The Stacks project

37.69 Characterizing pseudo-coherent complexes, III

In this section we discuss characterizations of pseudo-coherent complexes in terms of cohomology. This is a continuation of Derived Categories of Schemes, Section 36.34. A basic tool will be to reduce to the case of projective space using a derived version of Chow's lemma, see Lemma 37.69.2.

Lemma 37.69.1. Consider a commutative diagram of schemes

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

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

Proof. The question is local, hence we may assume all schemes are affine (some details omitted). Observe that

\[ \xymatrix{ X \times _ S Y \ar[r] \ar[d] & X' \times _{S'} Y' \ar[d] \\ X \ar[r] & X' } \]

is cartesian with flat vertical arrows. Write $X = \mathop{\mathrm{Spec}}(A)$, $X' = \mathop{\mathrm{Spec}}(A')$, $X' \times _{S'} Y' = \mathop{\mathrm{Spec}}(B')$. Then $X \times _ S Y = \mathop{\mathrm{Spec}}(A \otimes _{A'} B')$. Write $Z' = \mathop{\mathrm{Spec}}(C')$. We have to show

\[ \text{Tor}_ p^{B'}(A \otimes _{A'} B', C') = 0, \quad \text{for } p > 0 \]

Since $A' \to B'$ is flat we have $A \otimes _{A'} B' = A \otimes _{A'}^\mathbf {L} B'$. Hence

\[ (A \otimes _{A'} B') \otimes _{B'}^\mathbf {L} C' = (A \otimes _{A'}^\mathbf {L} B') \otimes _{B'}^\mathbf {L} C' = A \otimes _{A'}^\mathbf {L} C' = A \otimes _{A'} C' \]

The second equality by More on Algebra, Lemma 15.60.5. The last equality because $A' \to C'$ is flat. This proves the lemma. $\square$

Lemma 37.69.2 (Derived Chow's lemma). Let $A$ be a ring. Let $X$ be a separated scheme of finite presentation over $A$. Let $x \in X$. Then there exist an open neighbourhood $U \subset X$ of $x$, an $n \geq 0$, an open $V \subset \mathbf{P}^ n_ A$, a closed subscheme $Z \subset X \times _ A \mathbf{P}^ n_ A$, a point $z \in Z$, 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. $b : Z \to X$ is an isomorphism over $U$ and $b(z) = x$,

  3. $c : Z \to \mathbf{P}^ n_ A$ is a closed immersion over $V$,

  4. $b^{-1}(U) = c^{-1}(V)$, in particular $c(z) \in V$,

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

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

Proof. We can find a finite type $\mathbf{Z}$-subalgebra $A' \subset A$ and a scheme $X'$ separated and of finite presentation over $A'$ whose base change to $A$ is $X$. See Limits, Lemmas 32.10.1 and 32.8.6. Let $x' \in X'$ be the image of $x$. If we can prove the lemma for $x' \in X'/A'$, then the lemma follows for $x \in X/A$. Namely, if $U', n', V', Z', z', E'$ provide the solution for $x' \in X'/A'$, then we can let $U \subset X$ be the inverse image of $U'$, let $n = n'$, let $V \subset \mathbf{P}^ n_ A$ be the inverse image of $V'$, let $Z \subset X \times \mathbf{P}^ n$ be the scheme theoretic inverse image of $Z'$, let $z \in Z$ be the unique point mapping to $x$, and let $E$ be the derived pullback of $E'$. Observe that $E$ is pseudo-coherent by Cohomology, Lemma 20.47.3. It only remains to check (5). To see this set $W = b^{-1}(U) = c^{-1}(V)$ and $W' = (b')^{-1}(U) = (c')^{-1}(V')$ 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 37.69.1 the schemes $X \times _ A V$ and $W'$ are Tor independent over $X' \times _{A'} V'$. Hence the derived pullback of $(b', c')_*\mathcal{O}_{W'}$ to $X \times _ A V$ is $(b, c)_*\mathcal{O}_ W$ by Derived Categories of Schemes, Lemma 36.22.5. This also uses that $R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}$ because $(b', c')$ is a closed immersion and simiarly 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 affine open neighbourhood $U \subset X$ of $x$. 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

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

Since the projection $X \times \mathbf{P}^ n \to X$ is separated, we conclude from Morphisms, Lemma 29.6.8 that $b : Z \to X$ is an isomorphism over $U$. Let $z \in Z$ be the unique point lying over $x$.

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 the same lemma we used above. Choose $V \subset \mathbf{P}^ n_ A$ open with $V \cap Y = j(U)$. Then we see that (3) and (4) hold.

Because $A$ is Noetherian we see that $X$ and $X \times _ A \mathbf{P}^ n_ A$ are Noetherian schemes. Hence we can take $E = (b, c)_*\mathcal{O}_ Z$ in this case, see Derived Categories of Schemes, Lemma 36.10.3. This finishes the proof. $\square$

Lemma 37.69.3. Let $A$, $x \in X$, and $U, n, V, Z, z, E$ be as in Lemma 37.69.2. For any $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have

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

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 $U \to V$ is the finitely presented closed immersion $c \circ (b|_ U)^{-1}$.

Proof. Since $b^{-1}(U) = c^{-1}(V)$ and since $c$ is a closed immersion over $V$, we see that $c \circ (b|_ U)^{-1}$ is a closed immersion. It is of finite presentation because $U$ and $V$ are of finite presentation over $A$, see Morphisms, Lemma 29.21.11. 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. Set $W = b^{-1}(U) = c^{-1}(V)$ and denote $i : W \to X \times _ A V$ the closed immersion $i = (b, c)|_ W$. 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 Schemes, Lemma 36.22.1 we can rewrite this as

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

which is what we want. $\square$

Lemma 37.69.4. Let $A$ be a ring. Let $X$ be a scheme 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 $A$.

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 a neighbourhood of $x$ and this will prove the lemma.

Choose $U, n, V, Z, z, E$ as in Lemma 37.69.2. 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 Schemes, Lemma 36.22.1. By Derived Categories of Schemes, Lemma 36.30.5 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 37.69.3 we have

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

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

Lemma 37.69.5. Let $A$ be a ring. Let $X$ be a scheme 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 37.69.4, 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 Schemes, Proposition 36.40.5 it follows that $K \in D^-_\mathit{QCoh}(\mathcal{O}_ X)$. Now the result follows by Derived Categories of Schemes, Lemma 36.34.3. $\square$

Lemma 37.69.6. Let $A$ be a ring. Let $X$ be a scheme separated, of finite presentation, and flat over $A$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X).$ If $R \Gamma (X, E \otimes ^\mathbf {L} K)$ is perfect in $D(A)$ for every perfect $E \in D(\mathcal{O}_ X)$, then $K$ is $\mathop{\mathrm{Spec}}(A)$-perfect.

Proof. By Lemma 37.69.5, $K$ is pseudo-coherent relative to $A$. By Lemma 37.59.18, $K$ is pseudo-coherent in $D( \mathcal{O}_ X)$. By Derived Categories of Schemes, Proposition 36.40.6 we see that $K$ is in $D^-(\mathcal{O}_ X)$. Let $\mathfrak {p}$ be a prime ideal of $A$ and denote $i : Y \to X$ the inclusion of the scheme theoretic fibre over $\mathfrak {p}$, i.e., $Y$ is a scheme over $\kappa (\mathfrak p)$. By Derived Categories of Schemes, Lemma 36.35.13, we will be done if we can show $Li^*(K)$ is bounded below. Let $G \in D_{perf} (\mathcal{O}_ X)$ be a perfect complex which generates $D_\mathit{QCoh}(\mathcal{O}_ X)$, see Derived Categories of Schemes, Theorem 36.15.3. We have

\begin{align*} R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ Y}(Li^*(G), Li^*(K)) & = R\Gamma (Y, Li^*(G ^\vee \otimes ^\mathbf {L} K)) \\ & = R\Gamma (X, G^\vee \otimes ^{\mathbf{L}} K) \otimes ^\mathbf {L}_ A \kappa (\mathfrak {p}) \end{align*}

The first equality uses that $Li^*$ preserves perfect objects and duals and Cohomology, Lemma 20.50.5; we omit some details. The second equality follows from Derived Categories of Schemes, Lemma 36.22.5 as $X$ is flat over $A$. It follows from our hypothesis that this is a perfect object of $D(\kappa (\mathfrak {p}))$. The object $Li^*(G) \in D_{perf}(\mathcal{O}_ Y)$ generates $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Derived Categories of Schemes, Remark 36.16.4. Hence Derived Categories of Schemes, Proposition 36.40.6 now implies that $Li^*(K)$ is bounded below and we win. $\square$

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