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
Z \to X \times _ A \mathbf{P}^ n_ A is of finite presentation,
b : Z \to X is an isomorphism over U and b(z) = x,
c : Z \to \mathbf{P}^ n_ A is a closed immersion over V,
b^{-1}(U) = c^{-1}(V), in particular c(z) \in V,
E|_{X \times _ A V} \cong (b, c)_*\mathcal{O}_ Z|_{X \times _ A V},
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 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 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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