Flatness criteria and canonical epimorphisms from flat abelian sheaves

Assume $\mathcal{F}$ flat, so that each stalk ${\mathcal{F}}_x$ is a flat $\mathbb{Z}$-module [F12, F1]. Let $\phi :{\mathcal{G}}^{\mathrm{\prime }}\to \mathcal{G}$ be an injective morphism of abelian sheaves and let $\kappa :\mathcal{F}{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\mathrm{\prime }}\to \mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}$ be the induced morphism, whose stalk at $x$ is $\mathrm{id}\otimes {\phi }_x:{\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x^{\mathrm{\prime }}\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x$ by naturality of the stalk identification [F11]. Let $K:=\ker (\kappa )$ be the kernel sheaf; by [F2] its stalk is $K_x=\ker (\mathrm{id}\otimes {\phi }_x)$, and since ${\mathcal{G}}_x^{\mathrm{\prime }}\to {\mathcal{G}}_x$ is injective and ${\mathcal{F}}_x$ is flat, this kernel is zero [F1]. The zero morphism $K\to 0$ therefore has bijective stalks at every point (both are zero groups), so it is an isomorphism by [F4] and $K=0$; hence $\kappa$ is injective. Since $\mathcal{F}{\otimes }_{\mathbb{Z}}-$ is right exact [F11], it carries every short exact sequence to a short exact sequence and is exact; the argument in the other variable is identical.

The constant sheaf ${\mathbb{Z}}_X$ has stalk $\mathbb{Z}$ at every $x$, as the value at $x$ of the locally constant functions [F5], and $\mathbb{Z}$ is a free, hence flat, $\mathbb{Z}$-module [F6]; so ${\mathbb{Z}}_X$ is flat [F12].

Let $x\in X$. If $x\in U$, then for every open $V\subseteq U$ the support condition is vacuous and $(j_{U!}{\mathbb{Z}}_U)(V)={\mathbb{Z}}_U(V)$ [F7], so the restriction of $j_{U!}{\mathbb{Z}}_U$ to the open subspace $U$ is ${\mathbb{Z}}_U$ [F9]; by [F8] the stalk of that restriction at $x$ is the stalk of $j_{U!}{\mathbb{Z}}_U$ at $x$, which is therefore $\mathbb{Z}$ by [F5]. If $x\mathrm{\notin }U$ and $s\in (j_{U!}{\mathbb{Z}}_U)(V)$ is a section over an open $V\ni x$, then the support of $s$ is closed in $V$ and contained in $V\cap U$, so $x$ lies in the open subset $V\setminus \mathrm{Supp}(s)$ of $V$ on which $s$ vanishes [F7, F15]; hence the germ $s_x$ is zero and $(j_{U!}{\mathbb{Z}}_U{)}_x=0$. The stalk of $j_{U!}{\mathbb{Z}}_U$ is thus $\mathbb{Z}$ at points of $U$ and $0$ at points outside $U$, both flat $\mathbb{Z}$-modules [F1, F6], so $j_{U!}{\mathbb{Z}}_U$ is flat [F12].

Let $\mathcal{F}$ be an abelian sheaf. For each open $U\subseteq X$ and $s\in \mathcal{F}(U)$ define ${\phi }_{U,s}:j_{U!}{\mathbb{Z}}_U\to \mathcal{F}$ on a section $g\in (j_{U!}{\mathbb{Z}}_U)(V)$ as follows. The support $S=\mathrm{Supp}(g)$ is closed in $V$ and contained in $V\cap U$ [F7]. On $V\cap U$ use the section $g\cdot s$, defined locally where the integer-valued function $g$ is constant; these local products glue uniquely since they agree on overlaps [F5, F14]. On the open set $V\setminus S$ use the zero section. These opens cover $V$, and on their intersection $g=0$, so the sections agree and glue uniquely to ${\phi }_{U,s}(g)\in \mathcal{F}(V)$ [F14]. Addition and restriction of $g$ commute with both local formulas, hence with their unique gluing, so this is a morphism of sheaves. On $V=U$, the section $1_U\in (j_{U!}{\mathbb{Z}}_U)(U)$ maps to $s$. The construction is canonical in $(U,s)$; by the coproduct universal property [F13] the components induce $\mathrm{\Phi }:{⨁}_{(U,s)}{j}_{U!}{\mathbb{Z}}_U\to \mathcal{F}$.

Conversely assume $\mathcal{F}{\otimes }_{\mathbb{Z}}-$ exact, let $x\in X$, and let $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence of $\mathbb{Z}$-modules. Under the canonical isomorphism of [F5] the constant sheaves $M_X^{\mathrm{\prime }},M_X,M_X^{\mathrm{\prime }\mathrm{\prime }}$ are the sheaves of locally constant functions with values in $M^{\mathrm{\prime }},M,M^{\mathrm{\prime }\mathrm{\prime }}$, and a homomorphism $u$ of abelian groups induces the morphism of sheaves given pointwise on locally constant functions by $f\mapsto u\circ f$; these morphisms are compatible on stalks with $u$ under evaluation at $x$ [F5], so the induced sequence $0\to M_X^{\mathrm{\prime }}\to M_X\to M_X^{\mathrm{\prime }\mathrm{\prime }}\to 0$ has exact stalk sequence $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$ at every point and is therefore exact [F3]. Applying the exact functor $\mathcal{F}{\otimes }_{\mathbb{Z}}-$ and then taking the stalk at $x$, where the stalks of the tensor products are the tensor products of the stalks [F11] and the stalk of the constant sheaf $M_X$ is $M$ [F5], gives the exact sequence $0\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{M}^{\mathrm{\prime }}\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}M\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{M}^{\mathrm{\prime }\mathrm{\prime }}\to 0$. Hence ${\mathcal{F}}_x{\otimes }_{\mathbb{Z}}-$ is exact and ${\mathcal{F}}_x$ is flat [F1]; since $x$ was arbitrary, $\mathcal{F}$ is flat [F12]. With step 1.1 this proves clause 1, both equivalent formulations included.

Let $({\mathcal{F}}_i{)}_{i\in I}$ be a family of flat abelian sheaves with coproduct ${⨁}_{i\in I}{\mathcal{F}}_i$ [F13]. By [F11] the stalk of the coproduct at $x$ is ${⨁}_{i\in I}({\mathcal{F}}_i{)}_x$, a direct sum of flat $\mathbb{Z}$-modules, hence flat [F10]; therefore the coproduct is flat [F12]. This proves clause 2 together with [step 1.2] and [step 1.3].

The source of $\mathrm{\Phi }$ is a coproduct of the sheaves $j_{U!}{\mathbb{Z}}_U$, each flat by step 1.3, so it is flat by step 2.2. To see that $\mathrm{\Phi }$ is an epimorphism, compute stalks at $x$: by [F11] the stalk of the source is ${⨁}_{(U,s):x\in U}(j_{U!}{\mathbb{Z}}_U{)}_x\cong {⨁}_{(U,s):x\in U}\mathbb{Z}$, the pairs with $x\mathrm{\notin }U$ contributing $0$ by step 1.3. The $(U,s)$-component of the stalk map sends the canonical basis element to the germ $s_x$: for $V=U$ the function $g=1_U\in (j_{U!}{\mathbb{Z}}_U)(U)$ is locally constant with $V_1=U$ and $V_n=\mathrm{\varnothing }$ for $n\mathrm{\ne }1$, so the formula of step 1.4 gives ${\phi }_{U,s}(1_U)=s$ and hence germ $s_x$ at $x\in U$. Given $t_x\in {\mathcal{F}}_x$, represented by a section $t\in \mathcal{F}(W)$ over an open $W\ni x$ [F15], the pair $(W,t)$ is an index and its basis element maps to $t_x$; hence the stalk map is surjective in every degree, and $\mathrm{\Phi }$ is an epimorphism by [F3]. Therefore every abelian sheaf is a quotient of the flat abelian sheaf ${⨁}_{(U,s)}{j}_{U!}{\mathbb{Z}}_U$, canonically determined as in step 1.4. This is clause 3, and clauses 1 and 2 are step 2.1, step 1.2, step 1.3 and step 2.2. ∎