Graded Grothendieck groups, shift action, and Cartan map

By [F4], ${\mathrm{GrMod}}_0(A)$ is abelian and its degreewise kernels, cokernels, and finite biproducts have the stated graded structures. For a map between finite-dimensional modules, its kernel is a subspace of the finite-dimensional domain and its cokernel is a quotient of the finite-dimensional codomain; [F8] gives finite-dimensionality of kernels and images. Thus the full subcategory ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}^{\mathrm{g}\mathrm{r}}(A)$ is closed under kernels, cokernels, images, and finite biproducts. The ambient coimage and image of any map therefore remain in this full subcategory, as does their canonical isomorphism; hence [F10] makes it abelian. It is essentially small: each object has finite support and a finite homogeneous basis; for each finite-support dimension vector, the graded $k$-space with those dimensions has a fixed standard model, and its $A$-actions are a set of families of linear maps satisfying the algebra-action identities. Every object is isomorphic to one of these set-many models. This objectwise coordinate argument selects no basis simultaneously from an arbitrary family. Hence [F1] defines $G_0^{\mathrm{g}\mathrm{r}}(A)$.

Every finite-dimensional graded module has a finite homogeneous basis and so is finitely generated. Conversely, a finite graded projective has finitely many homogeneous generators by [F6]; sending the homogeneous generator of each matching shift $A\{r\}$ to a generator of $P$ gives a degree-zero surjection from a finite sum of shifts of $A$, so $P$ is finite-dimensional. The category ${\mathcal{P}}_{\mathrm{f}\mathrm{d}}^{\mathrm{g}\mathrm{r}}(A)$ is essentially small as a full subcategory of ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}^{\mathrm{g}\mathrm{r}}(A)$. It inherits preadditive hom groups from that abelian category. By [F5], the zero module and a finite direct sum of finite graded projectives are again finite graded projective; their module biproducts therefore make this an additive category by [F9]. Thus [F2] defines $K_0^{\mathrm{g}\mathrm{r}}(A)$.

Write $A^{\mathrm{♭}}$ for the underlying ungraded algebra given the trivial grading, and regard an ungraded module as an $A^{\mathrm{♭}}$-module concentrated in degree zero. This identifies the ungraded module category with a full subcategory of ${\mathrm{GrMod}}_0(A^{\mathrm{♭}})$ that is closed under kernels, cokernels, images, and finite biproducts by [F4]. Its finite-dimensional subcategory remains closed because kernels and images are subspaces and cokernels are quotients of finite-dimensional vector spaces. The ambient coimage-to-image isomorphisms remain in this full subcategory, so it is abelian by [F10]; it is essentially small by the same finite-dimensional action-matrix argument as in step 1.1. Therefore [F1] defines $G_0(A)$.

For each $r\in \mathbb{Z}$, shift sends a finite-dimensional graded module to a finite-dimensional graded module and leaves degree-zero maps degree-zero. It is exact because degreewise kernels and cokernels in [F4] are merely reindexed, and its inverse is shift by $-r$ by [F3]. Thus shift is an exact autoequivalence of ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}^{\mathrm{g}\mathrm{r}}(A)$, and [F7] induces an automorphism ${\sigma }_r$ of $G_0^{\mathrm{g}\mathrm{r}}(A)$ with ${\sigma }_r([M])=[M\{r\}]$.

If $P$ is finite graded projective, [F5] exhibits it as a summand of a finite direct sum of shifts $A\{s\}$. Shifting this splitting by $r$ exhibits $P\{r\}$ as a summand of a finite direct sum of shifts $A\{s+r\}$, so [F5] shows $P\{r\}$ is finite graded projective. The shift functor and its inverse are additive on this category; [F7] therefore induces inverse automorphisms ${\tau }_r$ and ${\tau }_{-r}$ on $K_0^{\mathrm{g}\mathrm{r}}(A)$, with ${\tau }_r([P])=[P\{r\}]$.

The class function $[P]\mapsto [P]$ from $\mathrm{Iso}({\mathcal{P}}_{\mathrm{f}\mathrm{d}}^{\mathrm{g}\mathrm{r}}(A))$ to $G_0^{\mathrm{g}\mathrm{r}}(A)$ is additive: the split sequence $0\to P\to P\oplus Q\to Q\to 0$ gives $[P\oplus Q]=[P]+[Q]$ by [F1]. Hence [F7] gives a unique homomorphism $c_A^{\mathrm{g}\mathrm{r}}$ with $c_A^{\mathrm{g}\mathrm{r}}([P])=[P]$.

The identities ${\sigma }_0={\tau }_0=1$, ${\sigma }_r{\sigma }_s={\sigma }_{r+s}$, and ${\tau }_r{\tau }_s={\tau }_{r+s}$ follow on generators from $(M\{s\})\{r\}=M\{r+s\}$ and $(P\{s\})\{r\}=P\{r+s\}$ in [F3]. Therefore $v^r\cdot x:={\sigma }_r(x)$ on $G_0^{\mathrm{g}\mathrm{r}}(A)$ and $v^r\cdot y:={\tau }_r(y)$ on $K_0^{\mathrm{g}\mathrm{r}}(A)$ extend by finite integer linear combinations to $\mathbb{Z}[v,v^{-1}]$-module structures.

On every projective-class generator, $c_A^{\mathrm{g}\mathrm{r}}(v[P])=c_A^{\mathrm{g}\mathrm{r}}([P\{1\}])=[P\{1\}]=v[P]=v{c}_A^{\mathrm{g}\mathrm{r}}([P])$ by [F3]. Since these generators generate $K_0^{\mathrm{g}\mathrm{r}}(A)$ as an abelian group and $v$ is invertible, $c_A^{\mathrm{g}\mathrm{r}}$ commutes with every Laurent polynomial action and is $\mathbb{Z}[v,v^{-1}]$-linear.

The ungraded class function $[P]\mapsto [P]$ from finite-dimensional projective left $A$-modules to $G_0(A)$ is additive by the split short exact sequence $0\to P\to P\oplus Q\to Q\to 0$ in ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}(A)$. By [F7] it factors uniquely through the split group $K_0(A)$, giving $c_A([P])=[P]$.