Lemma 76.17.8 (0CK8)—The Stacks project

Lemma 76.17.8. Let $S$ be a scheme. Consider a commutative diagram of first order thickenings

$\vcenter { \xymatrix{ (T_2 \subset T_2') \ar[d]_{(h, h')} \ar[rr]_{(a_2, a_2')} & & (X_2 \subset X_2') \ar[d]^{(f, f')} \\ (T_1 \subset T_1') \ar[rr]^{(a_1, a_1')} & & (X_1 \subset X_1') } } \quad \begin{matrix} \text{and a commutative} \\ \text{diagram} \end{matrix} \quad \vcenter { \xymatrix{ X_2' \ar[r] \ar[d] & B_2 \ar[d] \\ X_1' \ar[r] & B_1 } }$

of algebraic spaces over $S$ with $X_2 \to X_1$ and $B_2 \to B_1$ étale. For any $\mathcal{O}_{T_1}$ -linear map $\theta _1 : a_1^*\Omega _{X_1/B_1} \to \mathcal{C}_{T_1/T'_1}$ let $\theta _2$ be the composition

$\xymatrix{ a_2^*\Omega _{X_2/B_2} \ar@{=}[r] & h^*a_1^*\Omega _{X_1/B_1} \ar[r]^-{h^*\theta _1} & h^*\mathcal{C}_{T_1/T'_1} \ar[r] & \mathcal{C}_{T_2/T'_2} }$

(equality sign is explained in the proof). Then the diagram

$\xymatrix{ T_2' \ar[rr]_{\theta _2 \cdot a_2'} \ar[d] & & X'_2 \ar[d] \\ T_1' \ar[rr]^{\theta _1 \cdot a_1'} & & X'_1 }$

commutes where the actions $\theta _2 \cdot a_2'$ and $\theta _1 \cdot a_1'$ are as in Remark 76.17.3.

Proof. The equality sign comes from the identification $f^*\Omega _{X_1/S_1} = \Omega _{X_2/S_2}$ we get as the construction of the sheaf of differentials is compatible with étale localization (both on source and target), see Lemma 76.7.3. Namely, using this we have $a_2^*\Omega _{X_2/S_2} = a_2^*f^*\Omega _{X_1/S_1} = h^*a_1^*\Omega _{X_1/S_1}$ because $f \circ a_2 = a_1 \circ h$ . Having said this, the commutativity of the diagram may be checked on étale locally. Thus we may assume $T'_ i$ , $X'_ i$ , $B_2$ , and $B_1$ are schemes and in this case the lemma follows from More on Morphisms, Lemma 37.9.10. Alternative proof: using Lemma 76.9.2 it suffices to show a certain diagram of sheaves of rings on $X_1'$ is commutative; then argue exactly as in the proof of the aforementioned More on Morphisms, Lemma 37.9.10 to see that this is indeed the case. $\square$

The Stacks project

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