Stalks, coproducts and right exactness of the abelian sheaf tensor product

For open $U\ni x$ the germ maps $\mathcal{F}(U)\to {\mathcal{F}}_x$ and $\mathcal{G}(U)\to {\mathcal{G}}_x$ are compatible with the restriction maps of the neighbourhood diagram, so tensoring them gives maps $\mathcal{F}(U){\otimes }_{\mathbb{Z}}\mathcal{G}(U)\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x$ [F14] which form a cocone over that diagram. By the universal property of the filtered colimit [F2], applied to the tensor-product presheaf of [F15], they induce a canonical homomorphism $\phi :(\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G}{)}_x\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x$.

Let $({\mathcal{G}}_i{)}_{i\in I}$ be a family of abelian sheaves and let $P$ be the presheaf $P(U):={⨁}_{i\in I}{\mathcal{G}}_i(U)$ with the canonical injections ${\iota }_i:{\mathcal{G}}_i\to P$. Since colimits in the presheaf category are computed pointwise [F4], $P$ with the ${\iota }_i$ is the coproduct of the ${\mathcal{G}}_i$ among presheaves. I claim that $aP$ with the maps $\eta \circ {\iota }_i$, where $\eta :P\to aP$ is the sheafification map, is a coproduct in $\mathrm{A}\mathrm{b}(X)$: given a cocone $(H,f_i)$ with $H$ an abelian sheaf, the presheaf coproduct property gives a unique presheaf morphism $f:P\to H$ with $f{\iota }_i=f_i$, and by the sheafification universal property [F3] there is a unique morphism of sheaves $\bar{f}:aP\to H$ with $f=\bar{f}\eta$; then $\bar{f}\eta {\iota }_i=f_i$, and any competitor with this property induces a presheaf map $P\to H$ equal to $f$, hence equals $\bar{f}$.

Recall from [F15] that ${\mathrm{Tot}}^n({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet })={⨁}_{i+j=n}{\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j$ and that the differential $D$ on the summand ${\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j$ equals $d_{\mathcal{F}}^i\otimes \mathrm{id}+(-1{)}^i\mathrm{id}\otimes d_{\mathcal{G}}^j$. For $y\in {\mathcal{F}}^i$ and $z\in {\mathcal{G}}^j$ this gives $D(y\otimes z)=d_{\mathcal{F}}y\otimes z+(-1{)}^{i}y\otimes d_{\mathcal{G}}z$, hence $D^2(y\otimes z)=d_{\mathcal{F}}^{2}y\otimes z+(-1{)}^{i+1}{d}_{\mathcal{F}}y\otimes d_{\mathcal{G}}z+(-1{)}^i{d}_{\mathcal{F}}y\otimes d_{\mathcal{G}}z+(-1{)}^{2i}y\otimes d_{\mathcal{G}}^{2}z=0,$ because $d_{\mathcal{F}}^2=0=d_{\mathcal{G}}^2$ and $(-1{)}^{i+1}+(-1{)}^i=0$. The summands generate ${\mathrm{Tot}}^n$ and $D$ is additive, so $D^2=0$: the tensor-product total complex is a cochain complex.

Let $f\in {\mathbb{Z}}_X(U)$ be a locally constant function and $s\in \mathcal{G}(U)$. On an open $V\subseteq U$ on which $f$ is constant with value $n$, the assignment $x\mapsto f(x)s_x$ is the section $n\cdot s{\mathrm{\mid }}_V$; these local sections agree on overlaps because the constants agree there, so by unique gluing in the sheaf $\mathcal{G}$ [F18] they define a section $f\cdot s\in \mathcal{G}(U)$ with $(f\cdot s{)}_x=f(x)s_x$. The pairing $(f,s)\mapsto f\cdot s$ is additive in each variable and compatible with restrictions, so it defines a morphism of presheaves ${\mathbb{Z}}_X{\otimes }_{p,\mathbb{Z}}\mathcal{G}\to \mathcal{G}$; since $\mathcal{G}$ is a sheaf, this factors uniquely through the sheafification [F3] and yields $\lambda :{\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}\mathcal{G}\to \mathcal{G}$ with $\lambda (f\otimes s)=f\cdot s$ [F15]. At $x$ the stalk of ${\mathbb{Z}}_X$ is $\mathbb{Z}$ with $f\mapsto f(x)$ [F16] and the induced map is the unitor $n\otimes t\mapsto nt$ of [F12], an isomorphism; hence $\lambda$ is an isomorphism by [F8]. The same construction with $\mathcal{G}={\mathbb{Z}}_X$ gives ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X\cong {\mathbb{Z}}_X$. For sheaves concentrated in degree zero the total complex is the tensor product in degree zero [F15], so these identifications are the degree-zero identifications of clause 3.

Conversely let $s_x\in {\mathcal{F}}_x$ and $t_x\in {\mathcal{G}}_x$ be represented by sections $s\in \mathcal{F}(U)$ and $t\in \mathcal{G}(V)$; by [F6] the intersection $U\cap V$ is an open neighbourhood of $x$ and the restrictions $s{\mathrm{\mid }}_{U\cap V},t{\mathrm{\mid }}_{U\cap V}$ represent the same germs. The resulting class $[(s{\mathrm{\mid }}_{U\cap V})\otimes (t{\mathrm{\mid }}_{U\cap V})]$ does not depend on the choices: if $s^{\mathrm{\prime }}$ over $U^{\mathrm{\prime }}$ also represents $s_x$ and $t^{\mathrm{\prime }}$ over $V^{\mathrm{\prime }}$ also represents $t_x$, then by the equality criterion in filtered colimits [F5] and the stalk description [F2] there are neighbourhoods of $x$ on which $s$ agrees with $s^{\mathrm{\prime }}$ and $t$ with $t^{\mathrm{\prime }}$, and their intersection [F6] is a neighbourhood on which the two tensor representatives agree. The assignment $s_x\otimes t_x\mapsto [(s\otimes t)]$ is thus well defined on elementary tensors and is additive in each variable by the tensor relations [F13], so the universal property of the tensor product [F14] produces a homomorphism $\psi :{\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x\to (\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G}{)}_x$ with $\psi (s_x\otimes t_x)=[(s\otimes t)]$. On elementary tensors $\phi \psi$ is the identity and $\psi \phi$ fixes every class of an elementary tensor; these classes generate the filtered colimit and the elementary tensors generate ${\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x$ [F13], so $\phi$ and $\psi$ are mutually inverse.

At $x$ the stalk of $P$ is by [F2]. The canonical maps induce a homomorphism $\theta :P_x\to {⨁}_{i\in I}({\mathcal{G}}_i{)}_x$, which is bijective: a finite tuple of germs of sections over neighbourhoods $U_1,\dots ,U_k$ is represented by the restricted tuple over the open neighbourhood $U_1\cap \cdots \cap U_k$ [F6], so $\theta$ is surjective; and if two finite tuples over $U$ and $V$ have the same tuple of germs, then, component by component, the equality criterion in filtered colimits [F5] with the stalk description [F2] and finitely many intersections [F6] produce a common neighbourhood on which all components agree, so the two classes in $P_x$ coincide. Since the coproduct of the family in $\mathrm{A}\mathrm{b}(X)$ is $aP$ by [step 1.2] and sheafification preserves stalks by [F1], there is a canonical isomorphism $({⨁}_i{\mathcal{G}}_i{)}_x\cong P_x\cong {⨁}_i({\mathcal{G}}_i{)}_x$, which is clause 2.

By [F15] one has $\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}=a(\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G})$, and by [F1] the sheafification map induces a bijection on stalks, so composing its inverse with the isomorphism of step 2.1 gives a canonical isomorphism $(\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}{)}_x\cong {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x$. Every map entering its construction is induced by the functorial germ and restriction maps, so the isomorphism is natural in $\mathcal{F}$ and $\mathcal{G}$. This is clause 1.

In degree $n$ the canonical maps of clause 2 for the finite direct sum ${⨁}_{i+j=n}{\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j$ and of clause 1 for each summand identify the stalk $({\mathrm{Tot}}^n({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet }){)}_x$ with ${⨁}_{i+j=n}{\mathcal{F}}_x^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x^j={\mathrm{Tot}}^n({\mathcal{F}}_x^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x^{\bullet })$. These identifications are built from the natural stalk maps of [step 1.3] and [step 2.2], so they respect the Koszul differentials, which are given by the same formula on both sides [F17]; hence they assemble into an isomorphism of complexes. This is clause 3.

Let $0\to {\mathcal{G}}^{\mathrm{\prime }}\to \mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence of abelian sheaves and let $\mathcal{F}$ be an abelian sheaf. Applying $\mathcal{F}{\otimes }_{\mathbb{Z}}-$ degreewise gives the sequence $\mathcal{F}{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\mathrm{\prime }}\to \mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}\to \mathcal{F}{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$; by clause 1 and naturality of the stalk identification its stalk at $x$ is the sequence ${\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x^{\mathrm{\prime }}\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x\to {\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x^{\mathrm{\prime }\mathrm{\prime }}\to 0$, which is exact by the right exactness of the tensor product of $\mathbb{Z}$-modules [F9]. Since $x$ was arbitrary, the sheaf sequence is exact by the stalkwise exactness criterion [F7]. No injectivity on the left is claimed, and the argument in the first variable is symmetric.

The assignment $u\otimes v\mapsto v\otimes u$ on sections over an open set is additive in each variable and commutes with restrictions, so it defines a morphism of presheaves $\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G}\to \mathcal{G}{\otimes }_{p,\mathbb{Z}}\mathcal{F}$; by [F15] its sheafification is a morphism $\sigma :\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}\to \mathcal{G}{\otimes }_{\mathbb{Z}}\mathcal{F}$. At each $x$ its stalk is the swap map ${\mathcal{F}}_x{\otimes }_{\mathbb{Z}}{\mathcal{G}}_x\to {\mathcal{G}}_x{\otimes }_{\mathbb{Z}}{\mathcal{F}}_x$, an isomorphism by [F11], so $\sigma$ is an isomorphism by [F8].

For a family $({\mathcal{G}}_i)$ the maps $\mathcal{F}{\otimes }_{\mathbb{Z}}{\mathcal{G}}_i\to \mathcal{F}{\otimes }_{\mathbb{Z}}\left({⨁}_i{\mathcal{G}}_i\right)$ induced by the coproduct injections give a canonical map ${⨁}_i(\mathcal{F}{\otimes }_{\mathbb{Z}}{\mathcal{G}}_i)\to \mathcal{F}{\otimes }_{\mathbb{Z}}\left({⨁}_i{\mathcal{G}}_i\right)$. At $x$ the source identifies with ${⨁}_i({\mathcal{F}}_x{\otimes }_{\mathbb{Z}}({\mathcal{G}}_i{)}_x)$ by clause 2 and the target with ${\mathcal{F}}_x{\otimes }_{\mathbb{Z}}({⨁}_i({\mathcal{G}}_i{)}_x)$ by clause 1 and clause 2, and these agree by the module-level commutation of tensor products with direct sums [F10]; the comparison is the canonical one, hence an isomorphism by the stalkwise criterion [F8]. The first variable is treated symmetrically. Together with [step 4.1] this proves clause 4.

Clauses 1 to 5 are step 3.1, step 2.2, step 4.1, step 4.2 with step 5.1, and step 4.3 with step 1.4. Each isomorphism constructed is canonical: it is assembled from germ and restriction maps, the universal property of the filtered colimit [F2], the sheafification map [F1, F3], the module-level symmetry, unitor and distributivity isomorphisms [F10, F11, F12], and the gluing of compatible local sections [F18], all of which are unique once their data are specified; consequently the identifications are natural in their arguments and commute with the morphisms of clause 5. No choice principle is used: the only finitely many open sets intersected are canonical intersections [F6], the sheafification is the canonical double-plus construction [F15], the coproduct and its universal property are canonical [step 1.2], and every tensor element is a finite sum of elementary tensors [F13], so no infinite or arbitrary selection occurs. ∎