Projective Hom pairing descends and is graded sesquilinear

Put ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}(A)$ and ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}^{\mathrm{g}\mathrm{r}}(A)$ for the full subcategories of finite-dimensional modules in $A\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ and ${\mathrm{GrMod}}_0(A)$. The ambient categories are abelian by [F2] and [F5]. The finite full subcategories inherit preadditive Hom groups and composition from their ambient module categories. They are closed under kernels and cokernels: in the ungraded case these are a subspace of a finite-dimensional domain and a quotient of a finite-dimensional codomain; in the graded case [F5] computes them degreewise and their underlying spaces remain finite-dimensional. They also inherit finite biproducts, so they are additive by [F20]. For each map, its coimage and image remain in the finite subcategory because they are built from these kernels and cokernels; the ambient coimage-to-image isomorphism and its inverse are therefore morphisms in the full subcategory. By [F19], both finite subcategories are abelian. They are essentially small: for each ungraded dimension $n$, structures on the carrier $k^n$ form a subset of the set of functions $A\times k^n\to k^n$ satisfying the module identities; for graded modules, finite-support dimension vectors $\mathbf{n}:\mathbb{Z}\to \mathbb{N}$ with finite sum form a set, and the possible graded actions on ${⨁}_d{k}^{\mathbf{n}(d)}$ form a set of families of maps satisfying the action identities. Every object is isomorphic to one of these models by a finite (homogeneous) basis. These objectwise coordinate models prove essential smallness without simultaneous basis choices. The full projective subcategories inherit essential smallness.

Let $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be short exact in ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}(A)$. Centrality [F15] makes each $A$-linear map $k$-linear; sums and scalar multiples preserve $A$-linearity, so ${\mathrm{Hom}}_A(P,M)$ is a $k$-subspace of the full linear-map space [F16], hence finite-dimensional by [F17]. Since $P$ is projective by [F3], exactness of ${\mathrm{Hom}}_A(P,-)$ in the ambient abelian category [F4] gives $0\to {\mathrm{Hom}}_A(P,M^{\mathrm{\prime }})\to {\mathrm{Hom}}_A(P,M)\to {\mathrm{Hom}}_A(P,M^{\mathrm{\prime }\mathrm{\prime }})\to 0$. Its arrows are $k$-linear by [F15]. The last arrow is surjective and its kernel is the image of the first, isomorphic to ${\mathrm{Hom}}_A(P,M^{\mathrm{\prime }})$; rank-nullity [F14] gives $h(P,M)=h(P,M^{\mathrm{\prime }})+h(P,M^{\mathrm{\prime }\mathrm{\prime }})$, also when any of these spaces is zero.

For every integer $d$, shift by $d$ is an exact equivalence of ${\mathrm{GrMod}}_0(A)$: [F7] reindexes each homogeneous piece, so [F5] shows it preserves exact sequences, and its inverse is shift by $-d$. Thus if $q:E\twoheadrightarrow N$ is a degree-zero epimorphism, $q\{-d\}$ is an epimorphism. Given a degree-zero map $f:P\{d\}\to N$, shift it by $\{-d\}$ and lift the resulting map $P\to N\{-d\}$ through $q\{-d\}$ using the projectivity [F6]; shifting the lift back shows $P\{d\}$ is graded projective. It remains finite-dimensional because its underlying vector space is unchanged [F7].

The supports of finite-dimensional graded vector spaces $P$ and $M$ are finite: if a space of dimension $n$ had more than $n$ nonzero homogeneous components, choosing $n+1$ such components and one nonzero vector in each would contradict linear independence. If ${\mathrm{Hom}}_{A,d}(P,M)\mathrm{\ne }0$, a nonzero map has some element with nonzero image; decomposing it into homogeneous components shows some $P_i$ maps nontrivially into $M_{i+d}$. Hence $d\in \mathrm{supp}(M)-\mathrm{supp}(P)$, a finite set, and the sum defining $h_{\mathrm{g}\mathrm{r}}(P,M)$ in [F18] has finite support. If either module is zero, the sum is empty and equals zero. The argument makes only finite selections.

A function on the underlying modules is degree zero from $P\{d\}$ to $M$ exactly when it sends $P\{d{\}}_i=P_{i-d}$ into $M_i$ for every $i$; setting $j=i-d$ makes this precisely the degree-$d$ condition $P_j\to M_{j+d}$ [F8]. Therefore ${\mathrm{Hom}}_{A,d}(P,M)\cong {\mathrm{Hom}}_{A,0}(P\{d\},M)$. The degree-$d$ Hom space is a $k$-subspace of the full linear-map space, since its degree and $A$-linearity conditions are preserved by addition and scalar multiplication [F15, F16]; it is finite-dimensional by [F17].

A degree-$d$ map $P\{r\}\to M\{s\}$ is the same underlying $A$-linear function as a map $P\to M$ of degree $e=d+r-s$: from $P\{r{\}}_i=P_{i-r}$ its image lies in $M\{s{\}}_{i+d}=M_{i+d-s}$, which after $j=i-r$ is $M_{j+e}$. Thus ${\mathrm{Hom}}_{A,d}(P\{r\},M\{s\})\cong {\mathrm{Hom}}_{A,d+r-s}(P,M)$. Reindexing the finite sum from step 1.4 gives $h_{\mathrm{g}\mathrm{r}}(P\{r\},M\{s\})={\sum }_e{v}^{e-r+s}{\dim }_k{\mathrm{Hom}}_{A,e}(P,M)=v^{s-r}{h}_{\mathrm{g}\mathrm{r}}(P,M)$. The group actions [F12] identify these shifts with multiplication by $v^r$ and $v^s$, proving the displayed formula on generators.

Apply [F4] in ${\mathrm{GrMod}}_0(A)$ to the projective object $P\{d\}$ from step 1.3 and any short exact sequence of finite-dimensional graded modules. Step 2.1 identifies the resulting exact Hom sequence with the degree-$d$ Hom sequence, whose maps are $k$-linear by [F15] and whose spaces are finite-dimensional by step 2.1. Rank-nullity [F14] gives ${\dim }_k{\mathrm{Hom}}_{A,d}(P,M)={\dim }_k{\mathrm{Hom}}_{A,d}(P,M^{\mathrm{\prime }})+{\dim }_k{\mathrm{Hom}}_{A,d}(P,M^{\mathrm{\prime }\mathrm{\prime }})$. The three sums have finite support by step 1.4, so summing the coefficient identities gives $h_{\mathrm{g}\mathrm{r}}(P,M)=h_{\mathrm{g}\mathrm{r}}(P,M^{\mathrm{\prime }})+h_{\mathrm{g}\mathrm{r}}(P,M^{\mathrm{\prime }\mathrm{\prime }})$, including sequences with zero terms.

For fixed $P$, postcomposition by a module isomorphism $M\to M^{\mathrm{\prime }}$ identifies the ungraded Hom spaces; in the graded case a degree-zero isomorphism identifies every degree-$d$ Hom space [F8]. The generator values are therefore class functions. By steps 1.2 and 3.1 they are exact-sequence-additive. The $G_0$ presentation [F9] and universal property [F11] give unique homomorphisms ${\bar{h}}_P:G_0(A)\to \mathbb{Z}$ and ${\bar{h}}_P^{\mathrm{g}\mathrm{r}}:G_0^{\mathrm{g}\mathrm{r}}(A)\to R$ with the prescribed values on object classes [F1, F18].

If $P\cong P^{\mathrm{\prime }}$, precomposition with an isomorphism identifies their Hom spaces, preserving each degree in the graded case; hence the functions $[P]\mapsto {\bar{h}}_P$ and $[P]\mapsto {\bar{h}}_P^{\mathrm{g}\mathrm{r}}$ are class functions. The zero module is projective, and finite direct sums of projectives remain projective because lifts on the summands combine to a lift on the sum [F3, F6]. The direct-sum universal property [F13] gives ${\mathrm{Hom}}_A(P\oplus Q,M)\cong {\mathrm{Hom}}_A(P,M)\oplus {\mathrm{Hom}}_A(Q,M)$; in the graded case this decomposition preserves each degree because finite biproducts are computed degreewise [F5]. Thus the class functions are additive in the projective variable. The finite projective subcategories are essentially small by step 1.1; they are full preadditive subcategories and have a zero object and finite biproducts, so they are additive by [F20]. The split relation [F10] and universal property [F11], with target the abelian group of homomorphisms out of the corresponding $G_0$, factor these class functions uniquely through $K_0(A)$ and $K_0^{\mathrm{g}\mathrm{r}}(A)$. Evaluation defines the claimed pairings, which are $\mathbb{Z}$-bilinear and unique because object classes generate both groups; zero Hom spaces give zero on zero objects.

Bilinearity extends the generator shift identity to finite Laurent combinations. For $f={\sum }_r{a}_r{v}^r$ and $g={\sum }_s{b}_s{v}^s$, $⟨fx,gy{⟩}_{\mathrm{g}\mathrm{r}}={\sum }_{r,s}{a}_r{b}_s{v}^{s-r}⟨x,y{⟩}_{\mathrm{g}\mathrm{r}}=\bar{f}g⟨x,y{⟩}_{\mathrm{g}\mathrm{r}}$, where $\bar{f}(v)=f(v^{-1})$. Thus the graded pairing is sesquilinear, with no additional sign convention.