The Stacks project

Proof. Since $Y$ is quasi-compact and quasi-separated, the same is true for $X$. By Derived Categories of Spaces, Lemma 75.18.1 we can write $K = \text{hocolim} K_ n$ with $K_ n$ perfect and $K_ n \to K$ inducing an isomorphism on truncations $\tau _{\geq -n}$. Let $K_ n^\vee $ be the dual perfect complex (Cohomology on Sites, Lemma 21.48.4). We obtain an inverse system $\ldots \to K_3^\vee \to K_2^\vee \to K_1^\vee $ of perfect objects. By Lemma 76.52.5 we see that $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ is $Y$-perfect. Thus we may apply Lemma 76.52.10 to $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ and we obtain an inverse system

of perfect complexes on $Y$ with

Moreover, the formation of these complexes commutes with any base change, namely $Lg^*M_ n = Rf'_*((L(g')^*K_ n)^\vee \otimes _{\mathcal{O}_{X'}}^\mathbf {L} L(g')^*E) = Rf'_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L(g')^*K_ n, L(g')^*E)$.

As $K_ n \to K$ induces an isomorphism on $\tau _{\geq -n}$, we see that $K_ n \to K_{n + 1}$ induces an isomorphism on $\tau _{\geq -n}$. It follows that $K_{n + 1}^\vee \to K_ n^\vee $ induces an isomorphism on $\tau _{\leq n}$ as $K_ n^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, \mathcal{O}_ X)$. Suppose that $E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_ Y$-modules. Then the same is true after any base change, see Derived Categories of Spaces, Lemma 75.20.7. We find that $K_{n + 1}^\vee \otimes _{\mathcal{O}_ X} E \to K_ n^\vee \otimes _{\mathcal{O}_ X} E$ induces an isomorphism on $\tau _{\leq n + a}$ and the same is true after any base change. Applying the right derived functor $Rf_*$ we conclude the maps $M_{n + 1} \to M_ n$ induce isomorphisms on $\tau _{\leq n + a}$ and the same is true after any base change. Choose a distinguished triangle

Pick $y \in |Y|$. Choose an elementary étale neighbourhood $(U, u) \to (Y, y)$; this is possible by Decent Spaces, Lemma 68.11.4. Take $Y'$ equal to the spectrum of the residue field at $u$. Pull back to see that $C_ n|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \kappa (u)$ has nonzero cohomology only in degrees $\geq n + a$. By More on Algebra, Lemma 15.75.7 we see that the perfect complex $C_ n|_ U$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$ and after possibly shrinking $U$. Thus $C_ n$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$ (because $Y$ is quasi-compact). In particular, the dual perfect complex $C_ n^\vee $ has tor amplitude in $[-m_ n, -n - a]$.

Let $L_ n = M_ n^\vee $ be the dual perfect complex. The conclusion from the discussion in the previous paragraph is that $L_ n \to L_{n + 1}$ induces isomorphisms on $\tau _{\geq -n - a}$. Thus $L = \text{hocolim} L_ n$ is pseudo-coherent, see Derived Categories of Spaces, Lemma 75.18.1. Since we have

(Cohomology on Sites, Lemma 21.48.8) and since $R\mathop{\mathrm{lim}}\nolimits $ commutes with $Rf_*$ we find that

This proves the formula over $Y$. Since the construction of $M_ n$ is compatible with base chance, the formula continues to hold after any base change. $\square$