Exact adjoints induce adjoint operators on Grothendieck groups

The finite-dimensional left-module categories are essentially small abelian categories. In the ungraded case, [F5] makes the ambient module category abelian; kernels, cokernels, and finite biproducts of finite-dimensional modules remain finite-dimensional, so the full finite-dimensional subcategory is abelian by [F7]. For graded modules, [F6] gives the same conclusion degreewise. These categories are essentially small: every ungraded $n$-dimensional module is isomorphic to one on the standard vector space $k^n$, and its possible actions form a set of functions satisfying the module identities. Every finite-dimensional graded module is similarly isomorphic to a standard graded vector space specified by a finite-support dimension vector $\mathbb{Z}\to \mathbb{N}$; the possible graded actions on it form a set. Each such model uses bases only for one finite-dimensional object at a time. Morphisms are subsets of the set of linear maps between the underlying finite vector spaces, so the categories are locally small by [F4, F8, F12]. Their full subcategories of finite-dimensional projectives are essentially small and additive: zero objects and finite direct sums remain projective by the lifting properties in [F9, F10], and finite generation is preserved by taking the union of the finite generating families. Thus the universal group maps in [F2] apply to these categories.

For fixed $P,N$, the adjunction bijection [F3] is $\mathrm{\Phi }(u)=G(u)\circ {\eta }_P$. Since $G$ is $k$-linear by [F4] and composition is $k$-bilinear, $\mathrm{\Phi }$ preserves addition and scalar multiplication: $\mathrm{\Phi }(u+u^{\mathrm{\prime }})=\mathrm{\Phi }(u)+\mathrm{\Phi }(u^{\mathrm{\prime }})$ and $\mathrm{\Phi }(\lambda u)=\lambda \mathrm{\Phi }(u)$. Thus the set bijection [F3] is a $k$-linear isomorphism. Both Hom spaces are finite-dimensional because they are subspaces of the finite-dimensional linear-map spaces from [F12]. Consequently [F13] gives ${\dim }_k{\mathrm{Hom}}_B(FP,N)={\dim }_k{\mathrm{Hom}}_A(P,GN)$.

Exactness gives the induced maps on the two $G_0$ groups by [F2]. The functor $F$ is additive by [F9], and its projective-preservation hypothesis restricts it to an additive functor from finite projectives over $A$ to those over $B$; [F2] therefore gives $F_{\ast }$ on split $K_0$. The same reasoning applies in the graded categories. For every $r$, the natural shift isomorphisms identify $F(M\{r\})$ with $F(M)\{r\}$ and $G(N\{r\})$ with $G(N)\{r\}$, so their induced maps commute with multiplication by $v^r$. Hence all four induced maps are $\mathbb{Z}[v,v^{-1}]$-linear where applicable.

For every $d\in \mathbb{Z}$, a degree-$d$ map $F(P)\to N$ is the same underlying map as a degree-zero map $F(P)\{d\}\to N$, by [F11]. Use the natural shift isomorphism ${\alpha }_{P,d}:F(P\{d\})\stackrel{\cong }{\to }F(P)\{d\}$ and precompose with ${\alpha }_{P,d}$ to obtain a degree-zero map $F(P\{d\})\to N$. Apply the degree-zero adjunction bijection to the object $P\{d\}$; its output is a degree-zero map $P\{d\}\to G(N)$, which is exactly a degree-$d$ map $P\to G(N)$ under the same shift convention [F11]. Each operation is a $k$-linear bijection, so ${\mathrm{Hom}}_{B,d}(F(P),N)\cong {\mathrm{Hom}}_{A,d}(P,G(N))$ linearly. This constructs the degree-compatible graded transposition from the ordinary adjunction rather than replacing homogeneous Hom by all ungraded maps.

By [F1], the two sides of this dimension equality are respectively $⟨F_{\ast }[P],[N]{⟩}_B$ and $⟨[P],G_{\ast }[N]{⟩}_A$. This proves the ungraded identity on projective and module class generators. Every element of either Grothendieck group is a finite integer linear combination of such classes, so the $\mathbb{Z}$-bilinearity in [F1] extends the equality to all classes in the ungraded groups.

Taking dimensions in step 2.2 gives equality of every coefficient in $h_{\mathrm{g}\mathrm{r},B}(F(P),N)$ and $h_{\mathrm{g}\mathrm{r},A}(P,G(N))$. The sums are finite by [F1]. Thus the graded pairing identity holds on projective and module class generators, and the $\mathbb{Z}$-bilinearity in [F1] extends it to all $x\in K_0^{\mathrm{g}\mathrm{r}}(A)$ and $y\in G_0^{\mathrm{g}\mathrm{r}}(B)$. Step 2.1 gives Laurent-linearity of the induced operators.

In the tensor–Hom setting of [F14], the functors $F=M{\otimes }_A-$ and $G={\mathrm{HOM}}_B(M,-)$ have the required degree-zero adjunction interface. This theorem applies to that instance only after exactness on the finite categories, finite-dimensionality of outputs, projective preservation by $F$, and the stated shift conditions have each been verified.