The Stacks project

36.25 A projection formula for Ext

Lemma 36.25.3 (or similar results in the literature) is sometimes used to verify one of Artin's criteria for Quot functors, Hilbert schemes, and other moduli problems. Suppose that $f : X \to S$ is a proper, flat, finitely presented morphism of schemes and $E \in D(\mathcal{O}_ X)$ is perfect. Here the lemma says

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(E, f^*\mathcal{F}) = \mathop{\mathrm{Ext}}\nolimits ^ i_ S((Rf_*E^\vee )^\vee , \mathcal{F}) \]

for $\mathcal{F}$ quasi-coherent on $S$. Writing it this way makes it look like a projection formula for Ext and indeed the result follows rather easily from Lemma 36.21.1.

Lemma 36.25.1. Assumptions and notation as in Lemma 36.24.2. Then there are functorial isomorphisms

\[ H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \longrightarrow H^ i(X, E \otimes _{\mathcal{O}_ X}^\mathbf {L} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) \]

for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof).

Proof. We have

\[ \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F} = \mathcal{G}^\bullet \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F} \]

the first equality by Cohomology, Lemma 20.27.4, the second as $\mathcal{G}^ n$ is a flat $f^{-1}\mathcal{O}_ S$-module, and the third by definition of pullbacks. Hence we obtain

\begin{align*} H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) & = H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F}) \\ & = H^ i(S, Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*\mathcal{F})) \\ & = H^ i(S, Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet ) \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \\ & = H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \end{align*}

The first equality by the above, the second by Leray (Cohomology, Lemma 20.13.1), and the third equality by Lemma 36.21.1. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of quasi-coherent $\mathcal{O}_ S$-modules, the isomorphisms fit into commutative diagrams

\[ \xymatrix{ H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_3) \ar[r] \ar[d]_\delta & H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3)) \ar[d]^\delta \\ H^{i + 1}(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1) \ar[r] & H^{i + 1}(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1)) } \]

where the boundary maps come from the distinguished triangle

\[ K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_2 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_3 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1[1] \]

and the distinguished triangle in $D(\mathcal{O}_ X)$ associated to the short exact sequence

\[ 0 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3 \to 0 \]

of complexes of $\mathcal{O}_ X$-modules. This sequence is exact because $\mathcal{G}^ n$ is flat over $S$. We omit the verification of the commutativity of the displayed diagram. $\square$

Lemma 36.25.2. Assumptions and notation as in Lemma 36.24.3. Then there are functorial isomorphisms

\[ H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof).

Proof. As in the proof of Lemma 36.24.3 let $E^\vee $ be the dual perfect complex and recall that $K = Rf_*(E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet )$. Since we also have

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) = H^ i(X, E^\vee \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) \]

by construction of $E^\vee $, the existence of the isomorphisms follows from Lemma 36.25.1 applied to $E^\vee $ and $\mathcal{G}^\bullet $. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ then the isomorphisms fit into commutative diagrams

\[ \xymatrix{ H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_3) \ar[r] \ar[d]_\delta & \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3) \ar[d]^\delta \\ H^{i + 1}(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1) \ar[r] & \mathop{\mathrm{Ext}}\nolimits ^{i + 1}_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1) } \]

where the boundary maps come from the distinguished triangle

\[ K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_2 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_3 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1[1] \]

and the distinguished triangle in $D(\mathcal{O}_ X)$ associated to the short exact sequence

\[ 0 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3 \to 0 \]

of complexes. This sequence is exact because $\mathcal{G}$ is flat over $S$. We omit the verification of the commutativity of the displayed diagram. $\square$

Lemma 36.25.3. Let $f : X \to S$ be a morphism of schemes, $E \in D(\mathcal{O}_ X)$ and $\mathcal{G}^\bullet $ a complex of $\mathcal{O}_ X$-modules. Assume

  1. $S$ is Noetherian,

  2. $f$ is locally of finite type,

  3. $E \in D^-_{\textit{Coh}}(\mathcal{O}_ X)$,

  4. $\mathcal{G}^\bullet $ is a bounded complex of coherent $\mathcal{O}_ X$-modules flat over $S$ with support proper over $S$.

Then the following two statements are true

  1. for every $m \in \mathbf{Z}$ there exists a perfect object $K$ of $D(\mathcal{O}_ S)$ and functorial maps

    \[ \alpha ^ i_\mathcal {F} : \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \longrightarrow H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \]

    for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof) such that $\alpha ^ i_\mathcal {F}$ is an isomorphism for $i \leq m$

  2. there exists a pseudo-coherent $L \in D(\mathcal{O}_ S)$ and functorial isomorphisms

    \[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

    for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps.

Proof. Proof of (A). Suppose $\mathcal{G}^ i$ is nonzero only for $i \in [a, b]$. We may replace $X$ by a quasi-compact open neighbourhood of the union of the supports of $\mathcal{G}^ i$. Hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, -m - 1 + a)$ (possible by Theorem 36.13.6). Then the induced map

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(P, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

is an isomorphism for $i \leq m$. Namely, the kernel, resp. cokernel of this map is a quotient, resp. submodule of

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(C, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \quad \text{resp.}\quad \mathop{\mathrm{Ext}}\nolimits ^{i + 1}_{\mathcal{O}_ X}(C, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

where $C$ is the cone of $P \to E$. Since $C$ has vanishing cohomology sheaves in degrees $\geq -m - 1 + a$ these $\mathop{\mathrm{Ext}}\nolimits $-groups are zero for $i \leq m + 1$ by Derived Categories, Lemma 13.27.3. This reduces us to the case that $E$ is a perfect complex which is Lemma 36.25.2. The statement on boundaries is explained in the proof of Lemma 36.25.2.

Proof of (B). As in the proof of (A) we may assume $X$ is Noetherian. Observe that $E$ is pseudo-coherent by Lemma 36.9.3. By Lemma 36.18.1 we can write $E = \text{hocolim} E_ n$ with $E_ n$ perfect and $E_ n \to E$ inducing an isomorphism on truncations $\tau _{\geq -n}$. Let $E_ n^\vee $ be the dual perfect complex (Cohomology, Lemma 20.46.4). We obtain an inverse system $\ldots \to E_3^\vee \to E_2^\vee \to E_1^\vee $ of perfect objects. This in turn gives rise to an inverse system

\[ \ldots \to K_3 \to K_2 \to K_1\quad \text{with}\quad K_ n = Rf_*(E_ n^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]

perfect on $S$, see Lemma 36.24.2. By Lemma 36.25.2 and its proof and by the arguments in the previous paragraph (with $P = E_ n$) for any quasi-coherent $\mathcal{F}$ on $S$ we have functorial canonical maps

\[ \xymatrix{ & \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \ar[ld] \ar[rd] \\ H^ i(S, K_{n + 1} \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{F}) \ar[rr] & & H^ i(S, K_ n \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{F}) } \]

which are isomorphisms for $i \leq n + a$. Let $L_ n = K_ n^\vee $ be the dual perfect complex. Then we see that $L_1 \to L_2 \to L_3 \to \ldots $ is a system of perfect objects in $D(\mathcal{O}_ S)$ such that for any quasi-coherent $\mathcal{F}$ on $S$ the maps

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L_{n + 1}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L_ n, \mathcal{F}) \]

are isomorphisms for $i \leq n + a - 1$. This implies that $L_ n \to L_{n + 1}$ induces an isomorphism on truncations $\tau _{\geq -n - a + 2}$ (hint: take cone of $L_ n \to L_{n + 1}$ and look at its last nonvanishing cohomology sheaf). Thus $L = \text{hocolim} L_ n$ is pseudo-coherent, see Lemma 36.18.1. The mapping property of homotopy colimits gives that $\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L, \mathcal{F}) = \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L_ n, \mathcal{F})$ for $i \leq n + a - 3$ which finishes the proof. $\square$

Remark 36.25.4. The pseudo-coherent complex $L$ of part (B) of Lemma 36.25.3 is canonically associated to the situation. For example, formation of $L$ as in (B) is compatible with base change. In other words, given a cartesian diagram

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

of schemes we have canonical functorial isomorphisms

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{S'}}(Lg^*L, \mathcal{F}') \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(L(g')^*E, (g')^*\mathcal{G}^\bullet \otimes _{\mathcal{O}_{X'}} (f')^*\mathcal{F}') \]

for $\mathcal{F}'$ quasi-coherent on $S'$. Obsere that we do not use derived pullback on $\mathcal{G}^\bullet $ on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof.


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