Bounded Bimodule Complexes and Derived Tensor

For $a\in A$, right $A$-linearity of $d_F$ and left $A$-linearity of $d_G$ give $d((fa)\otimes g)=d_F(f)a\otimes g+(-1{)}^{p}fa\otimes d_G(g)=d_F(f)\otimes ag+(-1{)}^{p}f\otimes a{d}_G(g)=d(f\otimes ag)$; hence the differential descends to the balanced tensor, and additivity covers zero summands.

For $b\in B$ and $c\in C$, bimodule-linearity gives $d((bf)\otimes g)=bd(f\otimes g)$ and $d(f\otimes (gc))=d(f\otimes g)c$, so the outer actions commute with $d$; each differential preserves internal degree and the sign $(-1{)}^p$ depends only on cochain degree, while [F4] supplies the descended commuting outer actions.

Applying $d$ twice gives $d^2(f\otimes g)=d_F^2(f)\otimes g+((-1{)}^{p+1}+(-1{)}^p)d_F(f)\otimes d_G(g)+f\otimes d_G^2(g)=0$: the pure terms vanish by the complex identities and the mixed terms cancel, as in the ordinary calculation [F2]; the same formula covers zero differentials and zero summands.

If $F$ is supported in $[a,b]$ and $G$ in $[c,d]$, the total complex is supported in $[a+c,b+d]$ with finite diagonals; at the upper endpoint the differential has zero target and below the lower endpoint there is no preceding nonzero degree, so the totalization is bounded at both ends.

For internal-degree-zero bimodule chain maps $\varphi :F\to F^{\mathrm{\prime }}$ and $\psi :G\to G^{\mathrm{\prime }}$, their tensor is balanced and outer-linear, and $d(\varphi (f)\otimes \psi (g))=\varphi (d_{F}f)\otimes \psi (g)+(-1{)}^p\varphi (f)\otimes \psi (d_{G}g)=(\varphi \otimes \psi )d(f\otimes g)$ by the chain-map identities; identities and composition agree on elementary tensors and hence on the totalization.

Let $h:G\to G^{\mathrm{\prime }}$ be an internal-degree-zero bimodule homotopy of cochain degree $-1$ with ${\psi }_0-{\psi }_1=d_{G^{\mathrm{\prime }}}h+h{d}_G$; for $H(f\otimes g)=(-1{)}^{p}f\otimes h(g)$, the mixed terms in $dH+Hd$ have coefficients $(-1{)}^p$ and $(-1{)}^{p+1}$ and cancel, leaving $dH(f\otimes g)+Hd(f\otimes g)=f\otimes (d_{G^{\mathrm{\prime }}}h+h{d}_G)(g)=f\otimes ({\psi }_0-{\psi }_1)(g)$, so $H$ is a homotopy from $1_F\otimes {\psi }_0$ to $1_F\otimes {\psi }_1$ with the sign forced by [F1].

Let $k:F\to F^{\mathrm{\prime }}$ be an internal-degree-zero bimodule homotopy of cochain degree $-1$ with ${\varphi }_0-{\varphi }_1=d_{F^{\mathrm{\prime }}}k+k{d}_F$; for $K(f\otimes g)=k(f)\otimes g$, the mixed terms in $dK+Kd$ have coefficients $(-1{)}^{p-1}$ and $(-1{)}^p$ and cancel, leaving $dK(f\otimes g)+Kd(f\otimes g)=(d_{F^{\mathrm{\prime }}}k+k{d}_F)(f)\otimes g=({\varphi }_0-{\varphi }_1)(f)\otimes g$, so $K$ is a homotopy from ${\varphi }_0\otimes 1_G$ to ${\varphi }_1\otimes 1_G$.

Decompose ${\varphi }_0\otimes {\psi }_0-{\varphi }_1\otimes {\psi }_1=({\varphi }_0-{\varphi }_1)\otimes {\psi }_0+{\varphi }_1\otimes ({\psi }_0-{\psi }_1)$; postcomposing the homotopy in step 1.7 by $1_{F^{\mathrm{\prime }}}\otimes {\psi }_0$ handles the first summand and precomposing the homotopy in step 1.6 by ${\varphi }_1\otimes 1_G$ handles the second, so their sum proves well-definedness on homotopy classes in both variables. If an input is concentrated in one cochain degree the formulas reduce to one summand with no sign from its internal degree.

On a homogeneous triple tensor with cochain degrees $p,q,r$, the two parenthesizations have total degree $p+q+r$; the balanced associator sends $((f\otimes g)\otimes h)$ to $f\otimes (g\otimes h)$ and its inverse reverses this formula, while [L3] gives balanced well-definedness, internal degree zero, and compatibility with the outside $B$- and $E$-actions. Extending over the finite diagonals gives a degree-zero graded bimodule isomorphism in each cochain degree.

Applying [L1] to either parenthesization gives the coefficients $1,(-1{)}^p,(-1{)}^{p+q}$ on $d_F,d_G,d_H$. Thus the associator commutes with total differentials; its naturality follows from [L3] on each summand, so it is a natural chain isomorphism.

The maps $b\otimes f\mapsto bf$ and $f\otimes a\mapsto fa$ are the balanced unit isomorphisms of [L3], degree zero and outer-linear. The regular complexes have zero differential and lie in cochain degree zero, so the tensor differentials are respectively $1\otimes d_F$ and $d_F\otimes 1$; bimodule-linearity of $d_F$ makes both unit maps chain maps. Their canonical inverses and naturality are supplied by [L3].

On every pure tensor, either path around the associator pentagon sends the four factors to the same unparenthesized tensor. In each unit triangle, either path evaluates the unit factor by its module action and gives the same tensor. Pure tensors generate the balanced tensor products, so the coherence diagrams commute as chain maps; these formulas insert no sign because they change no cochain degree.

Reindexing the chain cone of [L4] by $n=-q$ gives $\mathrm{Cone}(u{)}^q=Y^q\oplus X^{q+1}$ and differential $(y,x)\mapsto (d_{Y}y+u^{q+1}x,-d_X^{q+1}x)$, with projection to $X[1{]}^q=X^{q+1}$. This fixes the inclusion and projection in the standard triangle [L5].

Distribute $F^p{\otimes }_A(Y^q\oplus X^{q+1})$ over its two summands. The right-variable comparison is the identity on the target summand and multiplication by $(-1{)}^p$ on the shifted-source summand; each component is balanced and bimodule-linear, and the inverse uses the same component formulas since $(-1{)}^{2p}=1$, making it an internal-degree-zero bimodule isomorphism in every total degree. These formulas commute with degree-zero maps in all inputs, so the comparison is natural.

For $f\in F^p$, $y\in Y^q$, and $x\in X^{q+1}$, the target component after the cone differential is $d_{F}f\otimes y+(-1{)}^{p}f\otimes d_{Y}y+(-1{)}^{p}f\otimes u(x)$, matching the target component obtained by first taking the tensor differential and then the comparison. On the shifted-source component the target cone differential is $-d_{F\otimes X}((-1{)}^{p}f\otimes x)=(-1{)}^{p+1}{d}_{F}f\otimes x-f\otimes d_{X}x$; applying the comparison after the source differential gives the same expression, since its sign on $d_{F}f$ is $(-1{)}^{p+1}$ and its sign on $(-1{)}^{p}f\otimes (-d_{X}x)$ is $-1$. Thus the comparison is a chain map.

The shift comparison $F{\otimes }_{A}X[1]\to (F{\otimes }_{A}X)[1]$ with value $(-1{)}^{p}f\otimes x$ is a chain isomorphism: the two total differentials agree because the shift negates $d_X$ on the first side and negates both terms of the total differential on the second. Under this comparison, the projection of $F\otimes \mathrm{Cone}(u)$ to $F\otimes X[1]$ equals the projection of $\mathrm{Cone}(1_F\otimes u)$ followed by the shift comparison, both sending $f\otimes x$ to $(-1{)}^{p}f\otimes x$; the inclusions of $F\otimes Y$ also agree. This identifies the entire standard cone triangles.

Write $\mathrm{Cone}(v{)}^p=F^{\mathrm{\prime }p}\oplus F^{p+1}$. After distributing the tensor with $X^q$, the left-variable comparison identifies these summands with $(F^{\mathrm{\prime }}{\otimes }_{A}X{)}^n$ and $(F{\otimes }_{A}X{)}^{n+1}$, respectively, for $n=p+q$, and is the identity on each part; it is natural since these components commute with all degree-zero maps.

On a target-part element $(f^{\mathrm{\prime }},f)\otimes x$, the target component of either differential is $d_{F^{\mathrm{\prime }}}{f}^{\mathrm{\prime }}\otimes x+(-1{)}^p{f}^{\mathrm{\prime }}\otimes d_{X}x+v(f)\otimes x$. On a shifted-source element $f\in F^{p+1}$ tensored with $x$, the source component on either side is $-d_{F}f\otimes x+(-1{)}^{p}f\otimes d_{X}x$, since the shifted total differential is $-d_{F\otimes X}$. Thus the identity comparison is a chain map, with inverse the identity on both summands.

The shift comparison $F[1]{\otimes }_{A}X\to (F{\otimes }_{A}X)[1]$ is identity on elementary tensors; for $f\in F^{p+1}$ its differential on either side is $-d_{F}f\otimes x+(-1{)}^{p}f\otimes d_{X}x$. The inclusion and projection commute with the identity-on-parts comparison, including projection to the shifted source under this shift comparison, so the entire standard triangle for $v$ tensors to that for $v{\otimes }_A{1}_X$.

The categories of graded bimodules used here are abelian: kernels and cokernels of degree-zero bimodule maps are computed in each internal degree and remain stable under both actions, and the canonical coimage-to-image map is an isomorphism degreewise. Applying [L6] makes their standard cone triangles distinguished in the homotopy categories; the comparisons already checked in both variables identify the image triangles with the respective standard cone triangles.

If $F$ is supported in $[a,b]$ and a bounded input $X$ is supported in $[c,d]$, then $(F{\otimes }_{A}X{)}^n={⨁}_{p+q=n}{F}^p{\otimes }_A{X}^q$ vanishes unless $a+c\le n\le b+d$, and each diagonal is finite. The signed differential is balanced and preserves the internal grading and outer $B$-action by [L1, L2]. Empty diagonals are zero; a zero factor gives the zero total complex. If $F$ is concentrated in degree $r$, its degree-$n$ term is $F^r{\otimes }_A{X}^{n-r}$ and the second-factor differential has sign $(-1{)}^r$; if $X$ is concentrated in degree $s$, its term is $F^{n-s}{\otimes }_A{X}^s$ with differential $d_F\otimes 1$.

Regard each $F^p$ as an ungraded right $A$-module. It is projective by hypothesis, hence flat by [L5]. Since $F$ is bounded, it is a bounded-above complex of right-flat modules.

For a degree-zero chain map $g:X\to Y$ of bounded graded left $A$-complexes, [L13] gives the chain map $1_F{\otimes }_{A}g$ and preserves identities and composition; [L12] shows it respects chain homotopy. Forgetting internal grading gives the same chain-map and homotopy formulas for ordinary module complexes. The totalization is additive on maps, so it defines additive functors on the graded and ungraded bounded homotopy categories.

Let $X$ have finite graded projective terms. For each $(p,q)$, [L3] with $M=F^p$ shows that $F^p{\otimes }_A{X}^q$ is finite graded projective over $B$. Each total degree is a finite direct sum of such terms, which is finite graded projective by [L4] after taking the direct sum of the finite shifted-free splittings. An empty diagonal is the zero module, a summand of the zero finite sum of shifts, and is finite graded projective. Thus $F{\otimes }_{A}X$ is bounded with finite graded projective terms, using the support bound of 1.1.

Applying [L6] to $F$ and any quasi-isomorphism of bounded left $A$-complexes proves that $F{\otimes }_A-$ preserves that quasi-isomorphism. In the graded case, the graded and ungraded balanced tensors impose the same relations $fa\otimes x=f\otimes ax$ on underlying elements, so forgetting internal grading identifies their underlying total complexes. A graded chain map that is a quasi-isomorphism is therefore an ungraded quasi-isomorphism, and its tensor remains one by [L6]; since the tensor differential preserves internal degree, vanishing of the underlying cohomology implies vanishing in each internal degree. This proves preservation of graded quasi-isomorphisms.

The full subcategory of bounded complexes with finite graded projective terms is closed under cochain shifts and mapping cones: shifts retain the same terms, and cone terms are finite direct sums of finite graded projectives. The cone-compatibility theorem supplies the natural shift isomorphism for $F{\otimes }_A-$ and identifies every cone triangle with the cone triangle of the tensored map. Together with additivity from 2.1 and closure from 2.2, the exact-functor criterion [L9] proves the functor is exact on $K^b({\mathrm{proj}}^{gr}A)\to K^b({\mathrm{proj}}^{gr}B)$.

By [L8], $D^b$ is the localization of the bounded homotopy category at quasi-isomorphisms. Since the functors in 2.3 send every inverted map to an isomorphism in the target localization, they induce functors on $D^b(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})\to D^b(B\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ and on the corresponding graded derived categories. The factorization is the localization property of the functor on the homotopy category, not a global choice of representatives.

View $F$ as a bounded-above right $A$-complex. The identity $F\to F$ is a supplied projective replacement, since every term is projective. Taking $P_N=F$ in [L7] represents $F{\otimes }_A^{\mathbf{L}}X$ by $\mathrm{Tot}(F{\otimes }_{A}X)$, and 2.3 proves directly that this value depends only on the derived object $X$. To verify the graded derived tensor model, let $Z$ be any acyclic graded left $A$-complex and fix a total degree $n$. If $F$ is supported in $[a,b]$, every component of a degree-$n$ element of $\mathrm{Tot}(F{\otimes }_{A}Z)$ has $Z$-degree at most $n-a$, its differential has $Z$-degree at most $n+1-a$, and every degree-$(n-1)$ potential boundary has $Z$-degree at most $n-1-a$. Choose $m\ge n+2-a$. Here ${\tau }_{\le m}Z$ denotes the good truncation equal to $Z^q$ for $q<m$, to $\ker (d_Z^m)$ in degree $m$, and to zero above $m$. It is bounded above and acyclic; the components and differentials just listed have $Z$-degrees at most $m-1$, so they are unchanged in $\mathrm{Tot}(F{\otimes }_A{\tau }_{\le m}Z)$. The map $0\to {\tau }_{\le m}Z$ is a quasi-isomorphism between bounded-above complexes, so [L6] makes the truncated total complex acyclic. The given cycle is thus a boundary there and in $\mathrm{Tot}(F{\otimes }_{A}Z)$. This proves $F$ is K-flat on graded modules and its signed tensor computes the graded derived tensor. No projective replacement of $X$ and no existence theorem for arbitrary projective resolutions is invoked; the conditional Axiom-of-Choice clause in [L7] is therefore not used.

The cone theorem of [L11] gives a natural shift comparison and sends each cone triangle to the corresponding cone triangle before localization. The localization triangulations are those of [L10], and the comparison descends along the localization in 3.2. Since the descended functors are additive by 2.1 and [L9] defines exactness by this shift comparison and triangle preservation, both derived functors are exact.

The bound in 1.1 proves the output is bounded. Step 2.2 proves finite graded projectivity for finite graded projective inputs; for general modules no finite-generation or finite-dimensionality conclusion is asserted. If both complexes are concentrated in degree zero, the formula reduces to module tensor; zero differentials and zero maps need no separate hypothesis, and endpoint degrees outside $[a+c,b+d]$ are zero. [step 1.1, step 2.2, step 2.3, step 4.1, step 3.3] $\mathrm{\square }$

For every bounded graded left $A$-complex $X$, define ${\eta }_X=u{\otimes }_A{1}_X:F{\otimes }_{A}X\to F^{\mathrm{\prime }}{\otimes }_{A}X$ and ${ϵ}_X=v{\otimes }_A{1}_X:F^{\mathrm{\prime }}{\otimes }_{A}X\to F{\otimes }_{A}X$. Since $u$ and $v$ are degree-zero bimodule chain maps, [L1] makes these balanced, internal-degree-zero $B$-linear chain maps.

Transfer $h$ to $H_X(f\otimes x)=h(f)\otimes x$. By [L2], $d{H}_X+H_{X}d=(d_{F}h+h{d}_F)\otimes 1_X=((vu-{\mathrm{id}}_F)\otimes 1_X)$, which is ${ϵ}_X{\eta }_X-\mathrm{id}$. Transferring $h^{\mathrm{\prime }}$ gives ${\eta }_X{ϵ}_X-\mathrm{id}=d{H}_X^{\mathrm{\prime }}+H_X^{\mathrm{\prime }}d$. These homotopies are natural in $X$, since for every $g:X\to Y$ both orders send $f\otimes x$ to $h(f)\otimes g(x)$, and likewise for $h^{\mathrm{\prime }}$. Thus $[\eta ]$ and $[ϵ]$ are mutually inverse natural isomorphisms in the homotopy categories.

For every degree-zero chain map $g:X\to Y$, the composites in the naturality square for $\eta$ both send $f\otimes x$ to $u(f)\otimes g(x)$; the same check with $v$ proves naturality of $ϵ$. The maps are well defined on homotopy classes by [L2], so these are natural transformations on the bounded homotopy categories.

If $X$ has finite graded projective terms, [L3] places both totalizations and both tensor maps in the stated bounded projective homotopy categories. Step 1.2 proves that their composites are the identity morphisms there, so $[\eta ]$ and $[ϵ]$ give inverse natural isomorphisms on $K^b({\mathrm{proj}}^{gr}A)$.

For bounded ordinary or graded module complexes, [L4] makes both totalization functors preserve quasi-isomorphisms and [L5] identifies their localizations with the derived tensor functors. The natural transformations of Step 2.1 therefore descend through the localizations, and the homotopies of Step 1.2 still make their composites identities. They are mutually inverse natural isomorphisms on both ordinary and graded bounded derived categories.

Empty diagonals and a zero input give zero total complexes, so the formulas remain valid there. For a one-term input, the same first-variable map and homotopy formulas apply; if either supplied homotopy is zero, its equation reduces to a strict inverse equation. Bounded endpoints add only zero components, and the homotopy formulas have no terms outside the given bounded supports. No map or representative is selected: all maps and homotopies are supplied in the hypotheses. The proposition is an implication, not an iff claim. [step 1.1, step 1.2, given, algebra] $\mathrm{\square }$

Apply [L5, L6] separately to $F$ and $G$. This defines the two tensor functors on the bounded homotopy categories of finite graded projectives and on ordinary and graded bounded derived categories; in each setting their values are represented by signed ordinary totalization.

For a bounded left $B$-complex $X$, define ${\eta }_X:F{\otimes }_A(G{\otimes }_{B}X)\to X$ by the inverse associator to $(F{\otimes }_{A}G){\otimes }_{B}X$, followed by $u{\otimes }_B{1}_X$ and the unit $B{\otimes }_{B}X\to X$. Define ${ϵ}_X:X\to F{\otimes }_A(G{\otimes }_{B}X)$ in reverse order using the inverse unit, $v{\otimes }_B{1}_X$, and the associator. By [L1, L2, L3] these are chain maps on the balanced total complexes.

For a bounded left $A$-complex $Y$, use the inverse associator to write $G{\otimes }_B(F{\otimes }_{A}Y)$ as $(G{\otimes }_{B}F){\otimes }_{A}Y$, then apply $u^{\mathrm{\prime }}{\otimes }_A{1}_Y$ and the unit $A{\otimes }_{A}Y\to Y$. The reverse natural map uses the inverse unit, $v^{\mathrm{\prime }}{\otimes }_A{1}_Y$, and the associator. These are chain maps by [L1, L2, L3].

If $h$ is a supplied homotopy between $vu$ and $1_H$, [L4] gives the homotopy $K_X(z\otimes x)=h(z)\otimes x$ after tensoring with $X$; the homotopy between $uv$ and $1_B$ transfers in the same way. Conjugating these homotopies by the associator and unit maps shows that ${ϵ}_X{\eta }_X$ and ${\eta }_X{ϵ}_X$ are homotopic to the respective identity maps. Thus the two maps are inverse in the homotopy category.

For a chain map $g:X\to X^{\mathrm{\prime }}$, the naturality square for $u\otimes 1$ commutes on each elementary tensor, since both routes send $z\otimes x$ to $u(z)\otimes g(x)$; the same holds for $v\otimes 1$. The associator and units are natural by [L1, L2], and the transferred homotopies are natural because $h(z)\otimes g(x)$ is independent of the order of applying $g$ and the homotopy. Hence $\eta$ and $ϵ$ are inverse natural isomorphisms on the bounded homotopy category of left $B$-complexes.

Transfer the supplied homotopies between $v^{\mathrm{\prime }}{u}^{\mathrm{\prime }}$ and $1_J$, and between $u^{\mathrm{\prime }}{v}^{\mathrm{\prime }}$ and $1_A$, by [L4]. They show that the maps of Step 1.3 are mutually inverse in the homotopy category. Naturality follows on elementary tensors exactly as in Step 3.1, so the two maps give inverse natural isomorphisms for the $G{\otimes }_B-$ and $F{\otimes }_A-$ composite on bounded left $A$-complexes.

By [L5], $F{\otimes }_A-$ and $G{\otimes }_B-$ restrict to the indicated bounded homotopy categories of finite graded projectives. Steps 2.1–4.1 give inverse natural isomorphisms there. Each functor is exact by [L5], so these are exact equivalences. The adjacent proposition [L7] assumes two-sided projectivity for both complexes being compared; that has not been included for $H$ or $J$, so it is not applied to them. No projectivity of $H$ or $J$ is needed because [L4] transfers the supplied homotopies directly in the first variable.

By [L6], each tensor functor preserves quasi-isomorphisms and its derived functor is represented by ordinary signed totalization. The natural transformations from Steps 3.1 and 4.1 commute with every quasi-isomorphism. After localization, the vertical maps in each such naturality square are invertible; the same square therefore commutes for the inverse of a quasi-isomorphism and hence for every morphism generated in the localization. The transformations descend, and their inverse identities from Steps 2.1 and 4.1 remain identities there. The two derived tensor functors are thus quasi-inverse exact equivalences in both ordinary and graded settings.

Empty or zero complexes give zero totalizations, where the maps and homotopies still satisfy the identity equations. One-term complexes and zero-differential complexes use the same formulas; bounded endpoints add only zero summands. All inverse maps and homotopies are supplied in the hypotheses, and the derived models use the supplied $F$ and $G$, so no family of choices or Axiom of Choice is used. The theorem is an implication and proves no biconditional. It gives inverse functors for the specified pair but proves no coherence for comparison isomorphisms indexed by a group. [step 1.2, step 2.1, step 1.3, step 4.1, step 5.1, step 5.2, given, algebra] $\mathrm{\square }$