Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?

Question

Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor $$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$ I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-colimit of this and obtain an equivalence of categories $$\operatorname{2-colim}_U\textsf{QCoh}(U) \xrightarrow{\sim} \textsf{Vect}(K).$$ If this is true, I wonder moreover if the natural functor $$\textsf{QCoh}(X) \to \operatorname{2-colim}_U\textsf{QCoh}(U)\cong \textsf{Vect}(K)$$ is the one which sends a quasi-coherent module over $X$ to its stalk on the generic point.

Answer 1

Let $x$ be a point in a scheme $X$. There are two posets, namely the poset of affine opens containing $x$, $A(x)$, and the poset of opens containing $x$, $O(x)$.

The inclusion $A(x)^{op} \to O(x)^{op}$ is (homotopy) cofinal - by Quillen's theorem A, it suffices to show that for any open $U$, $A(x)_{/U}$ is weakly contractible. But it is a co-directed poset : for any $V,W \to U$, $V\cap W$ is an open and therefore contains some affine open $x\in O\subset V\cap W\subset U$; therefore it is weakly contractible.

It follows that, in the $(2,1)$-category of presentable $1$-categories, $colim_{U\in O(x)^{op}}QCoh(U) \simeq colim_{U\in A(x)^{op}} QCoh(U)$.

Now note that the latter diagram in fact lives in the category of $E_0$ presentable categories, that is, presentable categories with a chosen basepoint (namely $O_U$) that are equivalent to something of the form $(Mod_R,R)$ for some (commutative) ring $R$.

By a $(2,1)$-categorical analogue of theorem 4.8.5.11. in Lurie's Higher algebra (which says that $R\mapsto (Mod_R,R)$, as a functor from rings to $E_0$ presentable categories, is fully faithful and colimit preserving) (it's actually not an analogue, it follows strictly from this theorem by specializing to $Set$-modules in presentable $(\infty,1)$-categories), and using the fact that $A(x)$ is weakly contractible, so colimits over it in $E_0$-objects are computed underlying, it follows that $colim_{U\in A(x)^{op}} QCoh(U)\simeq colim_{U\in A(x)^{op}} Mod_{O_U(U)}$ is equivalent to modules over $colim_{U \in A(x)^{op}} O_U(U)= colim_{U\in A(x)^{op}} O_X(U)= O_{X,x}$.

Note that here the colimit is taken in ordinary associative rings, but it's a filtered colimit, so it can also be taken in commutative rings.

In conclusion, for any point $x$, $colim_{U\in O(x)^{op}} QCoh(U)$ is equivalent to modules over $O_{X,x}$.

If $x$ is a generic point, then $O(x)$ is the category of all opens, and so you get the corresponding statement for $colim_U QCoh(U)$.

Note that here I've used the following version of "$2-\mathrm{colim}$": homotopy colimits in the $(2,1)$-category (in the sense of "$(\infty,1)$-categories with $1$-truncated mapping spaces") of presentable $1$-categories. I'm assuming that this shouldn't matter too much

Answer 2

I don't understand what kind of higher category magic went on in the other answer. For me, the answer should be "no", as taking direct limits of rings is only compatible in passing to categories of finitely presented modules.

Note that any vector space over the field of rational functions $K$ can be regarded as a quasi-coherent sheaf on $X$ (by taking pushforward along $\operatorname{Spec}(K)\to X$ - see Hartshorne AG Prop II 5.8). Denote by $\mathcal{K}$ the quasi-coherent sheaf on $X$ corresponding to $K$. Since the stalk of $\mathcal{K}$ at the generic point is $K$, which is the same as the stalk of $\mathcal{O}_X$, we have an isomorphism of stalks at the generic point $\mathcal{K}_\eta\to (\mathcal{O}_X)_\eta$. However, usually this isomorphism can not be spread out to any Zariski neighborhood of the generic point (except in trivial situations, for example if the point $\eta$ is already open). So the direct limit of categories does not see this map; in other words, $\mathcal{K}$ and $\mathcal{O}_X$ are not isomorphic in the direct limit of categories, but are isomorphic at the generic point.

As I hinted above, this issue is fixed if we work with coherent sheaves (assuming the schemes are Noetherian). Indeed, more generally if $R=\varinjlim R_i$ is a direct limit of rings (indexed by some filtering category), then the category $\operatorname{Mod}_{\rm fp}(R)$ of finitely presented modules is the direct limit of the $\operatorname{Mod}_{\rm fp}(R_i)$. To see the essential surjectivity of the map, take a finitely presented $R$-module $M$ and write is at the cokernel of a finite matrix $[r_{\alpha\beta}]$. We can find $i\gg 0$ such that all $r_{\alpha\beta}$ "come" from $R_i$, and take the cokernel there, obtaining a finitely presented $R_i$-module $M_i$ which is isomorphic to $M$ after we tensor with $R$ (because tensor product is right-exact). Full faithfulness can be proved analogously by presenting morphisms by lifting them to presentations.