A bounded two-sided projective bimodule complex defines exact derived tensor functors

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] $\mathrm{\square }$