Lemma 106.7.4. Let \mathcal{X} be an algebraic stack over a scheme S whose structure morphism \mathcal{X} \to S is locally of finite presentation. Let (A' \to A, x) be a deformation situation. If there exists a faithfully flat finitely presented A'-algebra B' and an object y' of \mathcal{X} over B' lifting x|_{B' \otimes _{A'} A}, then there exists an object x' over A' lifting x.
Proof. Let I = \mathop{\mathrm{Ker}}(A' \to A). Set B'_1 = B' \otimes _{A'} B' and B'_2 = B' \otimes _{A'} B' \otimes _{A'} B'. Let J = IB', J_1 = IB'_1, J_2 = IB'_2 and B = B'/J, B_1 = B'_1/J_1, B_2 = B'_2/J_2. Set y = x|_ B, y_1 = x|_{B_1}, y_2 = x|_{B_2}. Let F be the fppf sheaf of Lemma 106.7.2 (which applies, see footnote in the proof of Lemma 106.7.3). Thus we have an equalizer diagram
\xymatrix{ F(A') \ar[r] & F(B') \ar@<1ex>[r] \ar@<-1ex>[r] & F(B'_1) }
On the other hand, we have F(B') = \text{Lift}(y, B'), F(B'_1) = \text{Lift}(y_1, B'_1), and F(B'_2) = \text{Lift}(y_2, B'_2) in the terminology from Artin's Axioms, Section 98.21. These sets are nonempty and are (canonically) principal homogeneous spaces for T_ y(J), T_{y_1}(J_1), T_{y_2}(J_2), see Artin's Axioms, Lemma 98.21.2. Thus the difference of the two images of y' in F(B'_1) is an element
\delta _1 \in T_{y_1}(J_1) = T_ x(I) \otimes _ A B_1
The equality in the displayed equation holds by Lemma 106.7.3 applied to A' \to B'_1 and x and y_1, the flatness of A' \to B'_1 which also guarantees that J_1 = I \otimes _{A'} B'_1. We have similar equalities for B' and B'_2. A computation (omitted) shows that \delta _1 gives a 1-cocycle in the Čech complex
T_ x(I) \otimes _ A B \to T_ x(I) \otimes _ A B_1 \to T_ x(I) \otimes _ A B_2 \to \ldots
By Descent, Lemma 35.9.2 this complex is acyclic in positive degrees and has H^0 = T_ x(I). Thus we may choose an element in T_ x(I) \otimes _ A B = T_ y(J) whose boundary is \delta _1. Replacing y' by the result of this element acting on it, we find a new choice y' with \delta _1 = 0. Thus y' maps to the same element under the two maps F(B') \to F(B'_1) and we obtain an element o F(A') by the sheaf condition. \square
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