The Stacks project

76.51 Characterizing pseudo-coherent complexes, II

In this section we discuss a characterization of pseudo-coherent complexes in terms of cohomology. Earlier material on pseudo-coherent complexes on algebraic spaces may be found in Derived Categories of Spaces, Section 75.13 and in Derived Categories of Spaces, Section 75.18. The analogue of this section for schemes is More on Morphisms, Section 37.69. A basic tool will be to reduce to the case of projective space using a derived version of Chow's lemma, see Lemma 76.51.2.

Lemma 76.51.1. Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces

\[ \xymatrix{ Z' \ar[d] \ar[r] & Y' \ar[d] \\ X' \ar[r] & B' } \]

over $S$. Let $B \to B'$ be a morphism. Denote by $X$ and $Y$ the base changes of $X'$ and $Y'$ to $B$. Assume $Y' \to B'$ and $Z' \to X'$ are flat. Then $X \times _ B Y$ and $Z'$ are Tor independent over $X' \times _{B'} Y'$.

Proof. By Derived Categories of Spaces, Lemma 75.20.3 we may check tor independence étale locally on $X \times _ B Y$ and $Z'$. This1 reduces the lemma to the case of schemes which is More on Morphisms, Lemma 37.69.1. $\square$

Lemma 76.51.2 (Derived Chow's lemma). Let $A$ be a ring. Let $X$ be a separated algebraic space of finite presentation over $A$. Let $x \in |X|$. Then there exist an $n \geq 0$, a closed subspace $Z \subset X \times _ A \mathbf{P}^ n_ A$, a point $z \in |Z|$, an open $V \subset \mathbf{P}^ n_ A$, and an object $E$ in $D(\mathcal{O}_{X \times _ A \mathbf{P}^ n_ A})$ such that

  1. $Z \to X \times _ A \mathbf{P}^ n_ A$ is of finite presentation,

  2. $c : Z \to \mathbf{P}^ n_ A$ is a closed immersion over $V$, set $W = c^{-1}(V)$,

  3. the restriction of $b : Z \to X$ to $W$ is étale, $z \in W$, and $b(z) = x$,

  4. $E|_{X \times _ A V} \cong (b, c)_*\mathcal{O}_ Z|_{X \times _ A V}$,

  5. $E$ is pseudo-coherent and supported on $Z$.

Proof. We can find a finite type $\mathbf{Z}$-subalgebra $A' \subset A$ and an algebraic space $X'$ separated and of finite presentation over $A'$ whose base change to $A$ is $X$. See Limits of Spaces, Lemmas 70.7.1 and 70.6.9. Let $x' \in |X'|$ be the image of $x$. If we can prove the lemma for $(X'/A', x')$, then the lemma follows for $(X/A, x)$. Namely, if $n', Z', z', V', E'$ provide the solution for $(X'/A', x')$, then we can let $n = n'$, let $Z \subset X \times \mathbf{P}^ n$ be the inverse image of $Z'$, let $z \in Z$ be the unique point mapping to $x$, let $V \subset \mathbf{P}^ n_ A$ be the inverse image of $V'$, and let $E$ be the derived pullback of $E'$. Observe that $E$ is pseudo-coherent by Cohomology on Sites, Lemma 21.45.3. It only remains to check (5). To see this set $W = c^{-1}(V)$ and $W' = (c')^{-1}(V')$ and consider the cartesian square

\[ \xymatrix{ W \ar[d]_{(b, c)} \ar[r] & W' \ar[d]^{(b', c')} \\ X \times _ A V \ar[r] & X' \times _{A'} V' } \]

By Lemma 76.51.1 $X \times _ A V$ and $W'$ are tor-independent over $X' \times _{A'} V'$. Thus the derived pullback of $(b', c')_*\mathcal{O}_{W'}$ to $X \times _ A V$ is $(b, c)_*\mathcal{O}_ W$ by Derived Categories of Spaces, Lemma 75.20.4. This also uses that $R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}$ because $(b', c')$ is a closed immersion and similarly for $(b, c)_*\mathcal{O}_ Z$. Since $E'|_{U' \times _{A'} V'} = (b', c')_*\mathcal{O}_{W'}$ we obtain $E|_{U \times _ A V} = (b, c)_*\mathcal{O}_ W$ and (5) holds. This reduces us to the situation described in the next paragraph.

Assume $A$ is of finite type over $\mathbf{Z}$. Choose an étale morphism $U \to X$ where $U$ is an affine scheme and a point $u \in U$ mapping to $x$. Then $U$ is of finite type over $A$. Choose a closed immersion $U \to \mathbf{A}^ n_ A$ and denote $j : U \to \mathbf{P}^ n_ A$ the immersion we get by composing with the open immersion $\mathbf{A}^ n_ A \to \mathbf{P}^ n_ A$. Let $Z$ be the scheme theoretic closure of

\[ (\text{id}_ U, j) : U \longrightarrow X \times _ A \mathbf{P}^ n_ A \]

Let $z \in Z$ be the image of $u$. Let $Y \subset \mathbf{P}^ n_ A$ be the scheme theoretic closure of $j$. Then it is clear that $Z \subset X \times _ A Y$ is the scheme theoretic closure of $(\text{id}_ U, j) : U \to X \times _ A Y$. As $X$ is separated, the morphism $X \times _ A Y \to Y$ is separated as well. Hence we see that $Z \to Y$ is an isomorphism over the open subscheme $j(U) \subset Y$ by Morphisms of Spaces, Lemma 67.16.7. Choose $V \subset \mathbf{P}^ n_ A$ open with $V \cap Y = j(U)$. Then we see that (2) holds, that $W = (\text{id}_ U, j)(U)$, and hence that (3) holds. Part (1) holds because $A$ is Noetherian.

Because $A$ is Noetherian we see that $X$ and $X \times _ A \mathbf{P}^ n_ A$ are Noetherian algebraic spaces. Hence we can take $E = (b, c)_*\mathcal{O}_ Z$ in this case: (4) is clear and for (5) see Derived Categories of Spaces, Lemma 75.13.7. This finishes the proof. $\square$

Lemma 76.51.3. Let $X/A$, $x \in |X|$, and $n, Z, z, V, E$ be as in Lemma 76.51.2. For any $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have

\[ Rq_*(Lp^*K \otimes ^\mathbf {L} E)|_ V = R(W \to V)_*K|_ W \]

where $p : X \times _ A \mathbf{P}^ n_ A \to X$ and $q : X \times _ A \mathbf{P}^ n_ A \to \mathbf{P}^ n_ A$ are the projections and where the morphism $W \to V$ is the finitely presented closed immersion $c|_ W : W \to V$.

Proof. Since $W = c^{-1}(V)$ and since $c$ is a closed immersion over $V$, we see that $c|_ W$ is a closed immersion. It is of finite presentation because $W$ and $V$ are of finite presentation over $A$, see Morphisms of Spaces, Lemma 67.28.9. First we have

\[ Rq_*(Lp^*K \otimes ^\mathbf {L} E)|_ V = Rq'_*\left((Lp^*K \otimes ^\mathbf {L} E)|_{X \times _ A V}\right) \]

where $q' : X \times _ A V \to V$ is the projection because formation of total direct image commutes with localization. Denote $i = (b, c)|_ W : W \to X \times _ A V$ the given closed immersion. Then

\[ Rq'_*\left((Lp^*K \otimes ^\mathbf {L} E)|_{X \times _ A V}\right) = Rq'_*(Lp^*K|_{X \times _ A V} \otimes ^\mathbf {L} i_*\mathcal{O}_ W) \]

by property (5). Since $i$ is a closed immersion we have $i_*\mathcal{O}_ W = Ri_*\mathcal{O}_ W$. Using Derived Categories of Spaces, Lemma 75.20.1 we can rewrite this as

\[ Rq'_* Ri_* Li^* Lp^*K|_{X \times _ A V} = R(q' \circ i)_* Lb^*K|_ W = R(W \to V)_* K|_ W \]

which is what we want. (Note that restricting to $W$ and derived pulling back via $W \to X$ is the same thing as $W$ is étale over $X$.) $\square$

Lemma 76.51.4. Let $A$ be a ring. Let $X$ be an algebraic space separated and of finite presentation over $A$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$. If $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_ X)$, then $K$ is pseudo-coherent relative to $A$ (Definition 76.45.3).

Proof. Assume $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Let $x \in |X|$. We will show that $K$ is pseudo-coherent relative to $A$ in an étale neighbourhood of $x$. This will prove the lemma by our definition of relative pseudo-coherence.

Choose $n, Z, z, V, E$ as in Lemma 76.51.2. Denote $p : X \times \mathbf{P}^ n \to X$ and $q : X \times \mathbf{P}^ n \to \mathbf{P}^ n_ A$ the projections. Then for any $i \in \mathbf{Z}$ we have

\begin{align*} & R\Gamma (\mathbf{P}^ n_ A, Rq_*(Lp^*K \otimes ^\mathbf {L} E) \otimes ^\mathbf {L} \mathcal{O}_{\mathbf{P}^ n_ A}(i)) \\ & = R\Gamma (X \times \mathbf{P}^ n, Lp^*K \otimes ^\mathbf {L} E \otimes ^\mathbf {L} Lq^*\mathcal{O}_{\mathbf{P}^ n_ A}(i)) \\ & = R\Gamma (X, K \otimes ^\mathbf {L} Rq_*(E \otimes ^\mathbf {L} Lq^*\mathcal{O}_{\mathbf{P}^ n_ A}(i))) \end{align*}

by Derived Categories of Spaces, Lemma 75.20.1. By Derived Categories of Spaces, Lemma 75.25.5 the complex $Rq_*(E \otimes ^\mathbf {L} Lq^*\mathcal{O}_{\mathbf{P}^ n_ A}(i))$ is pseudo-coherent on $X$. Hence the assumption tells us the expression in the displayed formula is a pseudo-coherent object of $D(A)$. By Derived Categories of Schemes, Lemma 36.34.2 we conclude that $Rq_*(Lp^*K \otimes ^\mathbf {L} E)$ is pseudo-coherent on $\mathbf{P}^ n_ A$. By Lemma 76.51.3 we have

\[ Rq_*(Lp^*K \otimes ^\mathbf {L} E)|_{X \times _ A V} = R(W \to V)_*K|_ W \]

Since $W \to V$ is a closed immersion into an open subscheme of $\mathbf{P}^ n_ A$ this means $K|_ W$ is pseudo-coherent relative to $A$ for example by More on Morphisms, Lemma 37.59.18. $\square$

Lemma 76.51.5. Let $A$ be a ring. Let $X$ be an algebraic space separated and of finite presentation over $A$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$. If $R \Gamma (X, E \otimes ^{\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every perfect $E \in D(\mathcal{O}_ X)$, then $K$ is pseudo-coherent relative to $A$.

Proof. In view of Lemma 76.51.4, it suffices to show $R \Gamma (X, E \otimes ^{\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E \in D(\mathcal{O}_ X)$. By Derived Categories of Spaces, Proposition 75.29.3 it follows that $K \in D^-_\mathit{QCoh}(\mathcal{O}_ X)$. Now the result follows by Derived Categories of Spaces, Lemma 75.25.7. $\square$

[1] Here is the argument in more detail. Choose a surjective étale morphism $W' \to B'$ with $W'$ a scheme. Choose a surjective étale morphism $W \to B \times _{B'} W'$ with $W$ a scheme. Choose a surjective étale morphism $U' \to X' \times _{B'} W'$ with $U'$ a scheme. Choose a surjective étale morphism $V' \to Y' \times _{B'} W'$ with $V'$ a scheme. Observe that $U' \times _{W'} V' \to X' \times _{B'} Y'$ is surjective étale. Choose a surjective étale morphism $T' \to Z' \times _{X' \times _{B'} Y'} U' \times _{W'} V'$ with $T'$ a scheme. Denote $U$ and $V$ the base changes of $U'$ and $V'$ to $W$. Then the lemma says that $X \times _ B Y$ and $Z'$ are Tor independent over $X' \times _{B'} Y'$ as algebraic spaces if and only if $U \times _ W V$ and $T'$ are Tor independent over $U' \times _{W'} V'$ as schemes. Thus it suffices to prove the lemma for the square with corners $T', U', V', W'$ and base change by $W \to W'$. The flatness of $Y' \to B'$ and $Z' \to X'$ implies flatness of $V' \to W'$ and $T' \to U'$.

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