Projective–module Hom pairing on class generators

Every $A$-linear map between left $A$-modules is $k$-linear: for $\lambda \in k$, centrality of the scalar action gives $f(\lambda p)=f((\lambda 1_A)p)=(\lambda 1_A)f(p)=\lambda f(p)$. Sums and scalar multiples of $A$-linear maps remain $A$-linear, so ${\mathrm{Hom}}_A(P,M)$ is a $k$-vector subspace of ${\mathcal{L}}_k(P,M)$. Since $P,M$ are finite-dimensional, [F7] makes the ambient linear-map space finite-dimensional, and hence ${\mathrm{Hom}}_A(P,M)$ is finite-dimensional. Thus $h(P,M)$ is defined in $\mathbb{Z}$.

Each ${\mathrm{Hom}}_{A,d}(P,M)$ is also a $k$-vector subspace of ${\mathcal{L}}_k(P,M)$: the $A$-linearity and degree-$d$ conditions are preserved by addition and scalar multiplication, using [F4], [F5], and [F8]. Therefore every such homogeneous Hom space is finite-dimensional by [F6] and [F7].

The supports $I_P:=\{i:P_i\mathrm{\ne }0\}$ and $I_M:=\{j:M_j\mathrm{\ne }0\}$ are finite. Indeed, if a finite-dimensional graded space had more than $n$ nonzero components, where $n$ is its dimension, choosing one nonzero vector in each of $n+1$ distinct components would give $n+1$ linearly independent vectors; this is only a finite selection. If ${\mathrm{Hom}}_{A,d}(P,M)\mathrm{\ne }0$, choose a nonzero map $f$ in it. Since $f$ is nonzero, some $p\in P$ has $f(p)\mathrm{\ne }0$. Decompose $p$ into its finitely many homogeneous components. As $f$ is homogeneous and $f(p)\mathrm{\ne }0$, at least one component $p_i\in P_i$ has $f(p_i)\mathrm{\ne }0$. Then $i\in I_P$ and $i+d\in I_M$, so $d\in I_M-I_P$. This difference set is finite, hence only finitely many terms in $h_{\mathrm{g}\mathrm{r}}(P,M)$ can be nonzero. The exponent is this map degree $d$, consistent with the internal-shift normalization $v^d[M]=[M\{d\}]$ in [F3]. The formula is therefore a Laurent polynomial in $\mathbb{Z}[v,v^{-1}]$. If either module is zero, all homogeneous Hom spaces vanish and the sum is zero.

If $\alpha :P\to P^{\mathrm{\prime }}$ and $\beta :M\to M^{\mathrm{\prime }}$ are module isomorphisms, then $f\mapsto \beta f{\alpha }^{-1}$ is a $k$-linear isomorphism ${\mathrm{Hom}}_A(P,M)\to {\mathrm{Hom}}_A(P^{\mathrm{\prime }},M^{\mathrm{\prime }})$. For graded isomorphisms of degree zero it restricts, for every $d$, to an isomorphism of ${\mathrm{Hom}}_{A,d}$ spaces, since degree-zero maps preserve each homogeneous component. Hence both candidate values depend only on the object isomorphism classes.

The formulas in the Statement thus give well-defined functions on pairs of object isomorphism classes, with values in $\mathbb{Z}$ and $\mathbb{Z}[v,v^{-1}]$, respectively.

The groups in [F1] and [F2] impose additional short-exact-sequence and direct-sum relations. This Definition specifies only the object-level values; it makes no claim that they are additive for those relations or descend to $K_0\times G_0$.