Obstructions to abelian sheaf being quasi-coherent

Question

Let $X$ be a Noetherian spectral topological space. A necessary condition for an abelian sheaf on $X$ to be quasi-coherent with respect to some affine scheme structure on $X$ is that its higher cohomology vanishes.

I do not think that it is a sufficient condition. For example, take a single-point space $X=\{pt\}$ and a sheaf $F$ with $F(X)=\mathbb{Z}$. The higher cohomology of $F$ vanishes due to dimension considerations. If $F$ were quasi-coherent with respect to some affine scheme structure on $X$, it would be a module over a commutative unital ring $R$ with exactly one prime ideal. Since there are no non-zero nilpotents in $End(\mathbb{Z})$, elements of the prime ideal would annihilate $\mathbb{Z}$. Therefore, $\mathbb{Z}$ would also be a module over the quotient of $R$ by its prime ideal (which is a field). This is a contradiction because $\mathbb{Z}$ is free abelian.

The question is: what are some other reasonably-easy-to-formulate necessary conditions for an abelian sheaf to be quasi-coherent? Is it possible to give a not entirely tautological necessary and sufficient condition?

Answer 1

Here's an answer to the modified question in the comments:

Question: Let $X$ be an affine scheme, and $\mathcal F$ an $\mathcal O_X$-module. Under what conditions is $\mathcal F$ quasicoherent?

As observed by Thomason-Trobaugh (see Appendix B of "Higher Algebraic K-Theory of Schemes" in the Grothendieck Festschrift), for reasonable schemes the quasicoherent sheaves are coreflective in the category of $\mathcal O_X$-modules. For affine schemes, the coreflection is particularly easy to describe: it sends $\mathcal F$ to the sheafification of $\bar {\mathcal F}: U \mapsto \mathcal F(X) \otimes_{\mathcal O_X(X)} \mathcal O_X(U)$. That is,

Answer: $\mathcal F$ is quasicoherent if and only if the canonical map $\bar{\mathcal F} \to \mathcal F$ is an isomorphism after sheafification.

This can be checked on stalks without taking an explicit sheafification. For $p \in X$, the map $\bar {\mathcal F}_p \to \mathcal F_p$ is the map

$$\varinjlim_{U \ni p} \mathcal F(X) \otimes_{\mathcal O_X(X)} \mathcal O_X(U) \to \varinjlim_{U \ni p} \mathcal F(U)$$

• This map is surjective if and only if for all $U \ni p$ and for all $f \in \mathcal F(U)$ there exist $g_1,\dots, g_n \in \mathcal F(X)$, $U \supseteq V \ni p$, and $\varphi_1,\dots, \varphi_n \in \mathcal O_X(V)$ such that $f|_V = \sum_{i=1}^n \varphi_i g_i|_V$.

  The map is injective if and only if for all $g_1,\dots, g_n \in \mathcal F(X)$, $U \ni p$ and $\varphi_1,\dots, \varphi_n \in \mathcal O_X(U)$, if $\sum_{i=1}^n \varphi_i g_i = 0$ in $\mathcal F(U)$, then for some $U \supseteq V \ni p$, we have $\sum_{i=1}^n \varphi_i|_V g_i = 0$ in $\mathcal F(X) \otimes_{\mathcal O_X(X)} \mathcal O_X(V)$.

At least if $X$ is irreducible, these conditions may be rephrased as follows:

• Surjectivity: For all $U \ni p$ and all $f \in \mathcal F(U)$, there exists $g \in \mathcal F(X)$ and $\psi \in \mathcal O_X(X)$ not vanishing at $p$ such that $\psi|_U f = g|_U$.

  Injectivity: For all $f \in \mathcal F(X)$ and $U \ni p$, if $f|_U = 0$ in $\mathcal F(U)$, then there exists $\psi \in \mathcal O_X(X)$ not vanishing at $p$ such that $\psi f= 0$ in $\mathcal F(X)$.

Answer 2

1) In a similar spirit to your own example, here is a necessary condition for a sheaf of abelian groups $\mathscr{F}$ to be an $\mathcal{O}_X$-module for some scheme structure on $X$:

For each point $x \in X$, the stalk $\mathscr{F}_x$ has to admit a module structure over a local ring.

This is not too explicit, so one can replace "has to admit a module structure over a local ring" by some more concrete condition: the one that comes to mind is

Torsion part of $\mathscr{F}_x$ has to be a $p$-power torsion for a prime $p$ (in fact, $\mathscr{F}_x$ has to be uniquely $\ell$-divisible for every prime $\ell \neq p$)

Although this has no ambition of being close to sufficient, it already excludes a lot of sheaves (locally constant sheaves with a not-$p$-power torsion, for example).

2) The following necessary condition is somewhat similar, but this time it is an honest criterion for being quasi-coherent. The drawback is that it seems hard to check in general. One can use the fact is that for a quasi-coherent sheaf $\widetilde{M}$ on $X=\mathrm{Spec}\,R$, the canonical maps $\widetilde{M}(X) \rightarrow \widetilde{M}_x$ and $\widetilde{M}(X)\rightarrow \widetilde{M}(D_f)$ are localization morphisms. This property can be somewhat-accurately described without referencing the ring $R$, i.e. purely in language of sheaves of abelian groups, in the following way.

Lemma. Given an $R$-module $M$, where $R$ is a commutative ring, for $r \in R$ denote $r_M=[r\cdot -]|_M,$ i.e. the endomorphism of the abelian group $M$ given by multiplication by $r$ on $M$. For $S \subseteq R$ multiplicative set, denote $S_M=\{r_M\;|\; r \in S\}$. Then $M \rightarrow S^{-1}M(=M\otimes_RS^{-1}R)$ and $M \rightarrow S_M^{-1}M=M\otimes_{\mathbb{Z}[S_M]}S_M^{-1}\mathbb{Z}[S_M]$ are canonically isomorphic (as objects in the coslice category $M/\mathsf{Ab}$).

Call a map of abelian groups $M \rightarrow N$ endo-localization if $N$ is of the form $N=S_M^{-1}M$ for some subset $S_M \subseteq \mathrm{End}(M)$ of pairwise commuting endomorphisms (and the map $M \rightarrow N$ agrees with the localization map). Then

If $\mathscr{F}$ is a sheaf of Abelian groups on $X$ that is a quasi-coherent sheaf with respect to some affine scheme structure on $X$, then

1. Each of the maps to stalks $\mathscr{F}(X) \rightarrow \mathscr{F}_x$ is an endo-localization, with respect to multiplicative set of endomorphism containing (multiplication by) all but possibly one prime.

2. There is a basis of open sets $\{U_i\}_i$ of $X$containing $X$ such that $\mathscr{F}(U_i) \rightarrow \mathscr{F}(U_j)$ is an endo-localization whenever $U_j \subseteq U_i.$ with respect to multiplicative set generated by a single endomorphism.

Proof of Lemma: Both sides are/can be interpreted as:

1)$R$-modules on which $S$ acts by isomorphisms (and $R$-module map to it): For $ S_M^{-1}M$, the module structure is given on the left-hand side, i.e. $r(m\otimes f)=(rm)\otimes f$, and this makes the map $M \rightarrow S_M^{-1}M$ an $R$-module homomorphism. Also for $r \in S,$ $r(m\otimes f)=r_M(m)\otimes f=m \otimes r_Mf,$ showing that $r$ acts by isomorphism on $S_M^{-1}M$ (with the unique preimage to $\sum_i(m_i \otimes f_i)$ given by $\sum_i(m_i \otimes r_M^{-1}f_i)$).

2) $\mathbb{Z}[S_M]$-modules on which $S_M$ acts by isomorphisms: For $ S^{-1}M=M\otimes S^{-1}R$, let $S_M$ act on the left factor, and use $r_M(m\otimes f)=(r_M(m))\otimes f)=rm\otimes f=m\otimes rf$

Now use the universal properties of $S^{-1}M, S_M^{-1}M$ with respect to 1) and 2), resp., to conclude that these are canonically isomorphic.