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The Stacks project

Lemma 106.7.2. Let \mathcal{X} be an algebraic stack over a base scheme S. Assume \mathcal{I}_\mathcal {X} \to \mathcal{X} is locally of finite presentation. Let (A' \to A, x) be a deformation situation. Then the functor

F : B' \longmapsto \{ \text{lifts of }x|_{B' \otimes _{A'} A}\text{ to } B'\} /\text{isomorphisms}

is a sheaf on the site (\textit{Aff}/\mathop{\mathrm{Spec}}(A'))_{fppf} of Topologies, Definition 34.7.8.

Proof. Let \{ T'_ i \to T'\} _{i = 1, \ldots n} be a standard fppf covering of affine schemes over A'. Write T' = \mathop{\mathrm{Spec}}(B'). As usual denote

T'_{i_0 \ldots i_ p} = T'_{i_0} \times _{T'} \ldots \times _{T'} T'_{i_ p} = \mathop{\mathrm{Spec}}(B'_{i_0 \ldots i_ p})

where the ring is a suitable tensor product. Set B = B' \otimes _{A'} A and B_{i_0 \ldots i_ p} = B'_{i_0 \ldots i_ p} \otimes _{A'} A. Denote y = x|_ B and y_{i_0 \ldots i_ p} = x|_{B_{i_0 \ldots i_ p}}. Let \gamma _ i \in F(B'_ i) such that \gamma _{i_0} and \gamma _{i_1} map to the same element of F(B'_{i_0i_1}). We have to find a unique \gamma \in F(B') mapping to \gamma _ i in F(B'_ i).

Choose an actual object y'_ i of \textit{Lift}(y_ i, B'_ i) in the isomorphism class \gamma _ i. Choose isomorphisms \varphi _{i_0i_1} : y'_{i_0}|_{B'_{i_0i_1}} \to y'_{i_1}|_{B'_{i_0i_1}} in the category \textit{Lift}(y_{i_0i_1}, B'_{i_0i_1}). If the maps \varphi _{i_0i_1} satisfy the cocycle condition, then we obtain our object \gamma because \mathcal{X} is a stack in the fppf topology. The cocycle condition is that the composition

y'_{i_0}|_{B'_{i_0i_1i_2}} \xrightarrow {\varphi _{i_0i_1}|_{B'_{i_0i_1i_2}}} y'_{i_1}|_{B'_{i_0i_1i_2}} \xrightarrow {\varphi _{i_1i_2}|_{B'_{i_0i_1i_2}}} y'_{i_2}|_{B'_{i_0i_1i_2}} \xrightarrow {\varphi _{i_2i_0}|_{B'_{i_0i_1i_2}}} y'_{i_0}|_{B'_{i_0i_1i_2}}

is the identity. If not, then these maps give elements

\delta _{i_0i_1i_2} \in \text{Inf}_{y_{i_0i_1i_2}}(J_{i_0i_1i_2}) = \text{Inf}_ y(J) \otimes _ B B_{i_0i_1i_2}

Here J = \mathop{\mathrm{Ker}}(B' \to B) and J_{i_0 \ldots i_ p} = \mathop{\mathrm{Ker}}(B'_{i_0 \ldots i_ p} \to B_{i_0 \ldots i_ p}). The equality in the displayed equation holds by Lemma 106.7.1 applied to B' \to B'_{i_0 \ldots i_ p} and y and y_{i_0 \ldots i_ p}, the flatness of the maps B' \to B'_{i_0 \ldots i_ p} which also guarantees that J_{i_0 \ldots i_ p} = J \otimes _{B'} B'_{i_0 \ldots i_ p}. A computation (omitted) shows that \delta _{i_0i_1i_2} gives a 2-cocycle in the Čech complex

\prod \text{Inf}_ y(J) \otimes _ B B_{i_0} \to \prod \text{Inf}_ y(J) \otimes _ B B_{i_0i_1} \to \prod \text{Inf}_ y(J) \otimes _ B B_{i_0i_1i_2} \to \ldots

By Descent, Lemma 35.9.2 this complex is acyclic in positive degrees and has H^0 = \text{Inf}_ y(J). Since \text{Inf}_{y_{i_0i_1}}(J_{i_0i_1}) acts on morphisms (Artin's Axioms, Remark 98.21.4) this means we can modify our choice of \varphi _{i_0i_1} to get to the case where \delta _{i_0i_1i_2} = 0.

Uniqueness. We still have to show there is at most one \gamma restricting to \gamma _ i for all i. Suppose we have objects y', z' of \textit{Lift}(y, B') and isomorphisms \psi _ i : y'|_{B'_ i} \to z'|_{B'_ i} in \textit{Lift}(y_ i, B'_ i). Then we can consider

\psi _{i_1}^{-1} \circ \psi _{i_0} \in \text{Inf}_{y_{i_0i_1}}(J_{i_0i_1}) = \text{Inf}_ y(J) \otimes _ B B_{i_0i_1}

Arguing as before, the obstruction to finding an isomorphism between y' and z' over B' is an element in the H^1 of the Čech complex displayed above which is zero. \square


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