Kunneth Tor spectral sequence

Write the supplied resolutions as $P_{i,a}\to C_i$ and $Q_{j,b}\to D_j$, with commuting arrows $h_P,v_P$ and $h_Q,v_Q$ in complex and resolution degrees, as supplied in F1. Twisting the vertical arrows by ${\tilde{v}}_P=(-1{)}^i{v}_P$ and ${\tilde{v}}_Q=(-1{)}^j{v}_Q$ puts each commuting grid into anticommuting homological double-complex form, with signed totals $h_P+{\tilde{v}}_P$ and $h_Q+{\tilde{v}}_Q$. Form $P_{i,a}{\otimes }_R{Q}_{j,b}$, group by $q=i+j$, $p=a+b$, and take the Koszul total differential of F3. Its components lowering $q$ and $p$ are $d_q=h_P\otimes 1+(-1{)}^{i+a}1\otimes h_Q,d_p={\tilde{v}}_P\otimes 1+(-1{)}^{i+a}1\otimes {\tilde{v}}_Q,$ where $(-1{)}^{i+a}$ is the Koszul sign attached to the second factor of the tensor of the two signed totals. Then $d_q^2=d_p^2=0$ and $d_q{d}_p+d_p{d}_q=0$: the diagonal blocks vanish by the two internal anticommutations $h_P{\tilde{v}}_P+{\tilde{v}}_P{h}_P=0$ and $h_Q{\tilde{v}}_Q+{\tilde{v}}_Q{h}_Q=0$, while the mixed blocks cancel because the Koszul sign changes sign whenever the complex degree $i$ or the resolution degree $a$ changes. Hence $(P\otimes Q,d_q,d_p)$ is a homological double complex whose total differential is $d_q+d_p$, the Koszul differential of the tensor of the signed totals of $P$ and $Q$.

Filter increasingly by $p$. At a fixed pair $(a,b)$ the horizontal complexes $P_{\ast ,a},Q_{\ast ,b}$ split into stalks of their homology and two-term identity disks, by F1. Tensoring a disk with any complex remains contractible: if $s$ contracts the disk, $s\otimes 1$ contracts the first-factor tensor, and $(-1{)}^{i}1\otimes s$ contracts a second-factor disk. Substitution in F3 cancels the mixed terms. Consequently horizontal homology is canonically ${⨁}_{i+j=q}{H}_i(P_{\ast ,a}){\otimes }_R{H}_j(Q_{\ast ,b})$, via the tensor-of-cycles map. The splitting argument proves this canonical map is an isomorphism; it need not choose splittings naturally.

Both augmented totals $\mathrm{Tot}P\to C$ and $\mathrm{Tot}Q\to D$ are quasi-isomorphisms: filtering by the original complex degree gives first page equal to that complex in resolution degree zero and zero in higher degrees, by the exact augmented projective columns; the finite filtration comparison identifies the augmentation on homology. Their total terms are finite sums of projectives, hence flat by F4. A bounded-below complex $L$ of flat modules preserves quasi-isomorphisms on tensoring: tensor the acyclic mapping cone with each $L_i$, obtaining exact complexes by flatness, and use the finite-diagonal row filtration of F2 to get an acyclic total. This proves the assertion on either side, without exchanging right and left modules.

At fixed $(i,j)$ the resolution complexes $H_i(P_{\ast ,a})$ and $H_j(Q_{\ast ,b})$ are projective resolutions of $H_{i}C$ and $H_{j}D$. The remaining $d_1$ on the preceding horizontal homology is their tensor-resolution differential, with the harmless constant total sign for fixed $q$. F5 identifies its degree-$p$ homology with ${\mathrm{Tor}}_p^R(H_{i}C,H_{j}D)$. Taking the finite sum over $i+j=q$ gives exactly the displayed $E^2$.

If $C$ is flat degreewise, use the two quasi-isomorphisms $\mathrm{Tot}P\otimes \mathrm{Tot}Q\to C\otimes \mathrm{Tot}Q\to C\otimes D$; the first uses flatness of $\mathrm{Tot}Q$ and the second of $C$. If $D$ is flat instead, use $\mathrm{Tot}P\otimes \mathrm{Tot}Q\to \mathrm{Tot}P\otimes D\to C\otimes D$. Thus the target in F2 is the stated ordinary tensor homology. In degree $n$ the filtration has endpoints $-1$ and $n-c-d$, hence is finite and strongly convergent after translation. For a chain map $f:C\to C^{\mathrm{\prime }}$, pass to the opposite module categories: $f^{\mathrm{o}\mathrm{p}}:(C^{\mathrm{\prime }}{)}^{\mathrm{o}\mathrm{p}}\to C^{\mathrm{o}\mathrm{p}}$. The projective Cartan–Eilenberg resolutions become injective ones there, so F6 gives a comparison $(P^{\mathrm{\prime }}{)}^{\mathrm{o}\mathrm{p}}\to P^{\mathrm{o}\mathrm{p}}$ over $f^{\mathrm{o}\mathrm{p}}$, unique up to its stated vertical homotopy and compatible with both filtrations. Reversing arrows returns the required projective comparison $P\to P^{\mathrm{\prime }}$ over $f$; the same applies to $Q\to Q^{\mathrm{\prime }}$. Tensoring these maps and their homotopies gives the natural $E^2$ and target maps under DC or the corresponding supplied comparison data. Zero complexes, a zero homology summand and a one-degree complex satisfy the same calculation.