Lemma 106.7.1. Let \mathcal{X} be an algebraic stack over a scheme S. Assume \mathcal{I}_\mathcal {X} \to \mathcal{X} is locally of finite presentation. Let A \to B be a flat S-algebra homomorphism. Let x be an object of \mathcal{X} over A and set y = x|_ B. Then \text{Inf}_ x(M) \otimes _ A B = \text{Inf}_ y(M \otimes _ A B).
Proof. Recall that \text{Inf}_ x(M) is the set of automorphisms of the trivial deformation of x to A[M] which induce the identity automorphism of x over A. The trivial deformation is the pullback of x to \mathop{\mathrm{Spec}}(A[M]) via \mathop{\mathrm{Spec}}(A[M]) \to \mathop{\mathrm{Spec}}(A). Let G \to \mathop{\mathrm{Spec}}(A) be the automorphism group algebraic space of x (this exists because \mathcal{X} is an algebraic space). Let e : \mathop{\mathrm{Spec}}(A) \to G be the neutral element. The discussion in More on Morphisms of Spaces, Section 76.17 gives
\text{Inf}_ x(M) = \mathop{\mathrm{Hom}}\nolimits _ A(e^*\Omega _{G/A}, M)
By the same token
\text{Inf}_ y(M \otimes _ A B) = \mathop{\mathrm{Hom}}\nolimits _ B(e_ B^*\Omega _{G_ B/B}, M \otimes _ A B)
Since G \to \mathop{\mathrm{Spec}}(A) is locally of finite presentation by assumption, we see that \Omega _{G/A} is locally of finite presentation, see More on Morphisms of Spaces, Lemma 76.7.15. Hence e^*\Omega _{G/A} is a finitely presented A-module. Moreover, \Omega _{G_ B/B} is the pullback of \Omega _{G/A} by More on Morphisms of Spaces, Lemma 76.7.12. Therefore e_ B^*\Omega _{G_ B/B} = e^*\Omega _{G/A} \otimes _ A B. we conclude by More on Algebra, Lemma 15.65.4. \square
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