The Stacks project

Lemma 48.17.9. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Assume $f$ is perfect (e.g., flat). Then

  1. $f^!\mathcal{O}_ Y$ is in $D_{\textit{Coh}}^ b(\mathcal{O}_ X)$,

  2. $f^!\mathcal{O}_ Y$ has finite tor dimension in $D(f^{-1}\mathcal{O}_ Y)$,

  3. $\mathcal{O}_ X \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f^!\mathcal{O}_ Y, f^!\mathcal{O}_ Y)$ is an isomorphism,

  4. $f^!$ maps $D_{\textit{Coh}}^ b(\mathcal{O}_ Y)$ into $D_{\textit{Coh}}^ b(\mathcal{O}_ X)$,

  5. the map $\mu _{f, K} : Lf^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} f^!\mathcal{O}_ Y \to f^!K$ of Lemma 48.16.5 is an isomorphism for all $K \in D_\mathit{QCoh}^+(\mathcal{O}_ Y)$.

Proof. (A flat morphism of finite presentation is perfect, see More on Morphisms, Lemma 37.61.5.) Assertions (a), (b), and (c) are local on $X$. Thus we may assume $X$ and $Y$ are affine. Then Remark 48.17.5 turns (a), (b), and (c) into (1)(a), (1)(b), and (1)(c) of Dualizing Complexes, Lemma 47.25.2. (Use Derived Categories of Schemes, Lemmas 36.10.5, 36.10.1, 36.10.3 36.10.8 to match the assertions.)

To prove (d) and (e) we begin with a series of preliminary remarks.

  1. We already know that $f^!$ sends $D_{\textit{Coh}}^+(\mathcal{O}_ Y)$ into $D_{\textit{Coh}}^+(\mathcal{O}_ X)$, see Lemma 48.17.6.

  2. If $f$ is an open immersion, then $f^! = f^*$ and (d) and (e) are true because we can take $\overline{X} = Y$ in the construction of $f^!$ and $\mu _ f$, see Lemma 48.17.1.

  3. If $f$ is a perfect proper morphism, then (e) is true by Lemma 48.13.3.

  4. If there exists an open covering $X = \bigcup U_ i$ and (d) is true for $U_ i \to Y$, then (d) is true for $X \to Y$. Same for (e). This holds because the construction of $f^!$ and $\mu _ f$ commutes with passing to open subschemes.

  5. If $g : Y \to Z$ is a second perfect morphism in $\textit{FTS}_ S$ and (e) holds for $f$ and $g$, then $f^!g^!\mathcal{O}_ Z = Lf^*g^!\mathcal{O}_ Z \otimes _{\mathcal{O}_ X}^\mathbf {L} f^!\mathcal{O}_ Y$ and (e) holds for $g \circ f$ by the commutative diagram of Lemma 48.16.5.

  6. If (d) and (e) hold for both $f$ and $g$, then (d) and (e) hold for $g \circ f$. Namely, then $f^!g^!\mathcal{O}_ Z$ is bounded above (by the previous point) and $L(g \circ f)^*$ has finite cohomological dimension and (d) follows from (e) which we saw above.

From these points we see it suffices to prove (d) and (e) in case $X$ is affine. Choose an immersion $X \to \mathbf{A}^ n_ Y$ (Morphisms, Lemma 29.39.2) which we factor as $X \to U \to \mathbf{A}^ n_ Y \to Y$ where $X \to U$ is a closed immersion and $U \subset \mathbf{A}^ n_ Y$ is open. Note that $X \to U$ is a perfect closed immersion by More on Morphisms, Lemma 37.61.8. Thus it suffices to prove the lemma for a perfect closed immersion and for the projection $\mathbf{A}^ n_ Y \to Y$.

Let $f : X \to Y$ be a perfect closed immersion. We already know (d) holds. Let $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. Then $f^!K = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, K)$ (Lemma 48.17.4) and $f_*f^!K = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(f_*\mathcal{O}_ X, K)$. Since $f$ is perfect, the complex $f_*\mathcal{O}_ X$ is perfect and hence $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(f_*\mathcal{O}_ X, K)$ is bounded above. This proves that (d) holds. Some details omitted.

Let $f : \mathbf{A}^ n_ Y \to Y$ be the projection. Then (d) holds by repeated application of Lemma 48.17.3. Finally, (e) is true because it holds for $\mathbf{P}^ n_ Y \to Y$ (flat and proper) and because $\mathbf{A}^ n_ Y \subset \mathbf{P}^ n_ Y$ is an open. $\square$

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