Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes

By [F1] the canonical morphism ${\mathrm{\Phi }}_{\mathcal{F}}:G(\mathcal{F})\to \mathcal{F}$ of the covering flat sheaf is an epimorphism with $G(\mathcal{F})$ flat, and by [F2] it is canonically determined by $\mathcal{F}$. To obtain a criterion for stalkwise surjectivity, use the exact sequence $G(\mathcal{F})\to \mathcal{F}\to \mathrm{coker}({\mathrm{\Phi }}_{\mathcal{F}})\to 0$: by [F8] it is exact exactly when all its stalk sequences are exact, that is, when the cokernel of $({\mathrm{\Phi }}_{\mathcal{F}}{)}_x$ is $(\mathrm{coker}{\mathrm{\Phi }}_{\mathcal{F}}{)}_x$ for every $x$; by [F18] the epimorphism ${\mathrm{\Phi }}_{\mathcal{F}}$ has zero cokernel, and by [F19] a sheaf is zero exactly when all its stalks are zero, so $({\mathrm{\Phi }}_{\mathcal{F}}{)}_x:G(\mathcal{F}{)}_x\to {\mathcal{F}}_x$ is surjective for every $x\in X$, and conversely a morphism all of whose stalk maps are surjective is an epimorphism.

Fix a morphism $\phi :\mathcal{A}\to \mathcal{B}$ of abelian sheaves and $x\in X$. Applying [F8] to the exact sequences $0\to \ker \phi \to \mathcal{A}\to \mathcal{B}$, $\mathcal{A}\to \mathrm{im}\phi \to 0$ and $\mathcal{A}\to \mathcal{B}\to \mathrm{coker}\phi \to 0$ gives $(\ker \phi {)}_x=\ker ({\phi }_x)$, $(\mathrm{im}\phi {)}_x=\mathrm{im}({\phi }_x)$ and $(\mathrm{coker}\phi {)}_x=\mathrm{coker}({\phi }_x)$. By [F20] the stalk construction is functorial, so $(\psi \circ \phi {)}_x={\psi }_x\circ {\phi }_x$ for composable morphisms $\phi ,\psi$ of abelian sheaves. By [F7] the stalk of a coproduct of abelian sheaves is the coproduct of the stalks. Finally, for a cochain complex ${\mathcal{C}}^{\bullet }$ of abelian sheaves with cocycles $Z^n$ and coboundaries $B^n$ the identities just recorded give $(H^n({\mathcal{C}}^{\bullet }){)}_x\cong H^n({\mathcal{C}}_x^{\bullet })$ for every $n\in \mathbb{Z}$, because $H^n$ is the cokernel of $B^n\to Z^n$ by [F12] and the stalk of that cokernel is the cokernel of the stalk map.

${\mathrm{I}\mathrm{H}}_{a+1}$ holds with ${\mathcal{P}}^j:=0$, $d_{\mathcal{P}}^j:=0$ and ${\alpha }^j:=0$ for $j\ge a+1$: for such $j$ we have $j>a$, so ${\mathcal{C}}^j=0$ by the choice of $a$ [F9] and the maps are zero maps between zero sheaves, whence (i), (ii) and (iii); each ${\mathcal{P}}^j=0$ is flat [F3]; for (iv) let $j>a+1$, then ${\mathcal{P}}^j={\mathcal{P}}^{j+1}=0$, so $Z^j=B^j=0$ and $Z^j\mathrm{/}{B}^j=0$, while ${\mathcal{C}}^j={\mathcal{C}}^{j+1}=0$ gives $H^j({\mathcal{C}}^{\bullet })=0$ [F12], and the zero morphism between zero objects is an isomorphism; for (v) the induced morphism $\ker (d_{\mathcal{P}}^{a+1})\to \ker (d_{\mathcal{C}}^{a+1})$ is the zero morphism between the zero sheaves ${\mathcal{P}}^{a+1}$ and ${\mathcal{C}}^{a+1}$, which is surjective on stalks.

Now let ${\mathcal{K}}^{\bullet }$ be a bounded-above complex of flat abelian sheaves and let ${\mathcal{F}}^{\bullet }$ be an acyclic bounded-above complex of abelian sheaves, with ${\mathcal{F}}^n={\mathcal{K}}^n=0$ for all $n>b$ [F9]. Fix $x\in X$. The stalk complex ${\mathcal{F}}_x^{\bullet }$ is bounded above, since $({\mathcal{F}}^n{)}_x=0$ for $n>b$ (the zero sheaf has zero stalk by [F20]), and it is acyclic: acyclicity of ${\mathcal{F}}^{\bullet }$ means exactness at every degree [F17], and by [F8] every stalk sequence of ${\mathcal{F}}^{\bullet }$ is exact in every degree, that is $H^n({\mathcal{F}}_x^{\bullet })=0$ for all $n$ [F12]. The stalk complex ${\mathcal{K}}_x^{\bullet }$ is a bounded-above complex of flat $\mathbb{Z}$-modules, because each ${\mathcal{K}}^n$ is flat [F3] and ${\mathcal{K}}_x^n=0$ for $n>b$ [F20]. This is exactly the situation of [F10], whose bounded-above hypothesis is read through the cochain reindexing of the total complex as recorded in [F11]; the total complex reindexed there is the module total complex of [F6] with $d(x\otimes y)=d_{\mathcal{F}}x\otimes y+(-1{)}^{i}x\otimes d_{\mathcal{K}}y$, since under the reindexing $(-1{)}^p$ becomes $(-1{)}^i$. Applying [F10] with $R=\mathbb{Z}$, $A={\mathcal{F}}_x^{\bullet }$ and $P={\mathcal{K}}_x^{\bullet }$ therefore gives that $\mathrm{Tot}({\mathcal{F}}_x^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{K}}_x^{\bullet })$ is acyclic. By [F5] and [F6] the stalk of $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })$ at $x$ is canonically isomorphic to that module total complex, hence acyclic. As $x$ was arbitrary, each stalk of $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })$ is acyclic, so by [F8] the complex $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })$ is exact in every degree, i.e. acyclic [F17]. Since ${\mathcal{F}}^{\bullet }$ was an arbitrary acyclic bounded-above complex, ${\mathcal{K}}^{\bullet }$ is K-flat [F4]. This is clause 2.

The full source in [F1] splits as the coproduct of the summands with $s\mathrm{\ne }0$ and the summands with $s=0$. The latter map to zero under ${\mathrm{\Phi }}_{\mathcal{E}}$, so restricting to $G_{\mathrm{r}\mathrm{e}\mathrm{d}}(\mathcal{E})$ is still surjective on stalks: a nonzero germ is represented by a nonzero section, and the zero germ is hit by zero. Every summand is flat and coproducts of these are flat by [F1]. Thus this restriction is a canonical flat epimorphism, and $G_{\mathrm{r}\mathrm{e}\mathrm{d}}(0)=0$. For a sheaf map $f:\mathcal{E}\to {\mathcal{E}}^{\mathrm{\prime }}$, send the summand $(U,s)$ to $(U,f_U(s))$ by the identity if $f_U(s)\mathrm{\ne }0$, and by zero if $f_U(s)=0$. These prescriptions commute with the epimorphisms and preserve identities and composition, including the case where a section becomes zero.

Clause 1 is proved by descending induction using the reduced flat epimorphisms of step 2.1. Let us say that ${\mathrm{I}\mathrm{H}}_n$ holds, for an integer $n\le a+1$, when there are sheaves ${\mathcal{P}}^j$ for $j\ge n$, with ${\mathcal{P}}^j$ flat and ${\mathcal{P}}^j=0$ for every $j>a$, morphisms $d_{\mathcal{P}}^j:{\mathcal{P}}^j\to {\mathcal{P}}^{j+1}$ and ${\alpha }^j:{\mathcal{P}}^j\to {\mathcal{C}}^j$ for $j\ge n$, such that (i) each ${\alpha }^j$ is surjective on stalks, (ii) ${\alpha }^{j+1}\circ d_{\mathcal{P}}^j=d_{\mathcal{C}}^j\circ {\alpha }^j$ for $j\ge n$, (iii) $d_{\mathcal{P}}^{j+1}\circ d_{\mathcal{P}}^j=0$ for $j\ge n$, (iv) for every $j>n$ the morphism $Z^j({\mathcal{P}}^{\bullet })\mathrm{/}{B}^j({\mathcal{P}}^{\bullet })\to H^j({\mathcal{C}}^{\bullet })$ induced on quotients by ${\alpha }^j$ -- well defined by (ii) -- is an isomorphism, and (v) the morphism $\ker (d_{\mathcal{P}}^n)\to \ker (d_{\mathcal{C}}^n)$ induced by ${\alpha }^n$ -- well defined by (ii) and (iii), since $d_{\mathcal{C}}^n{\alpha }^n={\alpha }^{n+1}{d}_{\mathcal{P}}^n=0$ on $\ker (d_{\mathcal{P}}^n)$ [F12, F16] -- is surjective on stalks. The data of ${\mathrm{I}\mathrm{H}}_n$ at degrees $j\ge n^{\mathrm{\prime }}$ are the data of ${\mathrm{I}\mathrm{H}}_{n^{\mathrm{\prime }}}$ for $n^{\mathrm{\prime }}>n$, and the induction step below adds objects and maps only in degree $n-1$, so the hypotheses for different $n$ are compatible and yield a single cochain complex ${\mathcal{P}}^{\bullet }$.

Assume ${\mathrm{I}\mathrm{H}}_n$ with $n\le a+1$ and construct the data of ${\mathrm{I}\mathrm{H}}_{n-1}$ in degree $n-1$. Put ${\mathcal{K}}_{\mathcal{P}}:=\ker (d_{\mathcal{P}}^n:{\mathcal{P}}^n\to {\mathcal{P}}^{n+1})$ and ${\mathcal{K}}_{\mathcal{C}}:=\ker (d_{\mathcal{C}}^n:{\mathcal{C}}^n\to {\mathcal{C}}^{n+1})$ [F16]; by (v) the morphism $e:{\mathcal{K}}_{\mathcal{P}}\to {\mathcal{K}}_{\mathcal{C}}$ induced by ${\alpha }^n$ is surjective on stalks. Since coproducts exist [F22], form the direct sum ${\mathcal{C}}^{n-1}\oplus {\mathcal{K}}_{\mathcal{P}}$ with its two structure maps and define $\delta :{\mathcal{C}}^{n-1}\oplus {\mathcal{K}}_{\mathcal{P}}\to {\mathcal{C}}^n$ to be the morphism whose components are $d_{\mathcal{C}}^{n-1}$ and the negative of the composite ${\mathcal{K}}_{\mathcal{P}}\stackrel{e}{\to }{\mathcal{K}}_{\mathcal{C}}\hookrightarrow {\mathcal{C}}^n$. Put $\mathcal{Q}:=\ker \delta$ [F16], let ${\pi }_1:\mathcal{Q}\to {\mathcal{C}}^{n-1}$ and ${\pi }_2:\mathcal{Q}\to {\mathcal{K}}_{\mathcal{P}}$ be the composites of the inclusion $\mathcal{Q}\subseteq {\mathcal{C}}^{n-1}\oplus {\mathcal{K}}_{\mathcal{P}}$ with the two projections, and let ${\mathcal{P}}^{n-1}:=G_{\mathrm{r}\mathrm{e}\mathrm{d}}(\mathcal{Q})$ with $\beta :={\mathrm{\Phi }}_{\mathcal{Q}}{\mathrm{\mid }}_{G_{\mathrm{r}\mathrm{e}\mathrm{d}}(\mathcal{Q})}:{\mathcal{P}}^{n-1}\to \mathcal{Q}$ the reduced flat epimorphism of step 2.1; define ${\alpha }^{n-1}:={\pi }_1\circ \beta :{\mathcal{P}}^{n-1}\to {\mathcal{C}}^{n-1}$ and $d_{\mathcal{P}}^{n-1}:=\iota \circ {\pi }_2\circ \beta :{\mathcal{P}}^{n-1}\to {\mathcal{P}}^n$, where $\iota :{\mathcal{K}}_{\mathcal{P}}\hookrightarrow {\mathcal{P}}^n$ is the inclusion. The sheaf ${\mathcal{P}}^{n-1}$ is flat and canonically determined by $\mathcal{Q}$ by step 2.1. By step 2.1 the map $\beta$ is surjective on stalks, and by step 1.2 and [F7] the stalk of $\mathcal{Q}=\ker \delta$ at a point $x$ is ${\mathcal{Q}}_x=\{(a,y)\in ({\mathcal{C}}^{n-1}{)}_x\oplus ({\mathcal{K}}_{\mathcal{P}}{)}_x:(d_{\mathcal{C}}^{n-1}{)}_x(a)=({\alpha }^n{)}_x(y)\}$, with $({\pi }_1{)}_x,(d_{\mathcal{P}}^{n-1}{)}_x$ given by the two components composed with ${\beta }_x$ and ${\iota }_x$.

(i) for degree $n-1$: the map ${\alpha }^{n-1}$ is surjective on stalks. Let $x\in X$ and $a\in ({\mathcal{C}}^{n-1}{)}_x$. Since $d_{\mathcal{C}}^n{d}_{\mathcal{C}}^{n-1}=0$, the element $(d_{\mathcal{C}}^{n-1}{)}_x(a)$ lies in $\ker ((d_{\mathcal{C}}^n{)}_x)=({\mathcal{K}}_{\mathcal{C}}{)}_x$ [F8]; as $e_x:({\mathcal{K}}_{\mathcal{P}}{)}_x\to ({\mathcal{K}}_{\mathcal{C}}{)}_x$ is surjective by (v), there is $y\in ({\mathcal{K}}_{\mathcal{P}}{)}_x$ with $({\alpha }^n{)}_x(y)=(d_{\mathcal{C}}^{n-1}{)}_x(a)$, so $(a,y)\in {\mathcal{Q}}_x$ by step 3.2. Since ${\beta }_x$ is surjective by step 2.1 there is $p\in ({\mathcal{P}}^{n-1}{)}_x$ with ${\beta }_x(p)=(a,y)$, and then $({\alpha }^{n-1}{)}_x(p)=({\pi }_1{)}_x(a,y)=a$. Hence $({\alpha }^{n-1}{)}_x$ is surjective; as $x$ was arbitrary, ${\alpha }^{n-1}$ is an epimorphism by step 1.1 and surjective on stalks.

(ii) and (iii) in degree $n-1$: $d_{\mathcal{P}}^n\circ d_{\mathcal{P}}^{n-1}=d_{\mathcal{P}}^n\circ \iota \circ {\pi }_2\circ \beta =0$ because $d_{\mathcal{P}}^n\circ \iota =0$ by the definition of ${\mathcal{K}}_{\mathcal{P}}=\ker (d_{\mathcal{P}}^n)$ [F16]. For the cochain identity, both ${\alpha }^n\circ d_{\mathcal{P}}^{n-1}$ and $d_{\mathcal{C}}^{n-1}\circ {\alpha }^{n-1}$ are morphisms ${\mathcal{P}}^{n-1}\to {\mathcal{C}}^n$, and by [F19] it suffices to compare their stalk maps at every point. Let $x\in X$ and $p\in ({\mathcal{P}}^{n-1}{)}_x$ with ${\beta }_x(p)=(a,y)\in {\mathcal{Q}}_x$; by step 3.2 the defining relation of ${\mathcal{Q}}_x$ is $(d_{\mathcal{C}}^{n-1}{)}_x(a)=({\alpha }^n{)}_x(y)$, so the left-hand stalk map sends $p$ to $({\alpha }^n{)}_x({\iota }_x(y))=(d_{\mathcal{C}}^{n-1}{)}_x(a)$, which is the image of $p$ under the right-hand stalk map. Hence ${\alpha }^n\circ d_{\mathcal{P}}^{n-1}=d_{\mathcal{C}}^{n-1}\circ {\alpha }^{n-1}$, which is (ii) for $j=n-1$, while (ii) and (iii) for $j\ge n$ are those of ${\mathrm{I}\mathrm{H}}_n$; together with the computation of $d_{\mathcal{P}}^n\circ d_{\mathcal{P}}^{n-1}=0$ this is (ii) and (iii) for ${\mathrm{I}\mathrm{H}}_{n-1}$.

(iv) for degree $n$, and (iv) of ${\mathrm{I}\mathrm{H}}_{n-1}$: the morphism ${\phi }_n:Z^n({\mathcal{P}}^{\bullet })\mathrm{/}{B}^n({\mathcal{P}}^{\bullet })\to H^n({\mathcal{C}}^{\bullet })$ induced by ${\alpha }^n$ is an isomorphism. By [F19] it suffices to show that its stalk at every $x$ is bijective. By step 1.2 the stalk of $Z^n({\mathcal{P}}^{\bullet })\mathrm{/}{B}^n({\mathcal{P}}^{\bullet })$ is $\ker ((d_{\mathcal{P}}^n{)}_x)\mathrm{/}\mathrm{im}((d_{\mathcal{P}}^{n-1}{)}_x)$ and the stalk of $H^n({\mathcal{C}}^{\bullet })$ is $H^n({\mathcal{C}}_x^{\bullet })$, the map between them being induced by $({\alpha }^n{)}_x$ restricted to the cocycles; as ${\beta }_x$ is surjective and ${\iota }_x$ is injective, step 3.2 gives $\mathrm{im}((d_{\mathcal{P}}^{n-1}{)}_x)=({\pi }_2{)}_x({\mathcal{Q}}_x)=\{y\in ({\mathcal{K}}_{\mathcal{P}}{)}_x:({\alpha }^n{)}_x(y)\in \mathrm{im}((d_{\mathcal{C}}^{n-1}{)}_x)\}$. That displayed set is exactly the preimage $M$ of the coboundaries $\mathrm{im}((d_{\mathcal{C}}^{n-1}{)}_x)$ under the restriction of $({\alpha }^n{)}_x$ to $\ker ((d_{\mathcal{P}}^n{)}_x)$: the inclusion $\subseteq$ is the defining relation of ${\mathcal{Q}}_x$ in step 3.2, and conversely a cocycle $y$ with $({\alpha }^n{)}_x(y)=(d_{\mathcal{C}}^{n-1}{)}_x(a)$ for some $a$ satisfies $(a,y)\in {\mathcal{Q}}_x$. Hence the kernel of the induced map on quotients is $M\mathrm{/}\mathrm{im}((d_{\mathcal{P}}^{n-1}{)}_x)=0$, and the induced map is surjective because $e_x$ is surjective onto $\ker ((d_{\mathcal{C}}^n{)}_x)$ by (v); so the stalk map is bijective and ${\phi }_n$ is an isomorphism. For $j>n$ the maps of (iv) are the isomorphisms of ${\mathrm{I}\mathrm{H}}_n$, unchanged. This proves (iv) for ${\mathrm{I}\mathrm{H}}_{n-1}$.

(v) for degree $n-1$: the morphism $\ker (d_{\mathcal{P}}^{n-1})\to \ker (d_{\mathcal{C}}^{n-1})$ induced by ${\alpha }^{n-1}$ is surjective on stalks. Let $x\in X$ and $a\in \ker ((d_{\mathcal{C}}^{n-1}{)}_x)$, so that $(d_{\mathcal{C}}^{n-1}{)}_x(a)=0$. Then $(a,0)\in {\mathcal{Q}}_x$, since the defining relation of step 3.2 reads $(d_{\mathcal{C}}^{n-1}{)}_x(a)=({\alpha }^n{)}_x(0)=0$ and $0\in ({\mathcal{K}}_{\mathcal{P}}{)}_x$; by step 2.1 choose $p\in ({\mathcal{P}}^{n-1}{)}_x$ with ${\beta }_x(p)=(a,0)$. Then $(d_{\mathcal{P}}^{n-1}{)}_x(p)={\iota }_x({\pi }_2{)}_x(a,0)=0$, so $p\in \ker ((d_{\mathcal{P}}^{n-1}{)}_x)$, and by step 4.2 the induced map sends $p$ to $({\alpha }^{n-1}{)}_x(p)=({\pi }_1{)}_x(a,0)=a$. Hence the induced morphism is surjective on stalks.

By step 1.3 and step 3.2 the hypotheses ${\mathrm{I}\mathrm{H}}_n$ hold for every $n\le a+1$, with the compatibility recorded in step 3.1. Define ${\mathcal{P}}^j$ to be the sheaf attached to degree $j$ (this is unambiguous because the data in degree $j$ are fixed at every stage $n\le j$), and similarly $d_{\mathcal{P}}^j$ and ${\alpha }^j$; then ${\mathcal{P}}^j$ is flat for every $j$ and ${\mathcal{P}}^j=0$ for $j>a$ [F1, F3]; $d_{\mathcal{P}}^{j+1}{d}_{\mathcal{P}}^j=0$ for all $j$ by (iii), the case $j\ge a+1$ being a composite of zero maps; ${\alpha }^j$ is surjective on stalks for every $j$ by (i); and $d_{\mathcal{C}}^j{\alpha }^j={\alpha }^{j+1}{d}_{\mathcal{P}}^j$ for every $j$ by (ii), the case $j\ge a+1$ being zero maps, so $\alpha =({\alpha }^j)$ is a cochain map [F21]. For every $j$ the map $Z^j({\mathcal{P}}^{\bullet })\mathrm{/}{B}^j({\mathcal{P}}^{\bullet })\to H^j({\mathcal{C}}^{\bullet })$ induced by ${\alpha }^j$ is an isomorphism, by (iv) applied with $n=j-1$; since $Z^j\mathrm{/}{B}^j=H^j$ [F12], this is the map $H^j(\alpha )$ of [F13] read through the reindexing convention of [F15]. Hence $H^j(\alpha )$ is an isomorphism for every $j$ and $\alpha :{\mathcal{P}}^{\bullet }\to {\mathcal{C}}^{\bullet }$ is a quasi-isomorphism [F14, F15]. Finally, ${\mathcal{P}}^{\bullet }$ and $\alpha$ are canonically determined by ${\mathcal{C}}^{\bullet }$: increasing the starting bound adds only zero terms since the recursion has $\mathcal{Q}=0$ and $G_{\mathrm{r}\mathrm{e}\mathrm{d}}(0)=0$ above the original bound (step 2.1); at each step $\mathcal{Q}$ is the kernel of the morphism $\delta$ built from the previous data by the displayed formula [F16], and ${\mathcal{P}}^{n-1}=G_{\mathrm{r}\mathrm{e}\mathrm{d}}(\mathcal{Q})$ with its epimorphism is the canonical reduced flat epimorphism onto $\mathcal{Q}$ of step 2.1, so no selection is made anywhere in the recursion. This is clause 1.

Clause 1 is step 5.2: every bounded-above complex ${\mathcal{C}}^{\bullet }$ of abelian sheaves admits the canonically determined bounded-above, termwise surjective quasi-isomorphism $\alpha :{\mathcal{P}}^{\bullet }\to {\mathcal{C}}^{\bullet }$ out of a complex of flat sheaves, no choice principle entering. Clause 2 is step 1.4: every bounded-above complex of flat sheaves is K-flat, and so tensoring with it preserves quasi-isomorphisms by the K-flat criterion of the page. Together these are the two flatness inputs used by the derived tensor product of abelian sheaves. ∎