Projective and simple classes are dual bases under splitting

The category ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}(A)$ is abelian: it is the full subcategory of the abelian category of left $A$-modules [F5], and its finite-dimensional objects are closed under kernels, cokernels, and finite biproducts; the ambient coimage-to-image isomorphisms remain in the full subcategory, so [F6] applies. To verify essential smallness without choosing a skeleton, form the full subcategory whose objects are all module actions of $A$ on $k^n$ for $n\in \mathbb{N}$. This is a small category: for each $n$ its possible actions are a subset of the set of functions $A\times k^n\to k^n$, and all morphisms between these coordinate models form sets. Every finite-dimensional module is isomorphic to one such model by choosing a finite basis for that individual module, so the inclusion is fully faithful and essentially surjective. Thus ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}(A)$ is essentially small and the group in [F4] is defined. Every object has finite length by induction on its $k$-dimension: for nonzero $M$, choose a proper submodule $N$ of maximal dimension among the finite set of possible dimensions; then $M\mathrm{/}N$ is simple, and an induction series for $N$ extends by this quotient to one for $M$. This uses one submodule at a time and no global choice; [F7] records the length convention. Therefore [F2] applies to ${\mathcal{M}}_{\mathrm{f}\mathrm{d}}(A)$.

Every $A$-linear map between the finite modules is $k$-linear by [F15], and the Hom spaces are $k$-subspaces of the corresponding spaces of linear maps by [F16]; they are finite-dimensional by [F17]. Fix $i,j$ and let $f:P_i\to S_j$. If $f\mathrm{\ne }0$, simplicity [F11] makes it surjective, so $\ker f$ is maximal. The cover kernel $K_i=\ker q_i$ lies in every maximal submodule: otherwise $K_i+\ker f=P_i$, contradicting [F8]. Thus $K_i\subseteq \ker f$, and $f$ factors uniquely through $q_i$ as a map $S_i\to S_j$. Conversely every map $S_i\to S_j$ composes with $q_i$, so ${\mathrm{Hom}}_A(P_i,S_j)\cong {\mathrm{Hom}}_A(S_i,S_j)$. By [F10], this is zero for $i\mathrm{\ne }j$ and is ${\mathrm{End}}_A(S_i)$ for $i=j$.

For every $d\in \mathbb{Z}$, the shift convention [F12] identifies a degree-$d$ map $P_i\to S_j$ with a degree-zero map $P_i\{d\}\to S_j$, by [F13]. The shifted cover $p_i\{d\}:P_i\{d\}\twoheadrightarrow S_i\{d\}$ is again a finite graded projective cover: shifting is an exact equivalence with inverse $\{-d\}$ by [F12, F14], so it preserves projectivity, and a finite homogeneous generating family remains finite after reindexing by [F19]. Shifting back also takes graded submodules and the cover-kernel condition in [F9] to those for $p_i$. For a degree-zero map $g:P_i\{d\}\to S_j$, if $g\mathrm{\ne }0$ then it is epic, and its graded kernel is maximal by graded simplicity [F3] and [F14]. Superfluity of $\ker (p_i\{d\})$ forces that kernel into $\ker g$, so $g$ factors uniquely through $S_i\{d\}$. A nonzero map between graded-simple modules is an isomorphism, since its kernel and image are graded submodules.

By [F18], the ungraded pairing matrix has entry ${\dim }_k{\mathrm{End}}_A(S_i)$ on the diagonal and zero off the diagonal. Under the splitting hypothesis each diagonal entry is $1$. The bases in [F1] and [F2] are therefore dual in the split case; without splitting the displayed diagonal dimensions remain the exact pairing values.

If $S_i\{d\}\cong S_j$, the choice of one representative per shift orbit forces $i=j$. A finite-dimensional nonzero graded module has finite nonempty support, and an isomorphism $S_i\{d\}\cong S_i$ would make that support invariant under translation by $d$; its maximum then gives $d=0$. Therefore ${\mathrm{Hom}}_{A,d}(P_i,S_j)=0$ unless $i=j$ and $d=0$, while ${\mathrm{Hom}}_{A,0}(P_i,S_i)\cong {\mathrm{End}}_{A,0}(S_i)$ by step 1.3.

Taking the graded pairing coefficients in [F18] gives diagonal value ${\dim }_k{\mathrm{End}}_{A,0}(S_i)$ and zero off the diagonal, by steps 1.3 and 2.2. Under the splitting hypothesis the diagonal is $1$, and [F3] makes these bases dual over $R$. For nonsplit endomorphism rings the diagonal dimension remains as stated; neither this calculation nor the ungraded one determines the projective-to-module Cartan map or a projective/projective Cartan matrix. The chosen representative families are finite by [F1] and [F3], so these arguments use only finite choices and no axiom of choice.