The Stacks project

Lemma 91.13.1. Assume given a commutative diagram of morphisms ringed topoi
\begin{equation} \label{defos-equation-huge-1-ringed-topoi} \vcenter { \xymatrix{ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2) \ar[r]_{i_2} \ar[d]_{f_2} \ar[ddl]_ g & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_2), \mathcal{O}'_2) \ar[d]^{f'_2} \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}_2), \mathcal{O}_{\mathcal{B}_2}) \ar[r]^{t_2} \ar[ddl]|\hole & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) \ar[ddl] \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1), \mathcal{O}_1) \ar[r]_{i_1} \ar[d]_{f_1} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_1), \mathcal{O}'_1) \ar[d]^{f'_1} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}_1), \mathcal{O}_{\mathcal{B}_1}) \ar[r]^{t_1} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) } } \end{equation}

whose horizontal arrows are first order thickenings. Set $\mathcal{G}_ j = \mathop{\mathrm{Ker}}(i_ j^\sharp )$ and assume given a map of $g^{-1}\mathcal{O}_1$-modules $\nu : g^{-1}\mathcal{G}_1 \to \mathcal{G}_2$ giving rise to the commutative diagram
\begin{equation} \label{defos-equation-huge-2-ringed-topoi} \vcenter { \xymatrix{ & 0 \ar[r] & \mathcal{G}_2 \ar[r] & \mathcal{O}'_2 \ar[r] & \mathcal{O}_2 \ar[r] & 0 \\ & 0 \ar[r]|\hole & f_2^{-1}\mathcal{J}_2 \ar[u]_{c_2} \ar[r] & f_2^{-1}\mathcal{O}_{\mathcal{B}'_2} \ar[u] \ar[r]|\hole & f_2^{-1}\mathcal{O}_{\mathcal{B}_2} \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G}_1 \ar[ruu] \ar[r] & \mathcal{O}'_1 \ar[r] & \mathcal{O}_1 \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & f_1^{-1}\mathcal{J}_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] & f_1^{-1}\mathcal{O}_{\mathcal{B}'_1} \ar[ruu]|\hole \ar[u] \ar[r] & f_1^{-1}\mathcal{O}_{\mathcal{B}_1} \ar[ruu]|\hole \ar[u] \ar[r] & 0 } } \end{equation}

with front and back solutions to ( (The north-north-west arrows are maps on $\mathcal{C}_2$ after applying $g^{-1}$ to the source.)

  1. There exist a canonical element in $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_2}( Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)$ whose vanishing is a necessary and sufficient condition for the existence of a morphism of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_2), \mathcal{O}'_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_1), \mathcal{O}'_1)$ fitting into ( compatibly with $\nu $.

  2. If there exists a morphism $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_2), \mathcal{O}'_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_1), \mathcal{O}'_1)$ fitting into ( compatibly with $\nu $ the set of all such morphisms is a principal homogeneous space under

    \[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_1}( \Omega _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, g_*\mathcal{G}_2) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( g^*\Omega _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) = \mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_2}( Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2). \]

Proof. The proof of this lemma is identical to the proof of Lemma 91.7.1. We urge the reader to read that proof instead of this one. We will identify the underlying topoi for every thickening in sight (we have already used this convention in the statement). The equalities in the last statement of the lemma are immediate from the definitions. Thus we will work with the groups $\mathop{\mathrm{Ext}}\nolimits ^ k_{\mathcal{O}_2}( Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)$, $k = 0, 1$ in the rest of the proof. We first argue that we can reduce to the case where the underlying topos of all ringed topoi in the lemma is the same.

To do this, observe that $g^{-1}\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}$ is equal to the naive cotangent complex of the homomorphism of sheaves of rings $g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}_1} \to g^{-1}\mathcal{O}_1$, see Modules on Sites, Lemma 18.33.5. Moreover, the degree $0$ term of $\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}$ is a flat $\mathcal{O}_1$-module, hence the canonical map

\[ Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}} \longrightarrow g^{-1}\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}} \otimes _{g^{-1}\mathcal{O}_1} \mathcal{O}_2 \]

induces an isomorphism on cohomology sheaves in degrees $0$ and $-1$. Thus we may replace the Ext groups of the lemma with

\[ \mathop{\mathrm{Ext}}\nolimits ^ k_{g^{-1}\mathcal{O}_1}( g^{-1}\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) = \mathop{\mathrm{Ext}}\nolimits ^ k_{g^{-1}\mathcal{O}_1}( \mathop{N\! L}\nolimits _{g^{-1}\mathcal{O}_1/g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) \]

The set of morphism of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_2), \mathcal{O}'_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_1), \mathcal{O}'_1)$ fitting into ( compatibly with $\nu $ is in one-to-one bijection with the set of homomorphisms of $g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}'_1}$-algebras $g^{-1}\mathcal{O}'_1 \to \mathcal{O}'_2$ which are compatible with $f^\sharp $ and $\nu $. In this way we see that we may assume we have a diagram ( of sheaves on a site $\mathcal{C}$ (with $f_1 = f_2 = \text{id}$ on underlying topoi) and we are looking to find a homomorphism of sheaves of rings $\mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into it.

In the rest of the proof of the lemma we assume all underlying topological spaces are the same, i.e., we have a diagram ( of sheaves on a site $\mathcal{C}$ (with $f_1 = f_2 = \text{id}$ on underlying topoi) and we are looking for homomorphisms of sheaves of rings $\mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into it. As ext groups we will use $\mathop{\mathrm{Ext}}\nolimits ^ k_{\mathcal{O}_1}( \mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)$, $k = 0, 1$.

Step 1. Construction of the obstruction class. Consider the sheaf of sets

\[ \mathcal{E} = \mathcal{O}'_1 \times _{\mathcal{O}_2} \mathcal{O}'_2 \]

This comes with a surjective map $\alpha : \mathcal{E} \to \mathcal{O}_1$ and hence we can use $\mathop{N\! L}\nolimits (\alpha )$ instead of $\mathop{N\! L}\nolimits _{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}$, see Modules on Sites, Lemma 18.35.2. Set

\[ \mathcal{I}' = \mathop{\mathrm{Ker}}(\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}_1) \quad \text{and}\quad \mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}] \to \mathcal{O}_1) \]

There is a surjection $\mathcal{I}' \to \mathcal{I}$ whose kernel is $\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]$. We obtain two homomorphisms of $\mathcal{O}_{\mathcal{B}'_2}$-algebras

\[ a : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}'_1 \quad \text{and}\quad b : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}'_2 \]

which induce maps $a|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_1$ and $b|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_2$. Both $a$ and $b$ annihilate $(\mathcal{I}')^2$. Moreover $a$ and $b$ agree on $\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]$ as maps into $\mathcal{G}_2$ because the left hand square of ( is commutative. Thus the difference $b|_{\mathcal{I}'} - \nu \circ a|_{\mathcal{I}'}$ induces a well defined $\mathcal{O}_1$-linear map

\[ \xi : \mathcal{I}/\mathcal{I}^2 \longrightarrow \mathcal{G}_2 \]

which sends the class of a local section $f$ of $\mathcal{I}$ to $a(f') - \nu (b(f'))$ where $f'$ is a lift of $f$ to a local section of $\mathcal{I}'$. We let $[\xi ] \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_1}(\mathop{N\! L}\nolimits (\alpha ), \mathcal{G}_2)$ be the image (see below).

Step 2. Vanishing of $[\xi ]$ is necessary. Let us write $\Omega = \Omega _{\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}]/\mathcal{O}_{\mathcal{B}_1}} \otimes _{\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}]} \mathcal{O}_1$. Observe that $\mathop{N\! L}\nolimits (\alpha ) = (\mathcal{I}/\mathcal{I}^2 \to \Omega )$ fits into a distinguished triangle

\[ \Omega [0] \to \mathop{N\! L}\nolimits (\alpha ) \to \mathcal{I}/\mathcal{I}^2[1] \to \Omega [1] \]

Thus we see that $[\xi ]$ is zero if and only if $\xi $ is a composition $\mathcal{I}/\mathcal{I}^2 \to \Omega \to \mathcal{G}_2$ for some map $\Omega \to \mathcal{G}_2$. Suppose there exists a homomorphisms of sheaves of rings $\varphi : \mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into ( In this case consider the map $\mathcal{O}'_1[\mathcal{E}] \to \mathcal{G}_2$, $f' \mapsto b(f') - \varphi (a(f'))$. A calculation shows this annihilates $\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]$ and induces a derivation $\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}] \to \mathcal{G}_2$. The resulting linear map $\Omega \to \mathcal{G}_2$ witnesses the fact that $[\xi ] = 0$ in this case.

Step 3. Vanishing of $[\xi ]$ is sufficient. Let $\theta : \Omega \to \mathcal{G}_2$ be a $\mathcal{O}_1$-linear map such that $\xi $ is equal to $\theta \circ (\mathcal{I}/\mathcal{I}^2 \to \Omega )$. Then a calculation shows that

\[ b + \theta \circ d : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \longrightarrow \mathcal{O}'_2 \]

annihilates $\mathcal{I}'$ and hence defines a map $\mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into (

Proof of (2) in the special case above. Omitted. Hint: This is exactly the same as the proof of (2) of Lemma 91.2.1. $\square$

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