The Stacks project

Proof. Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc covering of schemes over $B$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and $\mathcal{F}_ i = u_{T_ i}$ and $\mathcal{G}_ i = \mathcal{G}_{T_ i}$. Note that $\{ X_ i \to X_ T\} _{i \in I}$ is an fpqc covering of $X_ T$, see Topologies on Spaces, Lemma 73.9.3. Thus a family of maps $u_ i : \mathcal{F}_ i \to \mathcal{G}_ i$ such that $u_ i$ and $u_ j$ restrict to the same map on $X_{T_ i \times _ T T_ j}$ comes from a unique map $u : \mathcal{F}_ T \to \mathcal{G}_ T$ by descent (Descent on Spaces, Proposition 74.4.1). $\square$

Proof. Choose a scheme $U$ and a surjective étale morphism $p : U \to T$. Let $R = U \times _ T U$ with projections $t, s : R \to U$.

Let $v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})$ be a natural transformation. Then $v(p)$ corresponds to a pair $(h_ U, u_ U)$ over $U$. As $v$ is a transformation of functors we see that the pullbacks of $(h_ U, u_ U)$ by $s$ and $t$ agree. Since $T = U/R$ (Spaces, Lemma 65.9.1), we obtain a morphism $h : T \to B$ such that $h_ U = h \circ p$. Then $\mathcal{F}_ U$ is the pullback of $\mathcal{F}_ T$ to $X_ U$ and similarly for $\mathcal{G}_ U$. Hence $u_ U$ descends to a $\mathcal{O}_{X_ T}$-module map $u : \mathcal{F}_ T \to \mathcal{G}_ T$ by Descent on Spaces, Proposition 74.4.1.

Conversely, let $(h, u)$ be a pair over $T$. Then we get a natural transformation $v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})$ by sending a morphism $a : T' \to T$ where $T'$ is a scheme to $(h \circ a, a^*u)$. We omit the verification that the construction of this and the previous paragraph are mutually inverse. $\square$

Proof. Let $u_ i : \mathcal{F}_{i, T} \to \mathcal{G}_{i, T}$ be an element in the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$. Since the base change of an fppf covering is an fppf covering (Topologies on Spaces, Lemma 73.7.3) we see that $\{ X_{i, T} \to X_ T\} _{i \in I}$ and $\{ X_{ijk, T} \to X_{i, T} \times _{X_ T} X_{j, T}\} $ are fppf coverings. Applying Descent on Spaces, Proposition 74.4.1 we first conclude that $u_ i$ and $u_ j$ restrict to the same morphism over $X_{i, T} \times _{X_ T} X_{j, T}$, whereupon a second application shows that there is a unique morphism $u : \mathcal{F}_ T \to \mathcal{G}_ T$ restricting to $u_ i$ for each $i$. This finishes the proof. $\square$

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $B$-schemes. We have to show that

Pick $0 \in I$. We may replace $B$ by $T_0$, $X$ by $X_{T_0}$, $\mathcal{F}$ by $\mathcal{F}_{T_0}$, $\mathcal{G}$ by $\mathcal{G}_{T_0}$, and $I$ by $\{ i \in I \mid i \geq 0\} $. See Remark 99.3.4. Thus we may assume $B = \mathop{\mathrm{Spec}}(R)$ is affine.

When $B$ is affine, then $X$ is quasi-compact and quasi-separated. Choose a surjective étale morphism $U \to X$ where $U$ is an affine scheme (Properties of Spaces, Lemma 66.6.3). Since $X$ is quasi-separated, the scheme $U \times _ X U$ is quasi-compact and we may choose a surjective étale morphism $V \to U \times _ X U$ where $V$ is an affine scheme. Applying Lemma 99.3.5 we see that $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is the equalizer of two maps between

This reduces us to the case that $X$ is affine.

In the affine case the statement of the lemma reduces to the following problem: Given a ring map $R \to A$, two $A$-modules $M$, $N$ and a directed system of $R$-algebras $C = \mathop{\mathrm{colim}}\nolimits C_ i$. When is it true that the map

is bijective? By Algebra, Lemma 10.127.5 this holds if $M \otimes _ R C$ is of finite presentation over $A \otimes _ R C$, i.e., when $M$ is of finite presentation over $A$. $\square$

Proof. Let $g : T \to B$ be a morphism where $T$ is a scheme. Denote $i_ T : X'_ T \to X_ T$ the base change of $i$. Denote $h : X_ T \to X$ and $h' : X'_ T \to X'$ the projections. Observe that $(h')^*i^*\mathcal{F} = i_ T^*h^*\mathcal{F}$. As a closed immersion is affine (Morphisms of Spaces, Lemma 67.20.6) we have $h^*i_*\mathcal{G} = i_{T, *}(h')^*\mathcal{G}$ by Cohomology of Spaces, Lemma 69.11.1. Thus we have

as desired. The middle equality follows from the adjointness of the functors $i_{T, *}$ and $i_ T^*$. $\square$

Proof. Under the assumptions of (2) we have $H^0(T, Lg^*K) = H^0(T, H^0(Lg^*K))$. Let us prove that the rule $T \mapsto H^0(T, H^0(Lg^*K))$ satisfies the sheaf property for the fppf topology. To do this assume we have an fppf covering $\{ h_ i : T_ i \to T\} $ of a scheme $g : T \to B$ over $B$. Set $g_ i = g \circ h_ i$. Note that since $h_ i$ is flat, we have $Lh_ i^* = h_ i^*$ and $h_ i^*$ commutes with taking cohomology. Hence

Similarly for the pullback to $T_ i \times _ T T_ j$. Since $Lg^*K$ is a pseudo-coherent complex on $T$ (Cohomology on Sites, Lemma 21.45.3) the cohomology sheaf $\mathcal{F} = H^0(Lg^*K)$ is quasi-coherent (Derived Categories of Spaces, Lemma 75.13.6). Hence by Descent on Spaces, Proposition 74.4.1 we see that

In this way we see that the rules in (1) and (2) satisfy the sheaf property for fppf coverings. This means we may apply Bootstrap, Lemma 80.11.2 to see it suffices to prove the representability étale locally on $B$. Moreover, we may check whether the end result is affine and of finite presentation étale locally on $B$, see Morphisms of Spaces, Lemmas 67.20.3 and 67.28.4. Hence we may assume that $B$ is an affine scheme.

Assume $B = \mathop{\mathrm{Spec}}(A)$ is an affine scheme. By the results of Derived Categories of Spaces, Lemmas 75.13.6, 75.4.2, and 75.13.2 we deduce that in the rest of the proof we may think of $K$ as a perfect object of the derived category of complexes of modules on $B$ in the Zariski topology. By Derived Categories of Schemes, Lemmas 36.10.1, 36.3.5, and 36.10.2 we can find a pseudo-coherent complex $M^\bullet $ of $A$-modules such that $K$ is the corresponding object of $D(\mathcal{O}_ B)$. Our assumption on pullbacks implies that $M^\bullet \otimes ^\mathbf {L}_ A \kappa (\mathfrak p)$ has vanishing $H^{-1}$ for all primes $\mathfrak p \subset A$. By More on Algebra, Lemma 15.76.4 we can write

with $\tau _{\geq 0}M^\bullet $ perfect with Tor amplitude in $[0, b]$ for some $b \geq 0$ (here we also have used More on Algebra, Lemmas 15.74.12 and 15.66.16). Note that in case (2) we also see that $\tau _{\leq - 1}M^\bullet = 0$ in $D(A)$ whence $M^\bullet $ and $K$ are perfect with tor amplitude in $[0, b]$. For any $B$-scheme $g : T \to B$ we have

(by the dual of Derived Categories, Lemma 13.16.1) hence we may replace $K$ by $\tau _{\geq 0}K$ and correspondingly $M^\bullet $ by $\tau _{\geq 0}M^\bullet $. In other words, we may assume $M^\bullet $ has tor amplitude in $[0, b]$.

Assume $M^\bullet $ has tor amplitude in $[0, b]$. We may assume $M^\bullet $ is a bounded above complex of finite free $A$-modules (by our definition of pseudo-coherent complexes, see More on Algebra, Definition 15.64.1 and the discussion following the definition). By More on Algebra, Lemma 15.66.2 we see that $M = \mathop{\mathrm{Coker}}(M^{- 1} \to M^0)$ is flat. By Algebra, Lemma 10.78.2 we see that $M$ is finite locally free. Hence $M^\bullet $ is quasi-isomorphic to

Note that this is a K-flat complex (Cohomology, Lemma 20.26.9), hence derived pullback of $K$ via a morphism $T \to B$ is computed by the complex

Thus it suffices to show that the functor

is representable by an affine scheme of finite presentation over $B$.

We may still replace $B$ by the members of an affine open covering in order to prove this last statement. Hence we may assume that $M$ is finite free (recall that $M^1$ is finite free to begin with). Write $M = A^{\oplus n}$ and $M^1 = A^{\oplus m}$. Let the map $M \to M^1$ be given by the $m \times n$ matrix $(a_{ij})$ with coefficients in $A$. Then $\widetilde{M} = \mathcal{O}_ B^{\oplus n}$ and $\widetilde{M^1} = \mathcal{O}_ B^{\oplus m}$. Thus the functor above is equal to the functor

Clearly this is representable by the affine scheme

and the lemma has been proved. $\square$

Proof. We may replace $X$ by a quasi-compact open neighbourhood of the support of $\mathcal{G}$, hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to \mathcal{F}$ by a perfect complex $P$ of the triple $(X, \mathcal{F}, -1)$, see Derived Categories of Spaces, Definition 75.14.1 and Theorem 75.14.7). Then the induced map

is an isomorphism because $P \to \mathcal{F}$ induces an isomorphism $H^0(P) \to \mathcal{F}$ and $H^ i(P) = 0$ for $i > 0$. Moreover, for any morphism $g : T \to B$ denote $h : X_ T = T \times _ B X \to X$ the projection and set $P_ T = Lh^*P$. Then it is equally true that

is an isomorphism, as $P_ T = Lh^*P \to Lh^*\mathcal{F} \to \mathcal{F}_ T$ induces an isomorphism $H^0(P_ T) \to \mathcal{F}_ T$ (because $h^*$ is right exact and $H^ i(P) = 0$ for $i > 0$). Thus it suffices to prove the result for the functor

By the Leray spectral sequence (see Cohomology on Sites, Remark 21.14.4) we have

where $f_ T : X_ T \to T$ is the base change of $f$. By Derived Categories of Spaces, Lemma 75.21.5 we have

By Derived Categories of Spaces, Lemma 75.22.3 the object $K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P, \mathcal{G})$ of $D(\mathcal{O}_ B)$ is perfect. This means we can apply Lemma 99.3.8 as long as we can prove that the cohomology sheaf $H^ i(Lg^*K)$ is $0$ for all $i < 0$ and $g : T \to B$ as above. This is clear from the last displayed formula as the cohomology sheaves of $Rf_{T, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)$ are zero in negative degrees due to the fact that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)$ has vanishing cohomology sheaves in negative degrees as $P_ T$ is perfect with vanishing cohomology sheaves in positive degrees. $\square$

Proof. By Lemma 99.3.2 the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ satisfies the sheaf property for fppf coverings. This mean we may¹ apply Bootstrap, Lemma 80.11.1 to check the representability étale locally on $B$. Moreover, we may check whether the end result is affine or of finite presentation étale locally on $B$, see Morphisms of Spaces, Lemmas 67.20.3 and 67.28.4. Hence we may assume that $B$ is an affine scheme.

Assume $B$ is an affine scheme. As $f$ is of finite presentation, it follows $X$ is quasi-compact and quasi-separated. Thus we can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a filtered colimit of $\mathcal{O}_ X$-modules of finite presentation (Limits of Spaces, Lemma 70.9.1). It is clear that

Hence if we can show that each $\mathit{Hom}(\mathcal{F}_ i, \mathcal{G})$ is representable by an affine scheme, then we see that the same thing holds for $\mathit{Hom}(\mathcal{F}, \mathcal{G})$. Use the material in Limits, Section 32.2 and Limits of Spaces, Section 70.4. Thus we may assume that $\mathcal{F}$ is of finite presentation.

Say $B = \mathop{\mathrm{Spec}}(R)$. Write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ with each $R_ i$ a finite type $\mathbf{Z}$-algebra. Set $B_ i = \mathop{\mathrm{Spec}}(R_ i)$. By the results of Limits of Spaces, Lemmas 70.7.1 and 70.7.2 we can find an $i$, a morphism of algebraic spaces $X_ i \to B_ i$, and finitely presented $\mathcal{O}_{X_ i}$-modules $\mathcal{F}_ i$ and $\mathcal{G}_ i$ such that the base change of $(X_ i, \mathcal{F}_ i, \mathcal{G}_ i)$ to $B$ recovers $(X, \mathcal{F}, \mathcal{G})$. By Limits of Spaces, Lemma 70.6.12 we may, after increasing $i$, assume that $\mathcal{G}_ i$ is flat over $B_ i$. By Limits of Spaces, Lemma 70.12.3 we may similarly assume the scheme theoretic support of $\mathcal{G}_ i$ is proper over $B_ i$. At this point we can apply Lemma 99.3.9 to see that $H_ i = \mathit{Hom}(\mathcal{F}_ i, \mathcal{G}_ i)$ is an algebraic space affine of finite presentation over $B_ i$. Pulling back to $B$ (using Remark 99.3.4) we see that $H_ i \times _{B_ i} B = \mathit{Hom}(\mathcal{F}, \mathcal{G})$ and we win. $\square$