Cyclotomic projective covers and dualities descend through the quotient

Base change of projectives and covers. A finite graded projective $P$ is a retract of a finite sum of shifted copies of $A$, so $B{\otimes }_{A}P$ is a retract of a finite sum of shifted copies of $B$: it is finite graded projective. The quotient adjunction identifies ${\mathrm{Hom}}_A(P,B{\otimes }_A{P}^{\mathrm{\prime }})={\mathrm{Hom}}_B(B{\otimes }_{A}P,B{\otimes }_A{P}^{\mathrm{\prime }})$, and every degree-zero endomorphism of $B{\otimes }_{A}P$ lifts through the canonical map $P\to B{\otimes }_{A}P$; endomorphisms preserve $JP$, so the induced map ${\mathrm{End}}_A^0(P)\to {\mathrm{End}}_B^0(B{\otimes }_{A}P)$ is a surjective ring homomorphism. If $P$ is indecomposable, its finite-dimensional local degree-zero endomorphism ring has no nontrivial idempotent, so a nonzero quotient has none either: $B{\otimes }_{A}P$ is zero or indecomposable. If $JS=0$, the old cover map factors through $P\mathrm{/}JP$ and remains essential (a submodule of $B{\otimes }_{A}P$ mapping onto $S$ pulls back to a submodule of $P$ mapping onto $S$; essentiality of the old cover makes it all of $P$), so $B{\otimes }_{A}P$ is the projective cover of $S$ with the same head. If $JS\mathrm{\ne }0$ and $B{\otimes }_{A}P\mathrm{\ne }0$, finite dimension [F3] supplies a simple quotient, which must be the unique head $S$; as a $B$-module that quotient is annihilated by $J$, forcing $JS=0$ and contradicting the assumption, so $B{\otimes }_{A}P=0$.

Every $B$-orbit arises and heads do not merge. Every simple $B$-module is a simple $A$-module by inflation, and its descended cover $B{\otimes }_{A}P(S)$ is a nonzero cover, so every $B$-simple/projective orbit arises this way. Distinct surviving heads remain distinct after quotienting because an isomorphism of the descended covers induces an isomorphism of their simple heads as $B$-modules, hence also as $A$-modules after inflation; hence the surviving projective-cover shift orbits cannot merge, and extension of scalars is surjective on projective $K_0$.

Dualities. For finite $P$, evaluation on shifted free modules and restriction to retracts identifies ${\mathrm{HOM}}_B(B{\otimes }_{A}P,B)={\mathrm{HOM}}_A(P,A){\otimes }_{A}B$, and applying the generator-fixing anti-involution $\psi$ gives $(B{\otimes }_{A}P{)}^{\mathrm{\#}}=B{\otimes }_A{P}^{\mathrm{\#}}$. Inflation does not change the finite vector duality $D_{\psi }$ of a simple, since the underlying vector space and the $\psi$-action are unchanged. The choice-free normalization clauses of [F2] therefore carry the unique $\mathrm{\#}$-self-dual generic covers and $D_{\psi }$-self-dual simples to unique normalized $B$-covers and simples.

The two pairings. For normalized labels, $h(P,S)=\mathrm{gdim}{\mathrm{HOM}}_B(P,S)=q^a$ in a single shift $a$ because $\mathrm{End}(S)=k$ absolutely, while distinct heads give zero; assigning the simple's shift makes $h={\delta }_{P,S}$. Similarly $T(P,S)=\mathrm{gdim}(P^{\psi }{\otimes }_{B}S)={\delta }_{P,S}$ by finite-projective evaluation and $D(P^{\psi }\otimes S)={\mathrm{HOM}}_B(P,D_{\psi }S)$ for evaluation of a tensor functional; $T$ is bilinear in the two arguments and $h(P,M)=T(P^{\mathrm{\#}},M)$ is first-bar sesquilinear. Extension of scalars induces the surjection on projective $K_0$ by step 2.1. No Frobenius, geometric basis or cyclotomic crystal statement is used.