The Stacks project

Lemma 105.7.1. 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 75.17 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 75.7.15. 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 75.7.12. 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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