Lemma 57.9.10. Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Let $K_0 \to K_1 \to K_2 \to \ldots $ be a system of objects of $D_{perf}(\mathcal{O}_{X \times Y})$ and $m \geq 0$ an integer such that
$H^ q(K_ i)$ is nonzero only for $q \leq m$,
for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ with $\dim (\text{Supp}(\mathcal{F})) = 0$ the object
\[ R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}}^\mathbf {L} K_ n) \]
has vanishing cohomology sheaves in degrees outside $[-m, m] \cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in degrees in $[-m, m]$.
Then $K_ n$ has vanishing cohomology sheaves in degrees outside $[-m, m] \cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in degrees in $[-m, m]$. Moreover, if $X$ and $Y$ are smooth over $k$, then for $n$ large enough we find $K_ n = K \oplus C_ n$ in $D_{perf}(\mathcal{O}_{X \times Y})$ where $K$ has cohomology only indegrees $[-m, m]$ and $C_ n$ only in degrees $[-m - n, m - n]$ and the transition maps define isomorphisms between various copies of $K$.
Proof.
Let $Z$ be the scheme theoretic support of an $\mathcal{F}$ as in (2). Then $Z \to \mathop{\mathrm{Spec}}(k)$ is finite, hence $Z \times Y \to Y$ is finite. It follows that for an object $M$ of $D_\mathit{QCoh}(\mathcal{O}_{X \times Y})$ with cohomology sheaves supported on $Z \times Y$ we have $H^ i(R\text{pr}_{2, *}(M)) = \text{pr}_{2, *}H^ i(M)$ and the functor $\text{pr}_{2, *}$ is faithful on quasi-coherent modules supported on $Z \times Y$; details omitted. Hence we see that the objects
\[ \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}}^\mathbf {L} K_ n \]
in $D_{perf}(\mathcal{O}_{X \times Y})$ have vanishing cohomology sheaves outside $[-m, m] \cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in $[-m, m]$. Let $z \in X \times Y$ be a closed point mapping to the closed point $x \in X$. Then we know that
\[ K_{n, z} \otimes _{\mathcal{O}_{X \times Y, z}}^\mathbf {L} \mathcal{O}_{X \times Y, z}/\mathfrak m_ x^ t\mathcal{O}_{X \times Y, z} \]
has nonzero cohomology only in the intervals $[-m, m] \cup [-m - n, m - n]$. We conclude by More on Algebra, Lemma 15.100.2 that $K_{n, z}$ only has nonzero cohomology in degrees $[-m, m] \cup [-m - n, m - n]$. Since this holds for all closed points of $X \times Y$, we conclude $K_ n$ only has nonzero cohomology sheaves in degrees $[-m, m] \cup [-m - n, m - n]$. In exactly the same way we see that the maps $K_ n \to K_{n + 1}$ are isomorphisms on cohomology sheaves in degrees $[-m, m]$ for $n > 2m$.
If $X$ and $Y$ are smooth over $k$, then $X \times Y$ is smooth over $k$ and hence regular by Varieties, Lemma 33.25.3. Thus we will obtain the direct sum decomposition of $K_ n$ as soon as $n > 2m + \dim (X \times Y)$ from Lemma 57.9.5. The final statement is clear from this.
$\square$
Comments (2)
Comment #5918 by Noah Olander on
Comment #6113 by Johan on