The Stacks project

36.34 Characterizing pseudo-coherent complexes, II

This section is a continuation of Section 36.19. In this section we discuss characterizations of pseudo-coherent complexes in terms of cohomology. More results of this nature can be found in More on Morphisms, Section 37.63.

Lemma 36.34.1. Let $A$ be a ring. Let $R$ be a (possibly noncommutative) $A$-algebra which is finite free as an $A$-module. Then any object $M$ of $D(R)$ which is pseudo-coherent in $D(A)$ can be represented by a bounded above complex of finite free (right) $R$-modules.

Proof. Choose a complex $M^\bullet $ of right $R$-modules representing $M$. Since $M$ is pseudo-coherent we have $H^ i(M) = 0$ for large enough $i$. Let $m$ be the smallest index such that $H^ m(M)$ is nonzero. Then $H^ m(M)$ is a finite $A$-module by More on Algebra, Lemma 15.63.3. Thus we can choose a finite free $R$-module $F^ m$ and a map $F^ m \to M^ m$ such that $F^ m \to M^ m \to M^{m + 1}$ is zero and such that $F^ m \to H^ m(M)$ is surjective. Picture:

\[ \xymatrix{ & F^ m \ar[d]^\alpha \ar[r] & 0 \ar[d] \ar[r] & \ldots \\ M^{m - 1} \ar[r] & M^ m \ar[r] & M^{m + 1} \ar[r] & \ldots } \]

By descending induction on $n \leq m$ we are going to construct finite free $R$-modules $F^ i$ for $i \geq n$, differentials $d^ i : F^ i \to F^{i + 1}$ for $i \geq n$, maps $\alpha : F^ i \to K^ i$ compatible with differentials, such that (1) $H^ i(\alpha )$ is an isomorphism for $i > n$ and surjective for $i = n$, and (2) $F^ i = 0$ for $i > m$. Picture

\[ \xymatrix{ & F^ n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \ar[r] & F^ i \ar[d]^\alpha \ar[r] & 0 \ar[d] \ar[r] & \ldots \\ M^{n - 1} \ar[r] & M^ n \ar[r] & M^{n + 1} \ar[r] & \ldots \ar[r] & M^ i \ar[r] & M^{i + 1} \ar[r] & \ldots } \]

The base case is $n = m$ which we've done above. Induction step. Let $C^\bullet $ be the cone on $\alpha $ (Derived Categories, Definition 13.9.1). The long exact sequence of cohomology shows that $H^ i(C^\bullet ) = 0$ for $i \geq n$. Observe that $F^\bullet $ is pseudo-coherent as a complex of $A$-modules because $R$ is finite free as an $A$-module. Hence by More on Algebra, Lemma 15.63.2 we see that $C^\bullet $ is $(n - 1)$-pseudo-coherent as a complex of $A$-modules. By More on Algebra, Lemma 15.63.3 we see that $H^{n - 1}(C^\bullet )$ is a finite $A$-module. Choose a finite free $R$-module $F^{n - 1}$ and a map $\beta : F^{n - 1} \to C^{n - 1}$ such that the composition $F^{n - 1} \to C^{n - 1} \to C^ n$ is zero and such that $F^{n - 1}$ surjects onto $H^{n - 1}(C^\bullet )$. Since $C^{n - 1} = M^{n - 1} \oplus F^ n$ we can write $\beta = (\alpha ^{n - 1}, -d^{n - 1})$. The vanishing of the composition $F^{n - 1} \to C^{n - 1} \to C^ n$ implies these maps fit into a morphism of complexes

\[ \xymatrix{ & F^{n - 1} \ar[d]^{\alpha ^{n - 1}} \ar[r]_{d^{n - 1}} & F^ n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\ \ldots \ar[r] & M^{n - 1} \ar[r] & M^ n \ar[r] & M^{n + 1} \ar[r] & \ldots } \]

Moreover, these maps define a morphism of distinguished triangles

\[ \xymatrix{ (F^ n \to \ldots ) \ar[r] \ar[d] & (F^{n - 1} \to \ldots ) \ar[r] \ar[d] & F^{n - 1} \ar[r] \ar[d]_\beta & (F^ n \to \ldots )[1] \ar[d] \\ (F^ n \to \ldots ) \ar[r] & M^\bullet \ar[r] & C^\bullet \ar[r] & (F^ n \to \ldots )[1] } \]

Hence our choice of $\beta $ implies that the map of complexes $(F^{n - 1} \to \ldots ) \to M^\bullet $ induces an isomorphism on cohomology in degrees $\geq n$ and a surjection in degree $n - 1$. This finishes the proof of the lemma. $\square$

Lemma 36.34.2. Let $A$ be a ring. Let $n \geq 0$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_{\mathbf{P}^ n_ A})$. The following are equivalent

  1. $K$ is pseudo-coherent,

  2. $R\Gamma (\mathbf{P}^ n_ A, E \otimes ^\mathbf {L} K)$ is a pseudo-coherent object of $D(A)$ for each pseudo-coherent object $E$ of $D(\mathcal{O}_{\mathbf{P}^ n_ A})$,

  3. $R\Gamma (\mathbf{P}^ n_ A, E \otimes ^\mathbf {L} K)$ is a pseudo-coherent object of $D(A)$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^ n_ A})$,

  4. $R\mathop{\mathrm{Hom}}\nolimits _{\mathbf{P}^ n_ A}(E, K)$ is a pseudo-coherent object of $D(A)$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^ n_ A})$,

  5. $R\Gamma (\mathbf{P}^ n_ A, K \otimes ^\mathbf {L} \mathcal{O}_{\mathbf{P}^ n_ A}(d))$ is pseudo-coherent object of $D(A)$ for $d = 0, 1, \ldots , n$.

Proof. Recall that

\[ R\mathop{\mathrm{Hom}}\nolimits _{\mathbf{P}^ n_ A}(E, K) = R\Gamma (\mathbf{P}^ n_ A, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\mathbf{P}^ n_ A}}(E, K)) \]

by definition, see Cohomology, Section 20.41. Thus parts (4) and (3) are equivalent by Cohomology, Lemma 20.47.5.

Since every perfect complex is pseudo-coherent, it is clear that (2) implies (3).

Assume (1) holds. Then $E \otimes ^\mathbf {L} K$ is pseudo-coherent for every pseudo-coherent $E$, see Cohomology, Lemma 20.44.5. By Lemma 36.30.5 the direct image of such a pseudo-coherent complex is pseudo-coherent and we see that (2) is true.

Part (3) implies (5) because we can take $E = \mathcal{O}_{\mathbf{P}^ n_ A}(d)$ for $d = 0, 1, \ldots , n$.

To finish the proof we have to show that (5) implies (1). Let $P$ be as in (36.20.0.1) and $R$ as in (36.20.0.2). By Lemma 36.20.1 we have an equivalence

\[ - \otimes ^\mathbf {L}_ R P : D(R) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_{\mathbf{P}^ n_ A}) \]

Let $M \in D(R)$ be an object such that $M \otimes ^\mathbf {L} P = K$. By Differential Graded Algebra, Lemma 22.35.4 there is an isomorphism

\[ R\mathop{\mathrm{Hom}}\nolimits (R, M) = R\mathop{\mathrm{Hom}}\nolimits _{\mathbf{P}^ n_ A}(P, K) \]

in $D(A)$. Arguing as above we obtain

\[ R\mathop{\mathrm{Hom}}\nolimits _{\mathbf{P}^ n_ A}(P, K) = R\Gamma (\mathbf{P}^ n_ A, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\mathbf{P}^ n_ A}}(E, K)) = R\Gamma (\mathbf{P}^ n_ A, P^\vee \otimes ^\mathbf {L}_{\mathcal{O}_{\mathbf{P}^ n_ A}} K). \]

Using that $P^\vee $ is the direct sum of $\mathcal{O}_{\mathbf{P}^ n_ A}(d)$ for $d = 0, 1, \ldots , n$ and (5) we conclude $R\mathop{\mathrm{Hom}}\nolimits (R, M)$ is pseudo-coherent as a complex of $A$-modules. Of course $M = R\mathop{\mathrm{Hom}}\nolimits (R, M)$ in $D(A)$. Thus $M$ is pseudo-coherent as a complex of $A$-modules. By Lemma 36.34.1 we may represent $M$ by a bounded above complex $F^\bullet $ of finite free $R$-modules. Then $F^\bullet = \bigcup _{p \geq 0} \sigma _{\geq p}F^\bullet $ is a filtration which shows that $F^\bullet $ is a differential graded $R$-module with property (P), see Differential Graded Algebra, Section 22.20. Hence $K = M \otimes ^\mathbf {L}_ R P$ is represented by $F^\bullet \otimes _ R P$ (follows from the construction of the derived tensor functor, see for example the proof of Differential Graded Algebra, Lemma 22.35.3). Since $F^\bullet \otimes _ R P$ is a bounded above complex whose terms are direct sums of copies of $P$ we conclude that the lemma is true. $\square$

Lemma 36.34.3. Let $A$ be a ring. Let $X$ be a scheme over $A$ which is quasi-compact and quasi-separated. 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 $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_ X)$.

Proof. There exists an integer $N$ such that $R\Gamma (X, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A)$ has cohomological dimension $N$ as explained in Lemma 36.4.1. Let $b \in \mathbf{Z}$ be such that $H^ i(K) = 0$ for $i > b$. Let $E$ be pseudo-coherent on $X$. It suffices to show that $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is $m$-pseudo-coherent for every $m$. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, m - N - 1 - b)$. This is possible by Theorem 36.14.6. Choose a distinguished triangle

\[ P \to E \to C \to P[1] \]

in $D_\mathit{QCoh}(\mathcal{O}_ X)$. The cohomology sheaves of $C$ are zero in degrees $\geq m - N - 1 - b$. Hence the cohomology sheaves of $C \otimes ^\mathbf {L} K$ are zero in degrees $\geq m - N - 1$. Thus the cohomology of $R\Gamma (X, C \otimes ^\mathbf {L} K)$ are zero in degrees $\geq m - 1$. Hence

\[ R\Gamma (X, P \otimes ^\mathbf {L} K) \to R\Gamma (X, E \otimes ^\mathbf {L} K) \]

is an isomorphism on cohomology in degrees $\geq m$. By assumption the source is pseudo-coherent. We conclude that $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is $m$-pseudo-coherent as desired. $\square$


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