The Stacks project

Lemma 105.16.6. Let $\mathcal{X}$ be an algebraic stack with affine diagonal. Let $B$ be a ring. Let $f_ i : \mathop{\mathrm{Spec}}(B) \to \mathcal{X}$, $i = 1, 2$ be two morphisms. Let $t : f_1^* \to f_2^*$ be an isomorphism of the tensor functors $f_ i^* : \text{QCoh}(\mathcal{O}_\mathcal {X}) \to \text{Mod}_ B$. Then there is a $2$-arrow $f_1 \to f_2$ inducing $t$.

Proof. Choose an affine scheme $U = \mathop{\mathrm{Spec}}(A)$ and a surjective smooth morphism $g : U \to \mathcal{X}$, see Properties of Stacks, Lemma 99.6.2. Since the diagonal of $\mathcal{X}$ is affine, we see that $U_ i = \mathop{\mathrm{Spec}}(B) \times _{f_ i, \mathcal{X}, g} U$ is affine. Say $U_ i = \mathop{\mathrm{Spec}}(C_ i)$. Then $C_ i$ is the $B$-algebra endowed with ring map $A \to C_ i$ constructed in Lemma 105.16.2 using the functor $F = f_ i^*$. Therefore $t$ induces an isomorphism $C_1 \to C_2$ of $B$-algebras, compatible with the ring maps $A \to C_1$ and $A \to C_2$. In other words, we have a commutative diagrams

\[ \xymatrix{ U_ i \ar[r] \ar[d] & U \ar[d]^ g \\ \mathop{\mathrm{Spec}}(B) \ar[r]^{f_ i} & \mathcal{X} } \quad \quad \xymatrix{ & U_2 \ar[ld] \ar[d]^{\cong } \ar[rd] \\ \mathop{\mathrm{Spec}}(B) & U_1 \ar[l] \ar[r] & U } \]

This already shows that the objects $f_1$ and $f_2$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(B)$ become isomorphic after the smooth covering $\{ U_1 \to \mathop{\mathrm{Spec}}(B)\} $. To show that this descends to an isomorphism of $f_1$ and $f_2$ over $\mathop{\mathrm{Spec}}(B)$, we have to show that our isomorphism (which comes from the commutative diagrams above) is compatible with the descent data over $U_1 \times _{\mathop{\mathrm{Spec}}(B)} U_1$. For this we observe that $U \times _\mathcal {X} U$ is affine too, that we have the morphism $g' : U \times _\mathcal {X} U \to \mathcal{X}$, and that

\[ U_ i \times _{\mathop{\mathrm{Spec}}(B)} U_ i = \mathop{\mathrm{Spec}}(B) \times _{f_ i, \mathcal{X}, g'} (U \times _\mathcal {X} U) \]

It follows that the isomorphism $C_1 \otimes _ B C_1 \to C_2 \otimes _ B C_2$ coming from the isomorphism $C_1 \to C_2$ is compatible with the morphisms $U_ i \times _{\mathop{\mathrm{Spec}}(B)} U_ i \to U \times _\mathcal {X} U$. Some details omitted. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.




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 0GRP. Beware of the difference between the letter 'O' and the digit '0'.