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Defining a quasi-coherent module $\mathcal{M}$ on a scheme $X$ to be a compatible family of modules $(\mathcal{M}(x))_{x \in X(A), A \in \textbf{Rings}}$ (as in here), is there a straightforward way to show the existence of (finite) limits (and that it forms an abelian category)?

One possible way, of course, should be to show that this definition gives rise to a category equivalent to the category of quasi-coherent sheaves of modules on the small Zariski-site associated to $X$, but that feels like a rather dirty solution.

The problem, I guess, is that taking pullbacks of sheaves of modules (generally) doesn't commute with taking limits so that the limit isn't defined "fibrewise"; colimits work fine for exactly that reason.

Another argument that a friend of mine explained to me seems to be that, denoting the in the above way defined category of modules as $\textbf{Mod}(X)$, one has

$$\textbf{Mod}(X) = \varprojlim_{A \in \textbf{Aff}/X} \textbf{Mod}(A)$$

where the ($2$-)limit is taken in the $(2,1)$-category of categories, functors and natural isomorphisms.

Now the argument would be that $\textbf{Mod}(A)$ is a locally presentable category, (certain?) limits of locally presentable categories are locally presentable, and locally presentable categories admit arbitrary limits.

I was still wondering whether there wouldn't be a more elementary way to for example directly construct kernels and finite products of modules when defined this way.

I'd appreciate any thoughts!

//Edit: Ok another way seems to be to first show that one can glue quasi-coherent modules along Zariski-coverings and then do everything locally. I guess that's fine for me, but I'd still be interested in seeing other elementary arguments if anyone has one!

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  • $\begingroup$ The 2-category of accessible categories and accessible functors between them is closed under PIE limits in the 2-category of categories, yes. This isn’t entirely trivial to prove. If you only care about finite limits then I would focus on the fact that a scheme is a colimit of a nice kind of diagram where all the arrows are flat morphisms of affine schemes. $\endgroup$
    – Zhen Lin
    Aug 20, 2020 at 15:11
  • $\begingroup$ @RizaHawkeye Is that correct though? That sounds like you're suggesting to take the limit fiber by fiber which should be wrong as pullbacks of modules don't commute with arbitrary limits... $\endgroup$
    – lush
    Aug 21, 2020 at 6:12
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    $\begingroup$ @lush By avoiding the use of Zariski covers, you are more or less trying to prove this statement for presheaves $X$ rather than schemes $X$. As you noted, the statement still holds true in this setting by generalities on presentable categories. However, the category in question is badly behaved for general presheaves - to construct a good category you need to work in a dg setting - and to the best of my knowledge is never used in this generality. So I recommend not avoiding Zariski covers. $\endgroup$
    – dhy
    Aug 21, 2020 at 15:43
  • $\begingroup$ @dhy Lurie uses it in this generality quite a bit in SAG in the chapters around Tannaka duality and some stuff much later on. You need to be able to talk about QCoh(X) for a space-valued functor on connective rings in order to even make sense of the generalized Artin representability theorem. $\endgroup$ Aug 23, 2020 at 13:31
  • $\begingroup$ @HarryGindi I'm not seeing where in SAG he uses the abelian category QCoh for a general prestack X (as opposed to in specific cases e.g. X is an actual geometric stack.) Can you give me the number? $\endgroup$
    – dhy
    Aug 23, 2020 at 23:40

2 Answers 2

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Here is the precise statement alluded to in the comments:

Let $C = \lim_i C_i$ be a limit of categories with projections $\pi_i : C \to C_i$. Let $\{X_j\}_j$ be a diagram in $C$. If for every $i$ the induced diagram $\{\pi_i(X_j)\}_j$ in $C_i$ has a limit $X_i$, and the transition functors $C_i \to C_{i'}$ send $X_i \mapsto X_{i'}$ for every morphism $i \to i'$ in the indexing category, then the original diagram $\{X_j\}_j$ in $C$ admits a limit $X$ such that $\pi_i(X) = X_i$ for every $i$.

For example, if $X$ is a scheme, then quasi-coherent $O_X$-modules can be defined as you did, except that you can require the maps $Spec(A) \to X$ to be Zariski immersions (since $X$ is a scheme). Then the transition functors are pullbacks along open immersions which are exact hence preserve finite limits.

Alternatively by descent, you can take a Zariski cover of $X$ by affines $U_i$, then $Mod(X)$ will be a limit of $Mod(U_i)$ and of the intersections (similar to the usual sheaf condition except that you have to go to the 3-way intersections since it is a sheaf of categories). Again you can then apply the same argument to say that limits will be computed on the $U_i$'s.

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  • $\begingroup$ Thanks for the elaboration! That explains what you had in mind, but I'm trying to use the definition of quasi-coherent modules as I wrote it down, so I guess it's a little more difficult to apply your argument (and the argument itself is also a little more involved than I hoped to find). This was still very useful an answer though, thanks for that! $\endgroup$
    – lush
    Aug 21, 2020 at 10:29
  • $\begingroup$ @lush That's why I gave the alternative argument, which doesn't use any particular definition of quasi-coherent modules. I'm just saying that you can compute limits of qcoh sheaves on X locally, i.e. on affine covers. I don't see what you find involved about this. If you're looking for something else, you'll have to clarify. $\endgroup$
    – user147129
    Aug 21, 2020 at 12:08
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So I was lush's friend who he had originally asked this question, and I had some concerns, specifically because I gave the same answer as Riza, then realized that it gave incorrect answers if you follow the direct nLab construction. The point is that the limit of a diagram in the limit has to be computed first pointwise in the lax limit as above, then you have to apply a coreflector into the actual limit.

For example, if I have a cartesian square of locally presentable categories

$$\begin{matrix} P&\xrightarrow{f^\prime_!}&C_1\\ g^\prime_!\downarrow &\ulcorner&\downarrow g_!\\ C_2&\xrightarrow{f_!}&C_0 \end{matrix}$$

and a diagram $d:D\to P$, I can compute $P$ as a colocalization of the lax limit of this diagram (the category of not-necessarily-cartesian sections of the associated cartesian fibration over the span category $\operatorname{Span}$). Let's denote this lax limit by $L$. Then we have an adjunction $P\leftrightarrows L$, where the left adjoint $P\to L$ is fully faithful. This tells us that the limit in $P$ is computed as the image under the coreflector $L\to P$ of the limit in $L$, which is actually indeed the pointwise limit together with the connecting maps

$$g_! \lim (f^\prime_! \circ d)\to \lim (g_! \circ f^\prime_! \circ d)=\lim (f_! \circ g^\prime_! \circ d) \leftarrow f_!\lim(g'_!\circ d).$$

So to form the true limit, I have to apply the coreflector to this formal diagram (viewed as an object of the lax limit).

This gives you a formula to compute the limit now of such a diagram, but actual existence of limits is following from the fact that this fibre product is presentable (plus the thing about arbitrary products still being presentable).

To finish working out the example, the coreflector then gives you the fibre product in $P$

$$ \lim(f^{\prime \ast}\lim (f^\prime_! \circ d)\to f^{\prime\ast}g^\ast\lim (g_! \circ f^\prime_! \circ d)=g^{\prime\ast}f^\ast\lim (f_! \circ g^\prime_! \circ d) \leftarrow g^{\prime\ast}\lim(g'_!\circ d)).$$

but in order for this formula to make sense, you first needed to know that limits in $P$ existed, and that's because $\operatorname{Pr}^L$ admits limits that agree with limits in $\mathbf{Cat}$.

Note: I've used the categorical convention for left and right adjoints (lower shriek and upper star, rather than upper star and lower star) in $\operatorname{Pr}^L$ rather than the algebro-geometric convention, because it is clearer in this case.

Edit: It looks like lush's question here was slightly different from the one we discussed in private. My mistake. Riza's answer is correct for flat covers (this is a theorem, but it is completely obvious for open immersions, as desired).

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    $\begingroup$ The existence of limits in $\operatorname{Pr}^L$ is probably somewhere in Adamek-Rosicky, but the proof in HTT 5.5.3.13 should be the same for 1-categories or ∞-categories. $\endgroup$ Aug 23, 2020 at 20:04

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