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I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack of elliptic curves. Then, they declare that $S \to S' \to \mathcal{M}_{1, 1}$ is commutative (there is an extra map $S \to \mathcal{M}_{1, 1}$ but I can't fit in since I don't know how to put commutative diagram here) when there is a morphism of elliptic curves $E \to E'$ over $S \to S'$. I think we should really add an extra assumption that the induced map $E \to E' \times_{S'} S$ is an isomorphism. At least, this would make sense when $\mathcal{M}_{1, 1}$ is a scheme and there exists a universal family of elliptic curves.

I hope someone can illuminate me with this. Thanks!

flag	asked Sep 11 at 23:07 QcH 325●1●7

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The title of this post doesn't really seem to match the body. – Qiaochu Yuan Sep 11 at 23:37

No, this should still be true in the world of stacks; saying the triangle commutes really means there's a 2-morphism there, which is an isomorphism of elliptic curves over $S$. You should write to one of the authors, I guess. – Aaron Mazel-Gee Sep 12 at 0:56

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Dear QcH, I think you are right. (This becomes most clear by taking $S = S' =$ a point, and just imagining that $E$ and $E'$ are distinct but isogenous curves.) Perhaps there is omitted hypothesis in what you are reading, or maybe you have just misunderstood some aspect of the set-up. If you can't resolve what's written there for yourself in finite time, I suggest that you write to Johan and ask him about it, because either there really is a mistake in the exposition, and he'll want to fix it, or else we're all being idiotic, and he will explain why! Regards, Matt – Emerton Sep 12 at 2:25

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@QcH: The zero map is another illustration of the problem. Johan indeed meant to say that the square he drew is cartesian (which is just another way to express the isomorphism to the pullback, as you have said it; it is also the way KM85 express things). Another bonus of writing to Johan is that he may still have some t-shirts lying around and so you may get a free one in this way. – grp Sep 12 at 4:04

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I emailed prof. de Jong and he already relied and fixed the typo. – QcH Sep 12 at 14:52

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