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Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between Boolean rings and Boolean algebras; the Boolean algebra corresponding to a Boolean ring $A$ (which I'll continue to call "$A$") has the same elements as $A$, and multiplication corresponds to "AND", while addition corresponds to "XOR".

Recall also that Stone duality gives an equivalence between the opposite of the category of Boolean algebras and totally disconnected compact Hausdorff spaces. Under this equivalence, a Boolean algebra $A$ is sent to the space $Spec A$ of ultrafilters on $A$, and $A$ is recovered as the algebra of clopen subspaces of $Spec A$.

Question: Let $A$ be a Boolean ring. Let $M$ be an $A$-module. How can the data of $M$ be described in terms of the Boolean algebra $A$, or better yet in terms of the topological space $Spec A$?

One thing to say is that $M$ is naturally an $\mathbb F_2$-vector space, and the $A$-module structure on $M$ corresponds to a representation of $A$ as a sublattice of the lattice of $\mathbb F_2$-subspaces of $M$. This is nice as far as it goes, but I'd really like a description which doesn't mention vector spaces at all, just like the usual definitions of Boolean algebras or totally disconnected compact Hausdorff spaces don't mention rings at all. For instance, it would be nice if this could be described as some kind of representation of the Boolean algebra $A$ on the powerset lattice of a set or something like that.

One possible direction: If $M$ is an $A$-module, then there is a natural preorder on $M$ where $m \leq m'$ iff there is $a \in A$ such that $m = am'$. The set $Spec M$ of ultrafilters on this preorder carries a natural topology with subbasis given by the sets $\hat m = \{p \in Spec M \mid m \in p\}$, for $m \in M$. There is a natural continuous map $Spec M \to Spec A$ given by $p \mapsto \{a \in A \mid \exists m \in p (am = m)\}$. This yields a faithful functor $Spec : Mod_A^{op} \to Top_{/Spec A}$. I wonder if there some additional structure / properties on $Spec M$ which can turn this functor in an equivalence?

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    $\begingroup$ Is "$M$ is a sheaf of $\mathbb{Z}/2\mathbb{Z}$-vector space on Spec $A$" an acceptable answer ? or do you really want something that is purely geometric and also apply some sort of Stone duality to $M$ ? $\endgroup$ – Simon Henry May 3 '20 at 15:42
  • $\begingroup$ In the second, case you could clarify the sort of description you would be happy with in the special case $A= \mathbb{Z}/2 \mathbb{Z}$. $\endgroup$ – Simon Henry May 3 '20 at 15:44
  • $\begingroup$ @SimonHenry I think a sheafy description would be great. Is it as simple as $M$ being an arbitrary sheaf of $\mathbb F_2$-vector spaces? $\endgroup$ – Tim Campion May 3 '20 at 15:47
  • $\begingroup$ It seems that Boolean algebra is used for "unital Boolean ring". So these categories are not equivalent (for instance in the category of Boolean rings, $\{0\}$ is both initial and terminal, while in the category of Boolean algebras, $\{0\}$ is terminal while $\mathbf{Z}/2\mathbf{Z}$ is initial). $\endgroup$ – YCor May 3 '20 at 18:22
  • $\begingroup$ @YCor I see. For me, rings are by default unital. I'll edit to clarify. $\endgroup$ – Tim Campion May 3 '20 at 18:23
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Theorem: Given $A$ a boolean ring/boolean algebra then there is an equivalence of categories between the category of $A$-modules and the category of sheaves of $\mathbb{F}_2$-vector spaces on Spec $A$. The equivalence sends every sheaf $\mathcal{M}$ of $\mathbb{F}_2$-vector space to its space of section, $\Gamma(\mathcal{M})$ which is a module over $\Gamma(\mathbb{F}_2) = A$.

Proof: Spec $A$ has a basis of clopen given by the elements of $A$. Using Grothendieck comparison lemma, this allows to give a more algebraic description of sheaves on Spec $A$ as:

For each $a \in A$ a set $F(a)$, with (functorial) restriction maps) $F(a) \to F(b)$ when $b \leqslant a$ such that the natural maps:

$$ F(a \cup b) \to F(a) \times F(b) $$

is an isomorphism when $a \cap b = 0$.

This allows to exhibit the inverse construction: Given a $A$-module $M$, we define

$$ \mathcal{M}(a) = aM $$

with the restriction map $aM \to bM$ being given by multiplication by $b$. One easily check that if $a \cap b = 0$ then $aM \times bM \simeq (a \cup b) M$ hence $\mathcal{M}$ is a sheaf of $\mathbb{F}_2$-vector space such that $\Gamma(\mathcal{M}) = \mathcal{M}(1) = M$.

Conversely, starting from any sheaf of $\mathbb{F}_2$-vector spaces $\mathcal{M}$ we have $\mathcal{M}(1) = \Gamma(\mathcal{M}) = \mathcal{M}(a) \times \mathcal{M}(\neg a)$, and the action of $a \in A$ on $\Gamma(\mathcal{M})$ is the identity on $\mathcal{M}(a)$ and zero on $\mathcal{M}(\neg a)$. It follows that $a\Gamma(\mathcal{M}) = \mathcal{M}(a)$, and from there we easily check that the two constructions are inverse of each others.

Note that the exact same argument proves more generally that:

Theorem: If $X$ is a stone space, and $\mathcal{A}$ is a sheaf of rings on $X$, then there is an equivalence of categories between sheaves of $\mathcal{A}$-modules and $\Gamma(\mathcal{A})$-modules.

In particular sheaves of abelian groups on $X$ corresponds to module over the ring of locally constant integer valued functions on $X$.

Even more generally (but the proof is more involved) the same conclusion holds if $X$ is an arbitrary locally compact space, $\mathcal{A}$ is a "c-soft" sheaf of rings and $\Gamma$ is replaced by the "compactly supported section" functor. I prove this as proposition 5.1 of this paper, which is about generalizing this sort of theorem when $X$ is not a space but a topos satisfying apropriate local finiteness assumption, but I'm convince this had been observed before, I just do not know a reference for it.

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  • $\begingroup$ Thanks! Just a note: we must use that $\mathcal M$ is a sheaf of $\mathbb F_2$-vector spaces rather than a general sheaf of abelian groups is in verifying that $\Gamma(\mathcal M)$ is an $A$-module. And indeed, the place it is needed is in verifying that $(a \mathrm{XOR} b)m = am + bm$ for all $a,b \in A, m \in \Gamma(\mathcal M)$. I will probably "accept" this answer, but I'll hold off for a bit in case there are other interesting descriptions which might come up. $\endgroup$ – Tim Campion May 3 '20 at 16:46
  • $\begingroup$ @TimCampion : Absolutely, though we do obtain a result for sheaves of abelian group, see the edit. $\endgroup$ – Simon Henry May 3 '20 at 16:55
  • $\begingroup$ More generally if K is any commutative ring and X is a stone space then modules for the ring of locally constant functions X to K correspond to sheaves of K-modules on X. These all are easy cases of Pierce's sheaf representation theory. $\endgroup$ – Benjamin Steinberg May 3 '20 at 19:19

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