Lemma 99.3.8. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $K$ be a pseudo-coherent object of $D(\mathcal{O}_ B)$.

1. If for all $g : T \to B$ in $(\mathit{Sch}/B)$ the cohomology sheaf $H^{-1}(Lg^*K)$ is zero, then the functor

$(\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, H^0(Lg^*K))$

is an algebraic space affine and of finite presentation over $B$.

2. If for all $g : T \to B$ in $(\mathit{Sch}/B)$ the cohomology sheaves $H^ i(Lg^*K)$ are zero for $i < 0$, then $K$ is perfect, $K$ locally has tor amplitude in $[0, b]$, and the functor

$(\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets},\quad (g : T \to B) \longmapsto H^0(T, Lg^*K)$

is an algebraic space affine and of finite presentation over $B$.

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

$H^0(T_ i, H^0(Lg_ i^*K)) = H^0(T_ i, H^0(h_ i^*Lg^*K)) = H^0(T, h_ i^*H^0(Lg^*K))$

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

$H^0(T, \mathcal{F}) = \mathop{\mathrm{Ker}}( \prod H^0(T_ i, h_ i^*\mathcal{F}) \to \prod H^0(T_ i \times _ T T_ j, (T_ i \times _ T T_ j \to T)^*\mathcal{F}))$

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

$M^\bullet = \tau _{\geq 0}M^\bullet \oplus \tau _{\leq - 1}M^\bullet$

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

$H^0(T, H^0(Lg^*K)) = H^0(T, H^0(Lg^*\tau _{\geq 0}K))$

(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

$M \to M^1 \to M^2 \to \ldots \to M^ d \to 0 \ldots$

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

$g^*\widetilde{M} \to g^*\widetilde{M^1} \to \ldots$

Thus it suffices to show that the functor

$(g : T \to B) \longmapsto \mathop{\mathrm{Ker}}( \Gamma (T,g^*\widetilde{M}) \to \Gamma (T, g^*(\widetilde{M^1}) )$

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

$(g : T \to B) \longmapsto \{ (f_1, \ldots , f_ n) \in \Gamma (T, \mathcal{O}_ T) \mid \sum g^\sharp (a_{ij})f_ i = 0,\ j = 1, \ldots , m\}$

Clearly this is representable by the affine scheme

$\mathop{\mathrm{Spec}}\left(A[x_1, \ldots , x_ n]/(\sum a_{ij}x_ i; j = 1, \ldots , m)\right)$

and the lemma has been proved. $\square$

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