Graded derived tensor equivalences induce Laurent-linear K0 and G0 maps

Since $B$ is finite dimensional over $k$, a graded left $B$-module generated by finitely many homogeneous elements is finite dimensional over $k$: the finitely many generators together with the finite-dimensional algebra act in only finitely many degrees and span a finite-dimensional space. Hence each term $F^p$, being finite graded projective as a left $B$-module [F2], is finite dimensional over $k$, and likewise each $G^q$. For a finite-dimensional graded left $A$-module $M$, each tensor $F^p{\otimes }_{A}M$ is a quotient of the finite-dimensional $k$-space $F^p{\otimes }_{k}M$ and is therefore finite dimensional. Consequently $F{\otimes }_A-$ carries bounded complexes of finite-dimensional graded left $A$-modules to bounded complexes of finite-dimensional graded left $B$-modules, and $G{\otimes }_B-$ does the same in the other direction.

The same published equivalence restricts on the projective side: by [F3] the tensor functors are mutually quasi-inverse exact equivalences between $K^b({\mathrm{proj}}^{gr}A)$ and $K^b({\mathrm{proj}}^{gr}B)$. The canonical functor $K^b({\mathrm{proj}}^{gr}A)\to D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}^{\mathrm{g}\mathrm{r}}(A)$ is fully faithful by the no-roof clause of [F6] and essentially surjective by the definition of graded perfectness, hence an equivalence of triangulated categories; combined with the graded clause of [F5] it identifies $K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(K^b({\mathrm{proj}}^{gr}A))$ with $K_0^{\mathrm{g}\mathrm{r}}(A)$, and similarly for $B$.

By [F2] both tensor functors preserve quasi-isomorphisms between bounded complexes, and by step 1.1 they preserve the full subcategories of bounded complexes of finite-dimensional modules in the graded and in the ungraded settings; hence they descend to exact functors $D^b({\mathcal{C}}_A^{\mathrm{g}\mathrm{r}})\to D^b({\mathcal{C}}_B^{\mathrm{g}\mathrm{r}})$ and back, where ${\mathcal{C}}^{\mathrm{g}\mathrm{r}}$ denotes finite-dimensional graded modules with degree-zero maps, and likewise ungraded. The chain-level unit and counit assembled in [F3] from the supplied bimodule maps, associators and unit maps are quasi-isomorphisms between bounded complexes of finite-dimensional modules when evaluated there, and the supplied homotopies show that their composites are homotopic to the identities; hence these descended functors are mutually quasi-inverse exact equivalences.

Applying [F5, F7] to the equivalence of step 2.1 gives mutually inverse isomorphisms $\bar{F}:G_0^{\mathrm{g}\mathrm{r}}(A)\to G_0^{\mathrm{g}\mathrm{r}}(B)$ and $\bar{G}:G_0^{\mathrm{g}\mathrm{r}}(B)\to G_0^{\mathrm{g}\mathrm{r}}(A)$: the abelian comparison identifies each graded $G_0^{\mathrm{g}\mathrm{r}}$ with the triangle group of the bounded derived category of finite-dimensional graded modules, the exact equivalence induces an isomorphism by [F7], and the two composite identifications are inverse because the functors are quasi-inverse. Likewise, by [F3, F5, F6, F7] and step 1.2, the projective-side equivalence induces mutually inverse isomorphisms ${\bar{F}}_K:K_0^{\mathrm{g}\mathrm{r}}(A)\to K_0^{\mathrm{g}\mathrm{r}}(B)$ and ${\bar{G}}_K$ in the other direction.

The maps of step 3.1 are $\mathbb{Z}[v,v^{-1}]$-linear. The degree-zero natural isomorphism $F{\otimes }_A(M\{r\})\cong (F{\otimes }_{A}M)\{r\}$ of [F4] exhibits the tensor functors as commuting with the internal-shift functors up to natural isomorphism, in both variables and for $G$ as well; since a natural isomorphism of exact functors induces the same map on triangle Grothendieck groups [F7], the induced maps on $K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}$ intertwine the maps induced by internal shift, and the comparisons of [F5] identify the latter with multiplication by $v$ in the sense of [F1]. The projective-side maps intertwine $[P]\mapsto [P\{r\}]$ in the same way, because $F{\otimes }_A(P\{r\})\cong (F{\otimes }_{A}P)\{r\}$ is an isomorphism of bounded complexes of finite graded projectives and therefore already an isomorphism in $K^b({\mathrm{proj}}^{gr})$. Extending by additivity gives $\bar{F}(vx)=v\bar{F}(x)$ for all $x$ and the analogous identity for $\bar{G}$.

By [F7] any supplied natural-isomorphism relation between composites of such exact shift-compatible functors gives equal induced maps on the triangle groups, hence by the identifications of step 3.1 equal maps $\bar{F},\bar{G}$ on $G_0^{\mathrm{g}\mathrm{r}}$ and on $K_0^{\mathrm{g}\mathrm{r}}$ and equal composites in the reverse direction; these are equalities of group homomorphisms only, and no coherent comparison isomorphisms between the underlying functors are produced. In the bases $\{[P_i]\}$ of $K_0^{\mathrm{g}\mathrm{r}}$ and $\{[S_i]\}$ of $G_0^{\mathrm{g}\mathrm{r}}$ from [F8], both groups are free $\mathbb{Z}[v,v^{-1}]$-modules, so the mutually inverse $\mathbb{Z}[v,v^{-1}]$-linear maps of steps 3.1 and 4.1 are represented by inverse matrices over $\mathbb{Z}[v,v^{-1}]$. This proves the stated mutual inversion, Laurent linearity, matrix and naturality assertions.