Lemma 15.98.1 (0A68)—The Stacks project

Lemma 15.98.1. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$ . the map

$R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(R\mathop{\mathrm{Hom}}\nolimits _ R(K, L), M)$

of Lemma 15.73.3 is an isomorphism in the following two cases

$K$ perfect, or
$K$ is pseudo-coherent, $L \in D^+(R)$ , and $M$ finite injective dimension.

Proof. Choose a K-injective complex $I^\bullet$ representing $M$ , a K-injective complex $J^\bullet$ representing $L$ , and a bounded above complex of finite projective modules $K^\bullet$ representing $K$ . Consider the map of complexes

$\text{Tot}(\mathop{\mathrm{Hom}}\nolimits ^\bullet (J^\bullet , I^\bullet ) \otimes _ R K^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , J^\bullet ), I^\bullet )$

of Lemma 15.71.6. Note that

$\left(\prod \nolimits _{p + r = t} \mathop{\mathrm{Hom}}\nolimits _ R(J^{-r}, I^ p)\right) \otimes _ R K^ s = \prod \nolimits _{p + r = t} \mathop{\mathrm{Hom}}\nolimits _ R(J^{-r}, I^ p) \otimes _ R K^ s$

because $K^ s$ is finite projective. The map is given by the maps

$c_{p, r, s} : \mathop{\mathrm{Hom}}\nolimits _ R(J^{-r}, I^ p) \otimes _ R K^ s \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(K^ s, J^{-r}), I^ p)$

which are isomorphisms as $K^ s$ is finite projective. For every element $\alpha = (\alpha ^{p, r, s})$ of degree $n$ of the left hand side, there are only finitely many values of $s$ such that $\alpha ^{p, r, s}$ is nonzero (for some $p, r$ with $n = p + r + s$ ). Hence our map is an isomorphism if the same vanishing condition is forced on the elements $\beta = (\beta ^{p, r, s})$ of the right hand side. If $K^\bullet$ is a bounded complex of finite projective modules, this is clear. On the other hand, if we can choose $I^\bullet$ bounded and $J^\bullet$ bounded below, then $\beta ^{p, r, s}$ is zero for $p$ outside a fixed range, for $s \gg 0$ , and for $r \gg 0$ . Hence among solutions of $n = p + r + s$ with $\beta ^{p, r, s}$ nonzero only a finite number of $s$ values occur. $\square$

The Stacks project

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