Exact projections onto linkage blocks preserve projectives

By [F1] every object $X$ is ${⨁}_C{X}_C$ with finitely many nonzero terms, the decomposition is functorial, and morphisms between objects supported on disjoint collections of classes vanish. Hence a morphism $f:X\to Y$ between decomposed objects is block diagonal, $f={⨁}_C{f}_C$ with $f_C:X_C\to Y_C$.

By [F2] the projectivity of $P$ in $\mathcal{O}$ says exactly that ${\mathrm{Hom}}_{\mathcal{O}}(P,-)$ is exact on $\mathcal{O}$, and the hypothesis on $Q$ says that ${\mathrm{Hom}}_{{\mathcal{O}}_C}(Q,-)$ is exact on ${\mathcal{O}}_C$.

Apply [F3] to the block-diagonal differentials of step 1.1: a short exact sequence $0\to A\to E\to B\to 0$ in $\mathcal{O}$ decomposes into the short exact sequences $0\to A_C\to E_C\to B_C\to 0$ of its components, and conversely exactness of all components gives exactness of the sequence. Therefore ${\mathrm{pr}}_C$ is an exact functor $\mathcal{O}\to {\mathcal{O}}_C$, and $i_C$ is exact as the inclusion of a full subcategory closed under subobjects and quotients.

For $A\in {\mathcal{O}}_C$ and any $X\in \mathcal{O}$, decomposing $X$ and using the vanishing of morphisms from $A$ into components supported on other classes gives a natural isomorphism ${\mathrm{Hom}}_{\mathcal{O}}(A,X)\cong {\mathrm{Hom}}_{{\mathcal{O}}_C}(A,{\mathrm{pr}}_{C}X)$; here ${\mathrm{Hom}}_{\mathcal{O}}(i_{C}A,X)$ is written ${\mathrm{Hom}}_{\mathcal{O}}(A,X)$. Similarly, for $Y\in {\mathcal{O}}_C$, decomposing $P$ gives ${\mathrm{Hom}}_{\mathcal{O}}(P,{\mathrm{pr}}_{C}Y)\cong {\mathrm{Hom}}_{\mathcal{O}}({\mathrm{pr}}_{C}P,{\mathrm{pr}}_{C}Y)$, because all components of $P$ other than ${\mathrm{pr}}_{C}P$ map to zero into the object ${\mathrm{pr}}_{C}Y$ of ${\mathcal{O}}_C$.

Let $P\in \mathcal{O}$ be projective. Composing the isomorphisms of step 2.2, for every $X\in \mathcal{O}$ there is a natural isomorphism ${\mathrm{Hom}}_{\mathcal{O}}({\mathrm{pr}}_{C}P,X)\cong {\mathrm{Hom}}_{{\mathcal{O}}_C}({\mathrm{pr}}_{C}P,{\mathrm{pr}}_{C}X)\cong {\mathrm{Hom}}_{\mathcal{O}}(P,{\mathrm{pr}}_{C}X)$. Now $X\mapsto {\mathrm{pr}}_{C}X$ is exact by step 2.1 and ${\mathrm{Hom}}_{\mathcal{O}}(P,-)$ is exact by step 1.2, so the composite functor $X\mapsto {\mathrm{Hom}}_{\mathcal{O}}({\mathrm{pr}}_{C}P,X)$ is exact. By [F2] applied in $\mathcal{O}$, ${\mathrm{pr}}_{C}P$ is projective in $\mathcal{O}$.

Conversely let $Q\in {\mathcal{O}}_C$ be projective in ${\mathcal{O}}_C$. By step 2.2, for every $X\in \mathcal{O}$ there is a natural isomorphism ${\mathrm{Hom}}_{\mathcal{O}}(Q,X)\cong {\mathrm{Hom}}_{{\mathcal{O}}_C}(Q,{\mathrm{pr}}_{C}X)$. The first functor is the composite of the exact functor ${\mathrm{pr}}_C$ of step 2.1 with the exact functor ${\mathrm{Hom}}_{{\mathcal{O}}_C}(Q,-)$ of step 1.2, hence is exact; by [F2], $Q$ is projective in $\mathcal{O}$.