Lemma 57.10.1. Let $R$ be a Noetherian ring. Let $X$, $Y$ be finite type schemes over $R$ having the resolution property. For any coherent $\mathcal{O}_{X \times _ R Y}$-module $\mathcal{F}$ there exist a surjection $\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module and $\mathcal{G}$ is a finite locally free $\mathcal{O}_ Y$-module.

## 57.10 Resolutions and bounds

The diagonal of a smooth proper scheme has a nice resolution.

**Proof.**
Let $U \subset X$ and $V \subset Y$ be affine open subschemes. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal sheaf of the reduced induced closed subscheme structure on $X \setminus U$. Similarly, let $\mathcal{I}' \subset \mathcal{O}_ Y$ be the ideal sheaf of the reduced induced closed subscheme structure on $Y \setminus V$. Then the ideal sheaf

satisfies $V(\mathcal{J}) = X \times _ R Y \setminus U \times _ R V$. For any section $s \in \mathcal{F}(U \times _ R V)$ we can find an integer $n > 0$ and a map $\mathcal{J}^ n \to \mathcal{F}$ whose restriction to $U \times _ R V$ gives $s$, see Cohomology of Schemes, Lemma 30.10.5. By assumption we can choose surjections $\mathcal{E} \to \mathcal{I}$ and $\mathcal{G} \to \mathcal{I}'$. These produce corresponding surjections

and hence a map $\mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{F}$ whose image contains the section $s$ over $U \times _ R V$. Since we can cover $X \times _ R Y$ by a finite number of affine opens of the form $U \times _ R V$ and since $\mathcal{F}|_{U \times _ R V}$ is generated by finitely many sections (Properties, Lemma 28.16.1) we conclude that there exists a surjection

where $\mathcal{E}_ j$ is finite locally free on $X$ and $\mathcal{G}_ j$ is finite locally free on $Y$. Setting $\mathcal{E} = \bigoplus \mathcal{E}_ j^{\otimes n_ j}$ and $\mathcal{G} = \bigoplus \mathcal{G}_ j^{\otimes n_ j}$ we conclude that the lemma is true. $\square$

Lemma 57.10.2. Let $R$ be a ring. Let $X$, $Y$ be quasi-compact and quasi-separated schemes over $R$ having the resolution property. For any finite type quasi-coherent $\mathcal{O}_{X \times _ R Y}$-module $\mathcal{F}$ there exist a surjection $\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module and $\mathcal{G}$ is a finite locally free $\mathcal{O}_ Y$-module.

**Proof.**
Follows from Lemma 57.10.1 by a limit argument. We urge the reader to skip the proof. Since $X \times _ R Y$ is a closed subscheme of $X \times _\mathbf {Z} Y$ it is harmless if we replace $R$ by $\mathbf{Z}$. We can write $\mathcal{F}$ as the quotient of a finitely presented $\mathcal{O}_{X \times _ R Y}$-module by Properties, Lemma 28.22.8. Hence we may assume $\mathcal{F}$ is of finite presentation. Next we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ of finite presentation over $\mathbf{Z}$ and similarly $Y = \mathop{\mathrm{lim}}\nolimits Y_ j$, see Limits, Proposition 32.5.4. Then $\mathcal{F}$ will descend to $\mathcal{F}_{ij}$ on some $X_ i \times _ R Y_ j$ (Limits, Lemma 32.10.2) and so does the property of having the resolution property (Derived Categories of Schemes, Lemma 36.36.9). Then we apply Lemma 57.10.1 to $\mathcal{F}_{ij}$ and we pullback.
$\square$

Lemma 57.10.3. Let $R$ be a Noetherian ring. Let $X$ be a separated finite type scheme over $R$ which has the resolution property. Set $\mathcal{O}_\Delta = \Delta _*(\mathcal{O}_ X)$ where $\Delta : X \to X \times _ R X$ is the diagonal of $X/k$. There exists a resolution

where each $\mathcal{E}_ i$ and $\mathcal{G}_ i$ is a finite locally free $\mathcal{O}_ X$-module.

**Proof.**
Since $X$ is separated, the diagonal morphism $\Delta $ is a closed immersion and hence $\mathcal{O}_\Delta $ is a coherent $\mathcal{O}_{X \times _ R X}$-module (Cohomology of Schemes, Lemma 30.9.8). Thus the lemma follows immediately from Lemma 57.10.1.
$\square$

Lemma 57.10.4. Let $X$ be a regular Noetherian scheme of dimension $d < \infty $. Then

for $\mathcal{F}$, $\mathcal{G}$ coherent $\mathcal{O}_ X$-modules we have $\mathop{\mathrm{Ext}}\nolimits ^ n_ X(\mathcal{F}, \mathcal{G}) = 0$ for $n > d$, and

for $K, L \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ and $a \in \mathbf{Z}$ if $H^ i(K) = 0$ for $i < a + d$ and $H^ i(L) = 0$ for $i \geq a$ then $\mathop{\mathrm{Hom}}\nolimits _ X(K, L) = 0$.

**Proof.**
To prove (1) we use the spectral sequence

of Cohomology, Section 20.40. Let $x \in X$. We have

see Cohomology, Lemma 20.48.4 (this also uses that $\mathcal{F}$ is pseudo-coherent by Derived Categories of Schemes, Lemma 36.10.3). Set $d_ x = \dim (\mathcal{O}_{X, x})$. Since $\mathcal{O}_{X, x}$ is regular the ring $\mathcal{O}_{X, x}$ has global dimension $d_ x$, see Algebra, Proposition 10.110.1. Thus $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x)$ is zero for $q > d_ x$. It follows that the modules $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})$ have support of dimension at most $d - q$. Hence we have $H^ p(X, \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})) = 0$ for $p > d - q$ by Cohomology, Proposition 20.20.7. This proves (1).

Proof of (2). We may use induction on the number of nonzero cohomology sheaves of $K$ and $L$. The case where these numbers are $0, 1$ follows from (1). If the number of nonzero cohomology sheaves of $K$ is $> 1$, then we let $i \in \mathbf{Z}$ be minimal such that $H^ i(K)$ is nonzero. We obtain a distinguished triangle

(Derived Categories, Remark 13.12.4) and we get the vanishing of $\mathop{\mathrm{Hom}}\nolimits (K, L)$ from the vanishing of $\mathop{\mathrm{Hom}}\nolimits (H^ i(K)[-i], L)$ and $\mathop{\mathrm{Hom}}\nolimits (\tau _{\geq i + 1}K, L)$ by Derived Categories, Lemma 13.4.2. Simlarly if $L$ has more than one nonzero cohomology sheaf. $\square$

Lemma 57.10.5. Let $X$ be a regular Noetherian scheme of dimension $d < \infty $. Let $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ and $a \in \mathbf{Z}$. If $H^ i(K) = 0$ for $a < i < a + d$, then $K = \tau _{\leq a}K \oplus \tau _{\geq a + d}K$.

**Proof.**
We have $\tau _{\leq a}K = \tau _{\leq a + d - 1}K$ by the assumed vanishing of cohomology sheaves. By Derived Categories, Remark 13.12.4 we have a distinguished triangle

By Derived Categories, Lemma 13.4.11 it suffices to show that the morphism $\delta $ is zero. This follows from Lemma 57.10.4. $\square$

Lemma 57.10.6. Let $k$ be a field. Let $X$ be a quasi-compact separated smooth scheme over $k$. There exist finite locally free $\mathcal{O}_ X$-modules $\mathcal{E}$ and $\mathcal{G}$ such that

in $D(\mathcal{O}_{X \times X})$ where the notation is as in Derived Categories, Section 13.36.

**Proof.**
Recall that $X$ is regular by Varieties, Lemma 33.25.3. Hence $X$ has the resolution property by Derived Categories of Schemes, Lemma 36.36.8. Hence we may choose a resolution as in Lemma 57.10.3. Say $\dim (X) = d$. Since $X \times X$ is smooth over $k$ it is regular. Hence $X \times X$ is a regular Noetherian scheme with $\dim (X \times X) = 2d$. The object

of $D_{perf}(\mathcal{O}_{X \times X})$ has cohomology sheaves $\mathcal{O}_\Delta $ in degree $0$ and $\mathop{\mathrm{Ker}}(\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \mathcal{E}_{2d-1} \boxtimes \mathcal{G}_{2d-1})$ in degree $-2d$ and zero in all other degrees. Hence by Lemma 57.10.5 we see that $\mathcal{O}_\Delta $ is a summand of $K$ in $D_{perf}(\mathcal{O}_{X \times X})$. Clearly, the object $K$ is in

which finishes the proof. (The reader may consult Derived Categories, Lemmas 13.36.1 and 13.35.7 to see that our object is contained in this category.) $\square$

Lemma 57.10.7. Let $k$ be a field. Let $X$ be a scheme proper and smooth over $k$. Then $D_{perf}(\mathcal{O}_ X)$ has a strong generator.

**Proof.**
Using Lemma 57.10.6 choose finite locally free $\mathcal{O}_ X$-modules $\mathcal{E}$ and $\mathcal{G}$ such that $\mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle $ in $D(\mathcal{O}_{X \times X})$. We claim that $\mathcal{G}$ is a strong generator for $D_{perf}(\mathcal{O}_ X)$. With notation as in Derived Categories, Section 13.35 choose $m, n \geq 1$ such that

This is possible by Derived Categories, Lemma 13.36.2. Let $K$ be an object of $D_{perf}(\mathcal{O}_ X)$. Since $L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} -$ is an exact functor and since

we conclude from Derived Categories, Remark 13.35.5 that

Applying the exact functor $R\text{pr}_{2, *}$ and observing that

by Derived Categories of Schemes, Lemma 36.22.1 we conclude that

The equality follows from the discussion in Example 57.9.6. Since $K$ is perfect, there exist $a \leq b$ such that $H^ i(X, K)$ is nonzero only for $i \in [a, b]$. Since $X$ is proper, each $H^ i(X, K)$ is finite dimensional. We conclude that the right hand side is contained in $smd(add(\mathcal{G}[-m + a, m + b])^{\star n})$ which is itself contained in $\langle \mathcal{G} \rangle _ n$ by one of the references given above. This finishes the proof. $\square$

Lemma 57.10.8. Let $k$ be a field. Let $X$ be a proper smooth scheme over $k$. There exists integers $m, n \geq 1$ and a finite locally free $\mathcal{O}_ X$-module $\mathcal{G}$ such that every coherent $\mathcal{O}_ X$-module is contained in $smd(add(\mathcal{G}[-m, m])^{\star n})$ with notation as in Derived Categories, Section 13.35.

**Proof.**
In the proof of Lemma 57.10.7 we have shown that there exist $m', n \geq 1$ such that for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$,

for any $a \leq b$ such that $H^ i(X, \mathcal{F})$ is nonzero only for $i \in [a, b]$. Thus we can take $a = 0$ and $b = \dim (X)$. Taking $m = \max (m', m' + b)$ finishes the proof. $\square$

The following lemma is the boundedness result referred to in the title of this section.

Lemma 57.10.9. Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $\mathcal{A}$ be an abelian category. Let $H : D_{perf}(\mathcal{O}_ X) \to \mathcal{A}$ be a homological functor (Derived Categories, Definition 13.3.5) such that for all $K$ in $D_{perf}(\mathcal{O}_ X)$ the object $H^ i(K)$ is nonzero for only a finite number of $i \in \mathbf{Z}$. Then there exists an integer $m \geq 1$ such that $H^ i(\mathcal{F}) = 0$ for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and $i \not\in [-m, m]$. Similarly for cohomological functors.

**Proof.**
Combine Lemma 57.10.8 with Derived Categories, Lemma 13.35.8.
$\square$

Lemma 57.10.10. 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

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

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 57.10.5. The final statement is clear from this. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)