Tensor product of abelian sheaves and its total complex

Let $X$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), and let $\mathrm{A}\mathrm{b}(X)$ be the abelian category of sheaves of abelian groups on $X$ (Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories). Write $A{\otimes }_{\mathbb{Z}}B$ for the tensor product of the abelian groups $A$ and $B$, that is, for The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums with $R=\mathbb{Z}$.

For sheaves $\mathcal{F},\mathcal{G}$ of abelian groups on $X$ the tensor-product presheaf $\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G}$ is the presheaf of abelian groups (Presheaves and sheaves of groups, rings, and modules) with $(\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G})(U):=\mathcal{F}(U){\otimes }_{\mathbb{Z}}\mathcal{G}(U)$ for every open $U\subseteq X$, whose restriction maps are induced by the restriction maps of $\mathcal{F}$ and $\mathcal{G}$; they are homomorphisms because the tensor product of abelian groups is functorial in each variable.

The tensor product of abelian sheaves is its sheafification (Sheafification of a presheaf) $\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}:=a\left(\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G}\right),$ a sheaf of abelian groups on $X$, and it is covariantly natural in $\mathcal{F}$ and $\mathcal{G}$: morphisms $f:\mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }}$ and $g:\mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }}$ induce a morphism $f{\otimes }_{\mathbb{Z}}g:\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}\to {\mathcal{F}}^{\mathrm{\prime }}{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\mathrm{\prime }}$ by functoriality of the presheaf tensor product and the sheafification adjunction (Sheafification is left adjoint to the inclusion of sheaves into presheaves). If abelian sheaves are regarded as modules over the constant sheaf of rings ${\mathbb{Z}}_X$ (The constant sheaf is the sheaf of locally constant functions), the two constructions $\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}$ and the tensor product over ${\mathbb{Z}}_X$ of Tensor product of sheaves of modules agree up to canonical isomorphism, because they have the same stalks by The stalk of a tensor product sheaf is the tensor product of the stalks.

For bounded-above cochain complexes ${\mathcal{F}}^{\bullet },{\mathcal{G}}^{\bullet }$ of abelian sheaves (Cochain complex in an abelian category, Bounded, bounded below, and bounded above complexes) the tensor-product total complex $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet })$ is the cochain complex with degree-$n$ term the direct sum (The direct sum of an indexed family of modules, $R=\mathbb{Z}$) ${\mathrm{Tot}}^n({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet }):={⨁}_{i+j=n}{\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j$ over the degree-$n$ diagonal, whose differential is the unique morphism that on the summand ${\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j$ equals $d_{\mathcal{F}}^i\otimes {\mathrm{id}}_{{\mathcal{G}}^j}+(-1{)}^i{\mathrm{id}}_{{\mathcal{F}}^i}\otimes d_{\mathcal{G}}^j\mathrm{.}$ If ${\mathcal{F}}^i=0$ for $i>a$ and ${\mathcal{G}}^j=0$ for $j>b$, then for each $n\le a+b$ only the finitely many pairs with $n-b\le i\le a$ contribute, so every diagonal is a finite direct sum and no infinite sum in a morphism group is required; consequently the total complex is again bounded above. For sheaves $\mathcal{F},\mathcal{G}$ concentrated in degree zero the total complex is $\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}$ in degree zero with zero terms elsewhere. The differential satisfies $d^2=0$, and the relation of this complex to the module-level tensor total complex is established in Stalks, coproducts and right exactness of the abelian sheaf tensor product.