The Stacks project

Lemma 36.35.11. Let $f : X \to S$ be a morphism of schemes. Let $E, K \in D(\mathcal{O}_ X)$. Assume

  1. $S$ is quasi-compact and quasi-separated,

  2. $f$ is proper, flat, and of finite presentation,

  3. $E$ is $S$-perfect,

  4. $K$ is pseudo-coherent.

Then there exists a pseudo-coherent $L \in D(\mathcal{O}_ S)$ such that

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O}_ S) \]

and the same is true after arbitrary base change: given

\[ \vcenter { \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } } \quad \quad \begin{matrix} \text{cartesian, then we have } \\ Rf'_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L(g')^*K, L(g')^*E) \\ = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lg^*L, \mathcal{O}_{S'}) \end{matrix} \]

Proof. Since $S$ is quasi-compact and quasi-separated, the same is true for $X$. By Lemma 36.19.1 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, Lemma 20.47.5). We obtain an inverse system $\ldots \to K_3^\vee \to K_2^\vee \to K_1^\vee $ of perfect objects. By Lemma 36.35.5 we see that $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ is $S$-perfect. Thus we may apply Lemma 36.35.10 to $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ and we obtain an inverse system

\[ \ldots \to M_3 \to M_2 \to M_1 \]

of perfect complexes on $S$ with

\[ M_ n = Rf_*(K_ n^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E) = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E) \]

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 Lemma 36.22.8. 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

\[ M_{n + 1} \to M_ n \to C_ n \to M_{n + 1}[1] \]

Take $S'$ equal to the spectrum of the residue field at a point $s \in S$ and pull back to see that $C_ n \otimes _{\mathcal{O}_ S}^\mathbf {L} \kappa (s)$ has nonzero cohomology only in degrees $\geq n + a$. By More on Algebra, Lemma 15.75.6 we see that the perfect complex $C_ n$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$. 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 Lemma 36.19.1. Since we have

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\text{hocolim} K_ n, E) = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E) = R\mathop{\mathrm{lim}}\nolimits K_ n^\vee \otimes _{\mathcal{O}_ X} E \]

(Cohomology, Lemma 20.48.1) and since $R\mathop{\mathrm{lim}}\nolimits $ commutes with $Rf_*$ we find that

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) = R\mathop{\mathrm{lim}}\nolimits M_ n = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L_ n, \mathcal{O}_ S) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O}_ S) \]

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


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