K-flat sheaf complexes preserve quasi-isomorphisms

For each $i$ the map $s^i\otimes {\mathrm{id}}_{{\mathcal{K}}^j}$ is defined on the summands ${\mathcal{A}}^i{\otimes }_{\mathbb{Z}}{\mathcal{K}}^j$ of $\mathrm{Tot}({\mathcal{A}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })$ and is compatible with the coproduct injections, so it induces a degree-zero morphism $\mathrm{Tot}(s\otimes \mathrm{id})$; it is a cochain map because $s$ is a cochain map [F7] and the Koszul differential [F4] has the same shape on both sides, the sign factor $(-1{)}^i$ being unchanged by $s^i$.

Assume the cone convention of [F3]. Define $\phi :\mathrm{Tot}(\mathrm{Cone}(s){\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })⟶\mathrm{Cone}(\mathrm{Tot}(s\otimes {\mathrm{id}}_{\mathcal{K}}))$ to be the identity on the summands ${\mathcal{B}}^p{\otimes }_{\mathbb{Z}}{\mathcal{K}}^j$ and the identity, after the identification of the $\mathcal{A}$-part of $\mathrm{Cone}(s{)}^p$ with ${\mathcal{A}}^{p+1}$, on the summands ${\mathcal{A}}^{p+1}{\otimes }_{\mathbb{Z}}{\mathcal{K}}^j$. Then $\phi$ is an isomorphism of graded groups, both sides having degree-$n$ term ${⨁}_{p+j=n}\left({\mathcal{B}}^p{\otimes }_{\mathbb{Z}}{\mathcal{K}}^j\right)\oplus {⨁}_{p+j=n}\left({\mathcal{A}}^{p+1}{\otimes }_{\mathbb{Z}}{\mathcal{K}}^j\right)$, and a direct check on generators shows that it commutes with the differentials: for $b\in {\mathcal{B}}^p$ the element $b\otimes y$ of $\mathrm{Cone}(s{)}^p\otimes {\mathcal{K}}^j$ has $d_{\mathrm{Cone}(s)}(b,0)=(d_{\mathcal{B}}b,0)$, so both sides give $d_{\mathcal{B}}b\otimes y+(-1{)}^{p}b\otimes d_{\mathcal{K}}y$; for $a\in {\mathcal{A}}^{p+1}$ the element $(0,a)\otimes y$ has $d_{\mathrm{Cone}(s)}(0,a)=(s(a),-d_{\mathcal{A}}a)$, so the left side is $s(a)\otimes y-d_{\mathcal{A}}a\otimes y+(-1{)}^{p}a\otimes d_{\mathcal{K}}y$, while the right side, by [F3] with $f=\mathrm{Tot}(s\otimes \mathrm{id})$ and by the Koszul differential of $\mathrm{Tot}({\mathcal{A}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })$ [F4], is $(s(a)\otimes y,-d_{\mathcal{A}}a\otimes y+(-1{)}^{p}a\otimes d_{\mathcal{K}}y)$, the same element. Hence $\phi$ is an isomorphism of complexes.

Since $s$ is a quasi-isomorphism, $\mathrm{Cone}(s)$ is acyclic [F2, F6]; it is a bounded-above complex of abelian sheaves because its terms ${\mathcal{B}}^n\oplus {\mathcal{A}}^{n+1}$ vanish for all sufficiently large $n$ by [F5], and ${\mathcal{K}}^{\bullet }$ is bounded above, so the tensor-product total complex $\mathrm{Tot}(\mathrm{Cone}(s){\otimes }_{\mathbb{Z}}{\mathcal{K}}^{\bullet })$ is acyclic by the K-flatness of ${\mathcal{K}}^{\bullet }$ [F1]. By the isomorphism of step 1.2 the cone $\mathrm{Cone}(\mathrm{Tot}(s\otimes {\mathrm{id}}_{\mathcal{K}}))$ of step 1.1 is acyclic, and therefore $\mathrm{Tot}(s\otimes {\mathrm{id}}_{\mathcal{K}})$ is a quasi-isomorphism by the cone criterion [F2]. This is clause 1.

For bounded-above ${\mathcal{F}}^{\bullet },{\mathcal{G}}^{\bullet }$ define $T:\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet })\to \mathrm{Tot}({\mathcal{G}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{F}}^{\bullet })$ on the summand ${\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j$ by $T(x\otimes y):=(-1{)}^{ij}y\otimes x$. This is an isomorphism of graded groups, and it commutes with the differentials: by [F4] the differential applied first gives $T(d_{\mathcal{F}}x\otimes y+(-1{)}^{i}x\otimes d_{\mathcal{G}}y)=(-1{)}^{(i+1)j}y\otimes d_{\mathcal{F}}x+(-1{)}^{i+i(j+1)}{d}_{\mathcal{G}}y\otimes x$, while applying the differential of $\mathrm{Tot}({\mathcal{G}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{F}}^{\bullet })$ first gives $(-1{)}^{ij}(d_{\mathcal{G}}y\otimes x+(-1{)}^{j}y\otimes d_{\mathcal{F}}x)$; the two expressions agree because $(-1{)}^{(i+1)j}=(-1{)}^{ij}(-1{)}^j$ and $(-1{)}^{i+i(j+1)}=(-1{)}^{ij}$; the Koszul signs used are those of the sheaf-level total complex, consistent with the module-level convention on stalks [F8]. Applying step 1.1 and step 2.1 to the swap of $s$ gives clause 2.

Clause 1 is step 2.1 and clause 2 is step 3.1. Both isomorphisms are canonical: the cone is the canonical cone of [F3], the identification of step 1.2 is the identity on the canonical summands, and the swap sign $(-1{)}^{ij}$ is forced by the Koszul convention [F4] through the computation of step 3.1; no selection is made beyond the hypotheses, and the K-flatness input [F1] is a universal statement about all acyclic bounded-above complexes. ∎