map $\mathcal{K}_2' \to \mathcal{K}_1'$ forming a commutative square. We form the pushout:

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2' \ar[r] \ar[d] & \mathcal{E}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}_1' \ar[r] & \mathcal{E}_1 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

There is a canonical $\mathcal{O}_1$-module structure on $\mathcal{E}_1$ and for this structure we have $c_{\mathcal{E}_1} = j_1 \circ c_1$ (this uses the commutativity of the diagram involving $c_1$ and $c_2$ above). The procedure of Lemma 91.10.4 tells us that $o(\mathcal{F}, \mathcal{K}_1, c_1)$ is the Yoneda extension class of the exact sequence of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}_1 \to \mathcal{K}_1' \to \mathcal{E}_1/\mathcal{K}_1 \to \mathcal{F} \to 0 \]

Since we have maps of exact sequences

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[d] \ar[r] & \mathcal{K}_2' \ar[d] \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K}_2 \ar[r] & \mathcal{K}_2' \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

we conclude that the claim is true.

Remark 91.10.10. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. We define a sequence of morphisms of first order thickenings

\[ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3) \]

of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ to be a complex if the corresponding maps between the ideal sheaves $\mathcal{I}_ i$ give a complex of $\mathcal{O}$-modules $\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3)$ factors through $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3)$, i.e., the first order thickening $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1)$ of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is trivial and comes with a canonical trivialization $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$.

We say a sequence of morphisms of first order thickenings

\[ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3) \]

of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is a short exact sequence if the corresponding maps between ideal sheaves is a short exact sequence

\[ 0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0 \]

of $\mathcal{O}$-modules.

Remark 91.10.11. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let

\[ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3) \]

be a complex first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, see Remark 91.10.10. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2, 3$ be pairs consisting of an $\mathcal{O}$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}_ i$. Assume given a short exact sequence of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0 \]

such that

\[ \vcenter { \xymatrix{ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathcal{I}_3 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_3} \ar[d] & \mathcal{K}_3 \ar[d] \\ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_2} & \mathcal{K}_2 } } \]

are commutative. Finally, assume given an extension

\[ 0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0 \]

as in (91.10.0.1) with $\mathcal{K} = \mathcal{K}_2$ of $\mathcal{O}'_2$-modules with $c_{\mathcal{F}'_2} = c_2$. In this situation we can apply the functoriality of Remark 91.10.7 to obtain an extension $\mathcal{F}'_1$ of $\mathcal{O}'_1$-modules (we'll describe $\mathcal{F}'_1$ in this special case below). By Remark 91.10.6 using the canonical splitting $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of Remark 91.10.10 we obtain $\xi _{\mathcal{F}'_1} \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}_1)$. Finally, we have the obstruction

\[ o(\mathcal{F}, \mathcal{K}_3, c_3) \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_3) \]

see Lemma 91.10.4. In this situation we claim that the canonical map

\[ \partial : \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}_1) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_3) \]

coming from the short exact sequence $0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0$ sends $\xi _{\mathcal{F}'_1}$ to the obstruction class $o(\mathcal{F}, \mathcal{K}_3, c_3)$.

To prove this claim choose an embedding $j : \mathcal{K}_3 \to \mathcal{K}$ where $\mathcal{K}$ is an injective $\mathcal{O}$-module. We can lift $j$ to a map $j' : \mathcal{K}_2 \to \mathcal{K}$. Set $\mathcal{E}'_2 = j'_*\mathcal{F}'_2$ equal to the pushout of $\mathcal{F}'_2$ by $j'$ so that $c_{\mathcal{E}'_2} = j' \circ c_2$. Picture:

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d]_{j'} & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{E}'_2 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

Set $\mathcal{E}'_3 = \mathcal{E}'_2$ but viewed as an $\mathcal{O}'_3$-module via $\mathcal{O}'_3 \to \mathcal{O}'_2$. Then $c_{\mathcal{E}'_3} = j \circ c_3$. The proof of Lemma 91.10.4 constructs $o(\mathcal{F}, \mathcal{K}_3, c_3)$ as the boundary of the class of the extension of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}/\mathcal{K}_3 \to \mathcal{E}'_3/\mathcal{K}_3 \to \mathcal{F} \to 0 \]

On the other hand, note that $\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3$ hence the class $\xi _{\mathcal{F}'_1}$ is the class of the extension

\[ 0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3 \to \mathcal{F} \to 0 \]

seen as a sequence of $\mathcal{O}$-modules using $\pi ^\sharp $ where $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}/\mathcal{K}_3 \ar[r] & \mathcal{E}'_3/\mathcal{K}_3 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

which is $\mathcal{O}$-linear (with the $\mathcal{O}$-module structures given above).

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