The Stacks project

Lemma 56.9.5. Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be schemes of finite type over $S$. Let $K \in D^ b_{\textit{Coh}}(\mathcal{O}_{X \times _ S Y})$. The corresponding Fourier-Mukai functor $\Phi _ K$ sends $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ if at least one of the following conditions is satisfied:

  1. the support of $H^ i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_ X)$,

  2. $K$ can be represented by a bounded complex $\mathcal{K}^\bullet $ of coherent $\mathcal{O}_{X \times _ S Y}$-modules, flat over $X$, with support proper over $Y$,

  3. the support of $H^ i(K)$ is proper over $Y$ for all $i$ and $X$ is a regular scheme,

  4. $K$ is perfect, the support of $H^ i(K)$ is proper over $Y$ for all $i$, and $Y \to S$ is flat.

Furthermore in each case the support condition is automatic if $X \to S$ is proper.

Proof. Let $M$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. In each case we will use Derived Categories of Schemes, Lemma 36.11.3 to show that

\[ \Phi _ K(M) = R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K) \]

is in $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. The derived tensor product $L\text{pr}_1^*M \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K$ is a pseudo-coherent object of $D(\mathcal{O}_{X \times _ S Y})$ (by Cohomology, Lemma 20.44.3, Derived Categories of Schemes, Lemma 36.10.3, and Cohomology, Lemma 20.44.5) whence has coherent cohomology sheaves (by Derived Categories of Schemes, Lemma 36.10.3 again). In each case the supports of the cohomology sheaves $H^ i(L\text{pr}_1^*M \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K)$ is proper over $Y$ as these supports are contained in the union of the supports of the $H^ i(K)$. Hence in each case it suffices to prove that this tensor product is bounded below.

Case (1). By Cohomology, Lemma 20.27.4 we have

\[ L\text{pr}_1^*M \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K \cong \text{pr}_1^{-1}M \otimes _{\text{pr}_1^{-1}\mathcal{O}_ X}^\mathbf {L} K \]

with obvious notation. Hence the assumption on tor dimension and the fact that $M$ has only a finite number of nonzero cohomology sheaves, implies the bound we want.

Case (2) follows because here the assumption implies that $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_ X)$ hence the argument in the previous paragraph applies.

In Case (3) it is also the case that $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_ X)$. Namely, choose affine opens $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(B)$ of $X$ and $Y$ mapping into the affine open $W = \mathop{\mathrm{Spec}}(R)$ of $S$. Then $K|_{U \times V}$ is given by a bounded complex of finite $A \otimes _ R B$-modules $M^\bullet $. Since $A$ is a regular ring of finite dimension we see that each $M^ i$ has finite projective dimension as an $A$-module (Algebra, Lemma 10.110.8) and hence finite tor dimension as an $A$-module. Thus $M^\bullet $ has finite tor dimension as a complex of $A$-modules (More on Algebra, Lemma 15.65.8). Since $X \times Y$ is quasi-compact we conclude there exist $[a, b]$ such that for every point $z \in X \times Y$ the stalk $K_ z$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{X, \text{pr}_1(z)}$. This implies $K$ has bounded tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_ X)$, see Cohomology, Lemma 20.45.5. We conclude as in the previous to paragraphs.

Case (4). With notation as above, the ring map $R \to B$ is flat. Hence the ring map $A \to A \otimes _ R B$ is flat. Hence any projective $A \otimes _ R B$-module is $A$-flat. Thus any perfect complex of $A \otimes _ R B$-modules has finite tor dimension as a complex of $A$-modules and we conclude as before. $\square$


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