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Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.

Context: Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with the fppf (or lisse-etale) topology. Because $\mathcal{G}$ is a $\mathbb{G}_m$-gerbe, there is an fppf (or etale) morphism of schemes $U \xrightarrow{i} X$ trivializing $\mathcal{G}$. So we have canonical isomorphism $\big(B\mathbb{G}_m\big)_U \cong U \times_X \mathcal{G}$ of $\mathbb{G}_m$-gerbes over $U$ and a morphism of stacks $\pi: \big(B\mathbb{G}_m\big)_U \to \mathcal{G}$.

What I want to know: If $\mathcal{F}$ is a quasi-coherent sheaf over $\mathcal{G}$ we often read the simplifying argument that because the question is etale local on $X$ one can assume $\mathcal{G} = B\mathbb{G}_m$. What is the justification for this?

Motivation for the question: I'm interested to prove some equivalence of categories of quasi-coherent sheaves with some extra property defined over some $\mathbb{G}_m$-gerbes and I know how to prove the result for $B\mathbb{G}_m$ (i.e. I know how to prove the equivalence once both gerbes have been trivialized over a common $X$-scheme). I would like to explain why this equivalence also holds over the non-trivial gerbes. I can be more explicit about this part if needed.

Attempt to answer the question I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over $X$. There is an etale surjective morphism $U \to X$ such that $U \times_X \mathcal{G} \cong (B\mathbb{G}_m)_U$.

Now I have an atlas $s: U \to (B\mathbb{G}_m)_U$ inducing an atlas $\pi \circ s : U \to \mathcal{G}$, where $\pi$ is the projection onto $\mathcal{G}$.

I then have a presentation of my gerbe

$$ U \times_\mathcal{G} U \times_\mathcal{G} U \to U \times_\mathcal{G} U \to U \xrightarrow{\pi \circ s} \mathcal{G} $$

and a sequence

$$ (B\mathbb{G}_m)_U \times_\mathcal{G} (B\mathbb{G}_m)_U \times_\mathcal{G} (B\mathbb{G}_m)_U \to (B\mathbb{G}_m)_U \times_\mathcal{G} (B\mathbb{G}_m)_U \to (B\mathbb{G}_m)_U \xrightarrow{\pi} \mathcal{G}. $$

If $\mathcal{F}$ is a quasi-coherent sheaf then on the atlas $U$ I get a quasi-coherent sheaf $(\pi \circ s)^* \mathcal{F}$ together with an isomorphism $\sigma: pr_1^*(\pi \circ s)^* \mathcal{F} \to pr_2^*(\pi \circ s)^* \mathcal{F}$ over $U \times_\mathcal{G} U$ which satisfies the cocycle condition over $U \times_\mathcal{G} U \times_\mathcal{G} U$.

Now I believe that we have

$$ s_*(\pi \circ s)^* \mathcal{F} = \pi^* \mathcal{F} $$

over $(B\mathbb{G}_m)_U$.

So I have this quasi-coherent sheaf together with an isomorphism $(s \times s)_*\sigma$ over $(B\mathbb{G}_m)_U \times_\mathcal{G} (B\mathbb{G}_m)_U$ (where $s \times s$ is the universal map induced by $s$) which satisfies the cocycle condition over triple fiber product.

This would be what I would call the descent data for the sheaf $\mathcal{F}$ and one could define the category of descent $QCoh \big((B\mathbb{G}_m)_U \to \mathcal{G} \big)$ with the objects similar to the one described above (obviously not necessarely assuming that it comes from a sheaf on the gerbe $\mathcal{G}$).

Question: Now this descent argument doesn't technically reduce to the case of $B\mathbb{G}_m$ but to a descent data of a sheaf on it. Is this what is usually intended?

Relation to the motivation:

Assume that what I said was correct, then would it be it exact to say the following: that to prove a given equivalence of categories as in the motivation I could equivalently have to prove that the categories of descent over the same $B\mathbb{G}_m$ are equivalent? Then in order to do that it would suffice to show that the given categories of quasi-coherent sheaves + extra property over $B\mathbb{G}_m$ are equivalent? (To justify the last part I think that if I can prove that the $QCoh(B\mathbb{G}_m)$ + property are equivalent, then the descent data would almost automatically correspond to one another through this equivalence? My property is some $\mathbb{G}_m$-action on the sheaves so it won't cause any problem on the descent data)

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  • $\begingroup$ I know what is a sheaf on a topological space, sheaf on a site (Category with Grothendieck topology)... How does a sheaf on a stack is defined? A stack (on a site) is given by a functor $\mathcal{D}\rightarrow \mathcal{C}$ (satisfying some property).. Is it that we pullback the Grotendieck topology on $\mathcal{C}$ to $\mathcal{D}$, to make it a site and then talk about sheaf on stack $\mathcal{D}$ as sheaf on the site $\mathcal{D}$? $\endgroup$ – Praphulla Koushik Aug 2 at 14:57
  • $\begingroup$ One possibility as you know is the induced topology math.stackexchange.com/questions/3163443/…. But say you have a big fppf (or étale) site on $Sch/S$ with $S$ a scheme, you can define the lisse-étale site on an algebraic stack $X$ where objects are pairs $(U,u)$ with $U$ an $S$-scheme and $u: U \to X$ a smooth morphism. A covering is a collection of maps $\{ (f_i,f_i^b) : (U_i,u_i) \to (U,u) \}$ where $\{ f_i : U_i \to U \}$ is an étale covering. You can also define similarly the flat-fppf site. So it doesn't have to be an induced site. $\endgroup$ – FelixBB Aug 3 at 13:09
  • $\begingroup$ By “it does not have to be an induced site” you mean it does not matter what covering I give? $\endgroup$ – Praphulla Koushik Aug 3 at 16:47
  • $\begingroup$ I meant that what you described in the answer you gave in the above link is not the only way to define a site on an algebraic stack. Indeed, I gave you the examples of lisse-étale site and flat-fppf site which are not of that form. $\endgroup$ – FelixBB Aug 3 at 16:52
  • $\begingroup$ The isomorphism $\big(B\mathbb{G}_m\big)_U \cong U \times_X \mathcal{G}$ of $\mathbb{G}_m$-gerbes over $U$ is not canonical. It is something that you have to choose. $\endgroup$ – S. Carnahan Aug 6 at 17:23
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I'm not sure how you have chosen to define "quasicoherent sheaf on the stack $\mathcal{G}$". One way to make a definition is to construct the fibered category $QCoh$ over affine schemes, whose objects are pairs $(X, \mathcal{F})$, where $X$ is an affine scheme, and $\mathcal{F}$ is a quasicoherent sheaf on $X$. Morphisms $(X, \mathcal{A}) \to (Y, \mathcal{B})$ are pairs given by a map $f: X \to Y$ of affine schemes and a map $f^\sharp: \mathcal{A} \to f^*\mathcal{B}$ of sheaves on $X$.

Then a quasicoherent sheaf over the gerbe $\mathcal{G}$ is just a morphism of fibered categories from $\mathcal{G}$ to $QCoh$, and these sheaves form a category whose morphisms are natural transformations. The fact that $QCoh$ is a stack in the étale topology implies that if we are given an étale cover $\big(B\mathbb{G}_m\big)_U \to \mathcal{G}$, then pullback induces an equivalence between the category of quasicoherent sheaves on $\mathcal{G}$ and the category of descent data for the cover. In other words, you may think of a quasicoherent sheaf on $\mathcal{G}$ as a quasicoherent sheaf $\mathcal{A}$ on $\big(B\mathbb{G}_m\big)_U$ equipped with an isomorphism $\phi: pr_0^* \mathcal{A} \to pr_1^*\mathcal{A}$ of quasicoherent sheaves on $\big(B\mathbb{G}_m\big)_{U \times_X U}$ satisfying the cocycle condition.

I think this is what you were trying to do with Approach 1 (now deleted), but I did not understand your $(T,\sigma)$ notation. Anyway, to answer your question, when people say that a question is étale local on $X$, they mean its truth value is unchanged by étale base change. In particular, its truth value respects étale descent.

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  • $\begingroup$ "I'm not sure how you have chosen to define "quasicoherent sheaf on the stack $\mathcal{G}$". " Once I have ringed site, I can always define a quasi-coherent sheaf on it. This is what I had in mind. See for example Tag 03DK. What I was trying to do in my question was to justify the descent using that we have an effective descent on the presentation of the algebraic stack. This then induces an effective descent for the corresponding sheaf on $B\mathbb{G}_m$. In approach 1 I wrote a description of the sheaf on the lisse-étale site and $(T,\sigma)$ is an object on that site. $\endgroup$ – FelixBB Aug 13 at 15:07
  • $\begingroup$ At any rate, thank you S. Carnahan for your answer. It is indeed helpful and clarifies most of my doubts. The last thing I would want to ask is: Don't you mean that because the sheaf satisfies étale descent then we can equally consider that sheaf after étale base change? $\endgroup$ – FelixBB Aug 14 at 5:16

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