## 57.15 A category of Fourier-Mukai kernels

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

Objects are proper smooth schemes over $S$.

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

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 36.30.4.

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 37.58.12 and the fact that $\Delta _{X/S}$ is a perfect morphism by Divisors, Lemma 31.22.11 and More on Morphisms, Lemma 37.58.7.

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 36.22.5). 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 36.22.1. 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 57.15.1. Let $S' \to S$ be a morphism of schemes. The rule which sends

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

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 36.30.4 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)

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