The Stacks project

56.17 A category of Fourier-Mukai kernels

Let $S$ be a scheme. We claim there is a category with

  1. Objects are proper smooth schemes over $S$.

  2. Morphisms from $X$ to $Y$ are isomorphism classes of objects of $D_{perf}(\mathcal{O}_{X \times _ S Y})$.

  3. Composition of the isomorphism class of $K \in D_{perf}(\mathcal{O}_{X \times _ S Y})$ and the isomorphism class of $K'$ in $D_{perf}(\mathcal{O}_{Y \times _ S Z})$ is the isomorphism class of

    \[ R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S Z}}^\mathbf {L} L\text{pr}_{23}^*K') \]

    which is in $D_{perf}(\mathcal{O}_{X \times _ S Z})$ by Derived Categories of Schemes, Lemma 36.28.4.

  4. The identity morphism from $X$ to $X$ is the isomorphism class of $\Delta _{X/S, *}\mathcal{O}_ X$ which is in $D_{perf}(\mathcal{O}_{X \times _ S X})$ by More on Morphisms, Lemma 37.53.12 and the fact that $\Delta _{X/S}$ is a perfect morphism by Divisors, Lemma 31.22.11 and More on Morphisms, Lemma 37.53.7.

Let us check that associativity of composition of morphisms holds; we omit verifying that the identity morphisms are indeed identities. To see this suppose we have $X, Y, Z, W$ and $c \in D_{perf}(\mathcal{O}_{X \times _ S Y})$, $c' \in D_{perf}(\mathcal{O}_{Y \times _ S Z})$, and $c'' \in D_{perf}(\mathcal{O}_{Z \times _ S W})$. Then we have

\begin{align*} c'' \circ (c' \circ c) & \cong \text{pr}^{134}_{14, *}( \text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *} \text{pr}^{1234}_{134, *} ((\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c'') \\ & \cong \text{pr}^{1234}_{14, *}( (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c”) \end{align*}

Here we use the notation

\[ p^{1234}_{134} : X \times _ S Y \times _ S Z \times _ S W \to X \times _ S Z \times _ S W \quad \text{and}\quad p^{134}_{14} : X \times _ S Z \times _ S W \to X \times _ S W \]

the projections and similarly for other indices. We also write $\text{pr}_*$ instead of $R\text{pr}_*$ and $\text{pr}^*$ instead of $L\text{pr}^*$ and we drop all super and sub scripts on $\otimes $. The first equality is the definition of the composition. The second equality holds because $\text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *} = \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}$ by base change (Derived Categories of Schemes, Lemma 36.21.5). The third equality holds because pullbacks compose correctly and pass through tensor products, see Cohomology, Lemmas 20.27.2 and 20.27.3. The fourth equality follows from the “projection formula” for $p^{1234}_{134}$, see Derived Categories of Schemes, Lemma 36.21.1. The fifth equality is that proper pushforward is compatible with composition, see Cohomology, Lemma 20.28.2. Since tensor product is associative this concludes the proof of associativity of composition.

Lemma 56.17.1. Let $S' \to S$ be a morphism of schemes. The rule which sends

  1. a smooth proper scheme $X$ over $S$ to $X' = S' \times _ S X$, and

  2. the isomorphism class of an object $K$ of $D_{perf}(\mathcal{O}_{X \times _ S Y})$ to the isomorphism class of $L(X' \times _{S'} Y' \to X \times _ S Y)^*K$ in $D_{perf}(\mathcal{O}_{X' \times _{S'} Y'})$

is a functor from the category defined for $S$ to the category defined for $S'$.

Proof. To see this suppose we have $X, Y, Z$ and $K \in D_{perf}(\mathcal{O}_{X \times _ S Y})$ and $M \in D_{perf}(\mathcal{O}_{Y \times _ S Z})$. Denote $K' \in D_{perf}(\mathcal{O}_{X' \times _{S'} Y'})$ and $M' \in D_{perf}(\mathcal{O}_{Y' \times _{S'} Z'})$ their pullbacks as in the statement of the lemma. The diagram

\[ \xymatrix{ X' \times _{S'} Y' \times _{S'} Z' \ar[r] \ar[d]_{\text{pr}'_{13}} & X \times _ S Y \times _ S Z \ar[d]^{\text{pr}_{13}} \\ X' \times _{S'} Z' \ar[r] & X \times _ S Z } \]

is cartesian and $\text{pr}_{13}$ is proper and smooth. By Derived Categories of Schemes, Lemma 36.28.4 we see that the derived pullback by the lower horizontal arrow of the composition

\[ R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S Z}}^\mathbf {L} L\text{pr}_{23}^*M) \]

indeed is (canonically) isomorphic to

\[ R\text{pr}'_{13, *}( L(\text{pr}'_{12})^*K' \otimes _{\mathcal{O}_{X' \times _{S'} Y' \times _{S'} Z'}}^\mathbf {L} L(\text{pr}'_{23})^*M') \]

as desired. Some details omitted. $\square$


Comments (2)

Comment #5255 by on

Roy Magen points out that in part (4) of the definition we need to replace the references to Lemmas 37.54.18 and 37.53.13 to more general references avoiding Noetherian hypotheses. Here are the details. The diagonal of a scheme smooth over over a scheme is a regular immersion by Lemma 31.22.11. A regular immersion is perfect by Lemma 37.53.7. And of course if is a perfect closed immersion of schemes, then is a perfect object of almost by definition of being a perfect morphism. We're going to have to add this as a lemma because it isn't a lemma yet as far as I can tell. I will fix this soon.


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