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 $X$.

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.

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.

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 $X$.