Existence of finite limits of quasi-coherent modules on a scheme

Question

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!

Answer 1

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.

Answer 2

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).