Lemma 105.7.3. Let $\mathcal{X}$ be an algebraic stack over a scheme $S$ whose structure morphism $\mathcal{X} \to S$ is locally of finite presentation. Let $A \to B$ be a flat $S$-algebra homomorphism. Let $x$ be an object of $\mathcal{X}$ over $A$. Then $T_ x(M) \otimes _ A B = T_ y(M \otimes _ A B)$.

Proof. Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$. We first reduce the lemma to the case where $x$ lifts to $U$. Recall that $T_ x(M)$ is the set of isomorphism classes of lifts of $x$ to $A[M]$. Therefore Lemma 105.7.21 says that the rule

$A_1 \mapsto T_{x|_{A_1}}(M \otimes _ A A_1)$

is a sheaf on the small étale site of $\mathop{\mathrm{Spec}}(A)$; the tensor product is needed to make $A[M] \to A_1[M \otimes _ A A_1]$ a flat ring map. We may choose a faithfully flat étale ring map $A \to A_1$ such that $x|_{A_1}$ lifts to a morphism $u_1 : \mathop{\mathrm{Spec}}(A_1) \to U$, see for example Sheaves on Stacks, Lemma 95.19.10. Write $A_2 = A_1 \otimes _ A A_1$ and set $B_1 = B \otimes _ A A_1$ and $B_2 = B \otimes _ A A_2$. Consider the diagram

$\xymatrix{ 0 \ar[r] & T_ y(M \otimes _ A B) \ar[r] & T_{y|_{B_1}}(M \otimes _ A B_1) \ar[r] & T_{y|_{B_2}}(M \otimes _ A B_2) \\ 0 \ar[r] & T_ x(M) \ar[r] \ar[u] & T_{x|_{A_1}}(M \otimes _ A A_1) \ar[r] \ar[u] & T_{x|_{A_2}}(M \otimes _ A A_2) \ar[u] }$

The rows are exact by the sheaf condition. We have $M \otimes _ A B_ i = (M \otimes _ A A_ i) \otimes _{A_ i} B_ i$. Thus if we prove the result for the middle and right vertical arrow, then the result follows. This reduces us to the case discussed in the next paragraph.

Assume that $x$ is the image of a morphism $u : \mathop{\mathrm{Spec}}(A) \to U$. Observe that $T_ u(M) \to T_ x(M)$ is surjective since $U \to \mathcal{X}$ is smooth and representable by algebraic spaces, see Criteria for Representability, Lemma 96.6.3 (see discussion preceding it for explanation) and More on Morphisms of Spaces, Lemma 75.19.6. Set $R = U \times _\mathcal {X} U$. Recall that we obtain a groupoid $(U, R, s, t, c, e, i)$ in algebraic spaces with $\mathcal{X} = [U/R]$. By Artin's Axioms, Lemma 97.21.6 we have an exact sequence

$T_{e \circ u}(M) \to T_ u(M) \oplus T_ u(M) \to T_ x(M) \to 0$

where the zero on the right was shown above. A similar sequence holds for the base change to $B$. Thus the result we want follows if we can prove the result of the lemma for $T_ u(M)$ and $T_{e \circ u}(M)$. This reduces us to the case discussed in the next paragraph.

Assume that $\mathcal{X} = X$ is an algebraic space locally of finite presentation over $S$. Then we have

$T_ x(M) = \mathop{\mathrm{Hom}}\nolimits _ A(x^*\Omega _{X/S}, M)$

by the discussion in More on Morphisms of Spaces, Section 75.17. By the same token

$T_ y(M \otimes _ A B) = \mathop{\mathrm{Hom}}\nolimits _ B(y^*\Omega _{X/S}, M \otimes _ A B)$

Since $X \to S$ is locally of finite presentation, we see that $\Omega _{X/S}$ is locally of finite presentation, see More on Morphisms of Spaces, Lemma 75.7.15. Hence $x^*\Omega _{X/S}$ is a finitely presented $A$-module. Clearly, we have $y^*\Omega _{X/S} = x^*\Omega _{X/S} \otimes _ A B$. we conclude by More on Algebra, Lemma 15.65.4. $\square$

[1] This lemma applies: $\Delta : \mathcal{X} \to \mathcal{X} \times _ S \mathcal{X}$ is locally of finite presentation by Morphisms of Stacks, Lemma 100.27.6 and the assumption that $\mathcal{X} \to S$ is locally of finite presentation. Therefore $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is locally of finite presentation as a base change of $\Delta$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DNT. Beware of the difference between the letter 'O' and the digit '0'.