The acyclic-resolution theorem for right derived functors

By exactness in [L1], let $Z^0=A$ and for each $q\ge 0$ let $Z^{q+1}$ fit into a short exact sequence $0\to Z^q\to J^q\to Z^{q+1}\to 0.$ Every $J^q$ is $F$-acyclic by [L1].

Apply [L4] to each short exact sequence from step 1.1 using the supplied injective resolutions of $Z^q$ and $Z^{q+1}$, which exist by the domain hypothesis in the statement. The result is a degreewise split short exact sequence of injective resolutions. By [L6], applying $F$ preserves its degreewise exactness, so [L5] gives a long exact cohomology sequence. The middle injective resolution supplied by the horseshoe construction may differ from the one fixed for $J^q$, but [L3] identifies their right derived objects. Its higher cohomology therefore vanishes because $J^q$ is $F$-acyclic. Using [L2] for degree $0$, we obtain exact sequences $0\to F(Z^q)\to F(J^q)\to F(Z^{q+1})\to R_I^{1}F(Z^q)\to 0$ and isomorphisms $R_I^{m+1}F(Z^q)\stackrel{\sim }{\to }{R}_I^{m}F(Z^{q+1})(m>0)\mathrm{.}$

Repeatedly applying the isomorphisms from step 2.1 gives $R_I^{n}F(A)=R_I^{n}F(Z^0)\stackrel{\sim }{\to }{R}_I^{1}F(Z^{n-1})(n>0)\mathrm{.}$

For $n>0$, the exact sequence $0\to F(Z^n)\to F(J^n)\to F(Z^{n+1})$ from step 2.1 shows that the $n$th cohomology of $F(J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet })$ is the cokernel of $F(J^{n-1})\to F(Z^n)$. The same step identifies that cokernel with $R_I^{1}F(Z^{n-1})$, so step 3.1 gives $H^n(F(J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))\cong R_I^{n}F(A)\mathrm{.}$

For $n=0$, step 2.1 with $q=0$ gives $0\to F(A)\to F(J^0)\to F(Z^1)$ exact, so $H^0(F(J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))\cong F(A)$. By [L2], $F(A)\cong R_I^{0}F(A)$. Together with step 4.1, this proves the theorem for all $n\ge 0$.