# Obstructions to abelian sheaf being quasi-coherent

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?

• if $F$ is an abelian sheaf on $X$, you at least need to give it the structure of an $\mathcal O_X$-module before you can ask whether it is quasicoherent. Are you fixing an $\mathcal O_X$-module structure, or are you quantifying over all possible $\mathcal O_X$-module structures? – Tim Campion Apr 14 '19 at 15:45
• @TimCampion yes, your point is valid. We are quantifying. – user137767 Apr 14 '19 at 16:14
• Consider the case where $X$ is a point. Then $F$ is an abelian group, and the question is whether this abelian group admits a vector space structure over any field. This just doesn't feel like the right question to be asking. – Tim Campion Apr 14 '19 at 17:23
• @TimCampion I respect your opinion. My feelings are somewhat different. I could add a bounty after 2 days if this question is too ugly otherwise. – user137767 Apr 14 '19 at 17:25
• though another reasonable question taking your suggestion into account would be: what obstructions are there for an $O_X$-module on a fixed affine scheme to be quasi-coherent (beyond the vanishing of cohomology). – user137767 Apr 14 '19 at 17:30

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 $$\mathcal F(X)$$.

• It does seem interesting to ask if there is a purely cohomological criterion for an $\mathcal O_X$-module to be quasicoherent... Since quasicoherence is a local condition, the criterion would presumably use local cohomology or something like that. The "surjectivity" condition seems to be an "almost flasque"-ness condition, and the "injectivity" condition seems to be a "controlled torsion" condition.... both of which sound "cohomological" in nature... – Tim Campion Apr 15 '19 at 17:03

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.