Derived adjunction for finite rings and closed immersions

On rings the adjunction is explicit: a map $u:M\to {\mathrm{Hom}}_A(B,I)$ corresponds to $m\mapsto u(m)(1)$, with inverse sending $v:M\to I$ to $m\mapsto (b\mapsto v(bm))$. Restriction of scalars is exact, so its right adjoint sends injectives to injectives: applying ${\mathrm{Hom}}_B(-,{\mathrm{Hom}}_A(B,I))={\mathrm{Hom}}_A(-,I)$ to an exact sequence proves this directly. Resolve $G$ by a bounded below complex $I^{\bullet }$ of injectives. Then ${\mathrm{Hom}}_A(B,I^{\bullet })$ is bounded below and injective over $B$; the displayed degreewise Hom identity, including the usual total-complex signs, computes the derived adjunction. Bounded below injective complexes compute these derived Homs because maps from acyclic complexes are null-homotopic, constructed successively from the lowest nonzero target degree using injectivity. The adjunction is natural in both arguments.

For a closed immersion, $i_{\ast }$ is exact on all module sheaves: at a point of $X$ its stalk is the original stalk, and outside $X$ the stalk vanishes. Its right adjoint is $i^{b}I=i^{-1}\mathcal{H}o{m}_Y(i_{\ast }{\mathcal{O}}_X,I)$, by the same evaluation formula, and the ideal defining $X$ annihilates this Hom. Since $i_{\ast }$ is exact and $i^b$ is its right adjoint, $i^b$ sends injectives to injectives: for injective $I$ the functor ${\mathrm{Hom}}_{{\mathcal{O}}_X}(-,i^{b}I)\cong {\mathrm{Hom}}_{{\mathcal{O}}_Y}(i_{\ast }-,I)$ is a composition of the exact functor $i_{\ast }$ with the exact functor ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(-,I)$. Restrictions of an injective module sheaf to an open subset remain injective, since extension by zero is exact and left adjoint to restriction. Applying the ringed-space adjunction on every open subset gives the internal Hom identity; applying it on all of $Y$ gives the global Hom identity. Injective resolutions now give the stated derived identities, and their counit is evaluation at $1$.

Extension by zero along a closed subset preserves flasque sheaves and is exact. Computing sheaf cohomology by flasque resolutions therefore identifies $R\mathrm{\Gamma }(Y,i_{\ast }K)$ with $R\mathrm{\Gamma }(X,K)$, first for sheaves and then for bounded complexes by totalizing their resolutions; this is the closed-immersion cohomology comparison of [F1], and the coherence of $i_{\ast }M$ for coherent $M$ is the same comparison read on an affine chart. AC enters through the resolution data; the adjunction formulas themselves involve no selections.