The Stacks project

56.10 Resolutions and bounds

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

Lemma 56.10.1. 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.

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

\[ \mathcal{J} = \mathop{\mathrm{Im}}(\text{pr}_1^*\mathcal{I} \otimes _{\mathcal{O}_{X \times _ R Y}} \text{pr}_2^*\mathcal{I}' \to \mathcal{O}_{X \times _ R Y}) \]

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 30.10.4. By assumption we can choose surjections $\mathcal{E} \to \mathcal{I}$ and $\mathcal{G} \to \mathcal{I}'$. These produce corresponding surjections

\[ \mathcal{E} \boxtimes \mathcal{G} \to \mathcal{J} \quad \text{and}\quad \mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{J}^ n \]

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

\[ \bigoplus \nolimits _{j = 1, \ldots , N} \mathcal{E}_ j^{\otimes n_ j} \boxtimes \mathcal{G}_ j^{\otimes n_ j} \to \mathcal{F} \]

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 56.10.2. 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 56.10.1 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 36.36.7). Then we apply Lemma 56.10.1 to $\mathcal{F}_{ij}$ and we pullback. $\square$

Lemma 56.10.3. 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

\[ \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0 \]

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 56.10.1. $\square$

Lemma 56.10.4. Let $X$ be a regular Noetherian scheme of dimension $d < \infty $. Then

  1. 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

  2. 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

\[ H^ p(X, \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})) \Rightarrow \mathop{\mathrm{Ext}}\nolimits ^{p + q}_ X(\mathcal{F}, \mathcal{G}) \]

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

\[ \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})_ x = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x) \]

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

\[ H^ i(K)[-i] \to K \to \tau _{\geq i + 1}K \]

(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 56.10.5. 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

\[ \tau _{\leq a}K \to K \to \tau _{\geq a + d}K \xrightarrow {\delta } (\tau _{\leq a}K)[1] \]

By Derived Categories, Lemma 13.4.11 it suffices to show that the morphism $\delta $ is zero. This follows from Lemma 56.10.4. $\square$

Lemma 56.10.6. 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

\[ \mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle \]

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 36.36.6. Hence we may choose a resolution as in Lemma 56.10.3. 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

\[ K = (\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0) \]

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 56.10.5 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

\[ \left\langle \bigoplus \nolimits _{i = 0, \ldots , 2d} \mathcal{E}_ i \boxtimes \mathcal{G}_ i \right\rangle \subset \left\langle \left(\bigoplus \nolimits _{i = 0, \ldots , 2d} \mathcal{E}_ i\right) \boxtimes \left(\bigoplus \nolimits _{i = 0, \ldots , 2d} \mathcal{G}_ i\right) \right\rangle \]

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 56.10.7. 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 56.10.6 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

\[ \mathcal{O}_\Delta \in smd(add(\mathcal{E} \boxtimes \mathcal{G}[-m, m])^{\star n}) \]

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

\[ L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} (\mathcal{E} \boxtimes \mathcal{G}) = (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \boxtimes \mathcal{G} \]

we conclude from Derived Categories, Remark 13.35.5 that

\[ L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} \mathcal{O}_\Delta \in smd(add( (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \boxtimes \mathcal{G}[-m, m])^{\star n}) \]

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

\[ R\text{pr}_{2, *} \left((K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \boxtimes \mathcal{G}\right) = R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \otimes _ k \mathcal{G} \]

by Derived Categories of Schemes, Lemma 36.22.1 we conclude that

\[ K = R\text{pr}_{2, *}(L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} \mathcal{O}_\Delta ) \in smd(add(R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{E}) \otimes _ k \mathcal{G}[-m, m])^{\star n}) \]

The equality follows from the discussion in Example 56.9.6. 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 56.10.8. 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 56.10.7 we have shown that there exist $m', n \geq 1$ such that for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$,

\[ \mathcal{F} \in smd(add(\mathcal{G}[-m' + a, m' + b])^{\star n}) \]

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 56.10.9. 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 56.10.8 with Derived Categories, Lemma 13.35.8. $\square$

Lemma 56.10.10. 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

  1. $H^ q(K_ i)$ is nonzero only for $q \leq m$,

  2. 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 15.99.2 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 56.10.5. The final statement is clear from this. $\square$


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