Bounded two-sided projective bimodule complexes act on C_m

Every tensor term is finite graded projective. Since $X\in C_m$ every term $X^q$ is a finite graded projective left $A_m$-module, and by hypothesis every $R^p$ is finite graded projective as a left $A_m$-module, so the projective-preservation clause of [L2] applied to the bimodule $R^p$ gives that $R^p{\otimes }_{A_m}{X}^q$ is a finite graded projective left $A_m$-module.

$R{\otimes }_{A_m}X$ lies in $C_m$. By [L3] the totalization is a bounded complex of graded left $A_m$-modules with terms the finite direct sums ${⨁}_{p+q=n}{R}^p{\otimes }_{A_m}{X}^q$; each summand is finite graded projective by step 1.1, and by [L1] a finite direct sum of modules that are degree-zero direct summands of finite direct sums of shifts of $A_m$ is again such a direct summand, hence finite graded projective, so every term of the totalization is a finite graded projective left $A_m$-module and the totalization is an object of $C_m$.

Functoriality and additivity. By [L3] a pair of chain maps acts on the totalization by $f\otimes g$ on the tensor factors, preserving the differentials and the identities and compositions; for fixed $R^{\bullet }$ this makes $X\mapsto R^{\bullet }{\otimes }_{A_m}X$ a functor on complexes. It is additive because on elementary tensors $(\mathrm{i}\mathrm{d}\otimes (f+g))(r\otimes x)=r\otimes (f+g)(x)=r\otimes fx+r\otimes gx=(\mathrm{i}\mathrm{d}\otimes f+\mathrm{i}\mathrm{d}\otimes g)(r\otimes x)$ and sums of elementary tensors span the total object, so the induced maps on morphism groups are group homomorphisms.

Commutation with the homological shift. For $X\in C_m$ define ${\xi }_X$ on the summand $R^p{\otimes }_{A_m}(X[1]{)}^q=R^p{\otimes }_{A_m}{X}^{q+1}$ of $(R{\otimes }_{A_m}X[1]{)}^n$ by ${\xi }_X(r\otimes x):=(-1{)}^{p}r\otimes x$, regarded as an element of $((R{\otimes }_{A_m}X)[1]{)}^n=(R{\otimes }_{A_m}X{)}^{n+1}$. Then ${\xi }_X$ is an isomorphism of graded left $A_m$-modules with inverse given by the same formula, and it is a chain map: on one hand ${\xi }_{X}d(r\otimes x)={\xi }_X(d_{R}r\otimes x+(-1{)}^{p}r\otimes (-d_{X}x))=(-1{)}^{p+1}{d}_{R}r\otimes x-r\otimes d_{X}x$, and on the other $d{\xi }_X(r\otimes x)=(-1{)}^{p}d(r\otimes x)=(-1{)}^p(d_{R}r\otimes x+(-1{)}^{p}r\otimes d_{X}x)=(-1{)}^p{d}_{R}r\otimes x+r\otimes d_{X}x$, the two expressions differing by the overall sign $-1$ that the shift of a complex carries by [L4]; hence ${\xi }_X{d}_{X[1]}=d_{(R{\otimes }_{A_m}X)[1]}{\xi }_X$ and ${\xi }_X$ is a degree-zero isomorphism of complexes, natural in $X$ because it acts on the tensor factors by the identity up to the fixed sign $(-1{)}^p$ and is therefore compatible with postcomposition by any chain map.

Commutation with cones. Let $f:X\to Y$ be a chain map in $C_m$. Identify the underlying graded groups of $R^{\bullet }{\otimes }_{A_m}\mathrm{Cone}(f)$ and $\mathrm{Cone}(R^{\bullet }{\otimes }_{A_m}f)$: in degree $n$ both are the direct sum of the groups $R^p{\otimes }_{A_m}{Y}^q$ with $p+q=n$ and the groups $R^p{\otimes }_{A_m}{X}^{q+1}$ with $p+q=n$, the former lying in the $Y$-part and the latter in the shifted $X$-part. Define $\theta$ to be the identity on the $Y$-parts and $(-1{)}^p$ times the identity on the $X$-parts. Then for $r\otimes y$ in a $Y$-part, $\theta d(r\otimes y)=d_{R}r\otimes y+(-1{)}^{p}r\otimes d_{Y}y=d\theta (r\otimes y)$ since a $Y$-part receives no contribution from the cone differential; and for $r\otimes x$ in an $X$-part, $d(r\otimes x)=d_{R}r\otimes x+(-1{)}^{p}r\otimes f(x)+(-1{)}^{p+1}r\otimes d_{X}x$, so $\theta d(r\otimes x)=(-1{)}^{p+1}{d}_{R}r\otimes x+(-1{)}^{p}r\otimes f(x)+(-1{)}^{2p+1}r\otimes d_{X}x$ and $d\theta (r\otimes x)=d((-1{)}^{p}r\otimes x)=(-1{)}^p(r\otimes f(x)-{\mathrm{\partial }}_{R\otimes X}(r\otimes x))$ by the cone formula of [L4], whose terms are $(-1{)}^{p}r\otimes f(x)$ and $-(-1{)}^p(d_{R}r\otimes x+(-1{)}^{p}r\otimes d_{X}x)$, and these agree with the previous display. Hence $\theta$ is an isomorphism of complexes $R^{\bullet }{\otimes }_{A_m}\mathrm{Cone}(f)\to \mathrm{Cone}(R^{\bullet }{\otimes }_{A_m}f)$ over the identities of the tensor factors.

Descent to homotopy classes. If $f,g:X\to Y$ are homotopic chain maps with $f-g=d_{Y}h+h{d}_X$, then by the homotopy clause of [L3] the induced maps satisfy $\mathrm{i}\mathrm{d}\otimes f-\mathrm{i}\mathrm{d}\otimes g=dH+Hd$ with $H=(-1{)}^p\mathrm{i}\mathrm{d}\otimes h$, so they are homotopic and define the same morphism of $C_m$; hence $R^{\bullet }{\otimes }_{A_m}-$ is a well-defined functor $C_m\to C_m$.

Cone triangles are sent to distinguished triangles. In $C_m$ the distinguished triangles are the cone triangles $X\stackrel{f}{\to }Y\to \mathrm{Cone}(f)\to X[1]$ and their isomorphic images by [L5]. By step 3.3 the functor carries the middle and third terms of the cone triangle on $f$ to the corresponding terms of the cone triangle on $R^{\bullet }{\otimes }_{A_m}f$, and by step 3.2 it carries the connecting morphism, which is induced by the projection of the cone onto the shift of $X$, to the connecting morphism induced by the projection of $\mathrm{Cone}(R^{\bullet }{\otimes }_{A_m}f)$ onto $(R^{\bullet }{\otimes }_{A_m}X)[1]$ composed with ${\xi }_X$; hence the image triangle is isomorphic, as a triangle, to the distinguished cone triangle on $R^{\bullet }{\otimes }_{A_m}f$ and is therefore distinguished by the first axiom of [L5].

The functor is additive. Additivity on homotopy classes follows from additivity on chain maps by step 3.1, because the homotopy class of a sum is the sum of the homotopy classes; the construction fixes the zero object and finite direct sums termwise, so $R^{\bullet }{\otimes }_{A_m}-$ is an additive functor in the sense required by [L8].

Flatness and quasi-isomorphism preservation. Every $R^p$ is a finitely generated graded projective right $A_m$-module by hypothesis, hence flat as a right $A_m$-module by [F7], so $R^{\bullet }$ is a bounded, in particular bounded-above, complex of flat right $A_m$-modules; by [L6] tensoring it with a bounded-above acyclic complex of left $A_m$-modules gives an acyclic total complex, and applying this to the cone of a quasi-isomorphism between bounded complexes of left $A_m$-modules shows that $R^{\bullet }{\otimes }_{A_m}-$ sends quasi-isomorphisms between such complexes to quasi-isomorphisms.

Exactness. The functor $R^{\bullet }{\otimes }_{A_m}-$ is additive by step 5.1, carries the shift by the natural isomorphism $\xi$ of step 3.2 and carries distinguished triangles to distinguished triangles by step 4.2, so $(R^{\bullet }{\otimes }_{A_m}-,\xi )$ satisfies the definition of an exact functor between triangulated categories of [L8].

Agreement with the derived tensor product. For $X\in C_m$ the complex $X$ is a bounded complex of graded projective left $A_m$-modules, hence already a projective replacement of itself via the identity, and $R^{\bullet }$ is by the previous step a bounded complex of flat right $A_m$-modules and, being termwise projective on the right by hypothesis, already a projective replacement of itself via the identity; so the representatives supplied in the definition of the derived tensor product [L9] may be taken to be $R^{\bullet }$ and $X$ themselves, and $\mathrm{Tot}(R^{\bullet }{\otimes }_{A_m}X)=R^{\bullet }{\otimes }_{A_m}X$ by [L3]. Consequently the class of $R^{\bullet }{\otimes }_{A_m}X$ in $D^b(A_m\text{-mod})$ is the derived tensor product $R^{\bullet }{\otimes }_{A_m}^{\mathbf{L}}X$, and by step 5.2 it depends on $X$ only through its quasi-isomorphism class, so $\mathrm{\Theta }(R^{\bullet }{\otimes }_{A_m}X)$ represents that derived tensor product.

Conclusion. For every $X\in C_m$ the signed totalization $R^{\bullet }{\otimes }_{A_m}X$ is an object of $C_m$ by step 2.1, the assignment is an additive functor on homotopy classes by steps 3.1 and 4.1 with the shift isomorphism $\xi$ of step 3.2, it sends distinguished triangles to distinguished triangles by step 4.2 and hence is an exact functor of triangulated categories by step 6.1, and it is identified with the derived tensor product by step 6.2, the identity replacements being available because $X$ is projective and $R^{\bullet }$ is termwise projective on each side. Left projectivity of the terms of $R^{\bullet }$ enters only through step 1.1 and right projectivity only through steps 5.2 and 6.2, the two shifts and the cone formula are those of $C_m$ fixed in [L4], all replacements and signs are finite and explicit, and no Axiom of Choice, no dependent choice and no enough-projectives hypothesis is used.