The Stacks project

36.10 Pseudo-coherent and perfect complexes

In this section we make the connection between the general notions defined in Cohomology, Sections 20.43, 20.44, 20.45, and 20.46 and the corresponding notions for complexes of modules in More on Algebra, Sections 15.63, 15.65, and 15.73.

Lemma 36.10.1. Let $X$ be a scheme. If $E$ is an $m$-pseudo-coherent object of $D(\mathcal{O}_ X)$, then $H^ i(E)$ is a quasi-coherent $\mathcal{O}_ X$-module for $i > m$ and $H^ m(E)$ is a quotient of a quasi-coherent $\mathcal{O}_ X$-module. If $E$ is pseudo-coherent, then $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. Locally on $X$ there exists a strictly perfect complex $\mathcal{E}^\bullet $ such that $H^ i(E)$ is isomorphic to $H^ i(\mathcal{E}^\bullet )$ for $i > m$ and $H^ m(E)$ is a quotient of $H^ m(\mathcal{E}^\bullet )$. The sheaves $\mathcal{E}^ i$ are direct summands of finite free modules, hence quasi-coherent. The lemma follows. $\square$

Lemma 36.10.2. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M^\bullet $ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_ X)$. Then $E$ is an $m$-pseudo-coherent (resp. pseudo-coherent) as an object of $D(\mathcal{O}_ X)$ if and only if $M^\bullet $ is $m$-pseudo-coherent (resp. pseudo-coherent) as a complex of $A$-modules.

Proof. It is immediate from the definitions that if $M^\bullet $ is $m$-pseudo-coherent, so is $E$. To prove the converse, assume $E$ is $m$-pseudo-coherent. As $X = \mathop{\mathrm{Spec}}(A)$ is quasi-compact with a basis for the topology given by standard opens, we can find a standard open covering $X = D(f_1) \cup \ldots \cup D(f_ n)$ and strictly perfect complexes $\mathcal{E}_ i^\bullet $ on $D(f_ i)$ and maps $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ inducing isomorphisms on $H^ j$ for $j > m$ and surjections on $H^ m$. By Cohomology, Lemma 20.43.8 after refining the open covering we may assume $\alpha _ i$ is given by a map of complexes $\mathcal{E}_ i^\bullet \to \widetilde{M^\bullet }|_{U_ i}$ for each $i$. By Modules, Lemma 17.14.6 the terms $\mathcal{E}_ i^ n$ are finite locally free modules. Hence after refining the open covering we may assume each $\mathcal{E}_ i^ n$ is a finite free $\mathcal{O}_{U_ i}$-module. From the definition it follows that $M^\bullet _{f_ i}$ is an $m$-pseudo-coherent complex of $A_{f_ i}$-modules. We conclude by applying More on Algebra, Lemma 15.63.14.

The case “pseudo-coherent” follows from the fact that $E$ is pseudo-coherent if and only if $E$ is $m$-pseudo-coherent for all $m$ (by definition) and the same is true for $M^\bullet $ by More on Algebra, Lemma 15.63.5. $\square$

Lemma 36.10.3. Let $X$ be a Noetherian scheme. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. For $m \in \mathbf{Z}$ the following are equivalent

  1. $H^ i(E)$ is coherent for $i \geq m$ and zero for $i \gg 0$, and

  2. $E$ is $m$-pseudo-coherent.

In particular, $E$ is pseudo-coherent if and only if $E$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_ X)$.

Proof. As $X$ is quasi-compact we see that in both (1) and (2) the object $E$ is bounded above. Thus the question is local on $X$ and we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ for some Noetherian ring $A$. In this case $E$ corresponds to a complex of $A$-modules $M^\bullet $ by Lemma 36.3.5. By Lemma 36.10.2 we see that $E$ is $m$-pseudo-coherent if and only if $M^\bullet $ is $m$-pseudo-coherent. On the other hand, $H^ i(E)$ is coherent if and only if $H^ i(M^\bullet )$ is a finite $A$-module (Properties, Lemma 28.16.1). Thus the result follows from More on Algebra, Lemma 15.63.17. $\square$

Lemma 36.10.4. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M^\bullet $ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_ X)$. Then

  1. $E$ has tor amplitude in $[a, b]$ if and only if $M^\bullet $ has tor amplitude in $[a, b]$.

  2. $E$ has finite tor dimension if and only if $M^\bullet $ has finite tor dimension.

Proof. Part (2) follows trivially from part (1). In the proof of (1) we will use the equivalence $D(A) = D_\mathit{QCoh}(X)$ of Lemma 36.3.5 without further mention. Assume $M^\bullet $ has tor amplitude in $[a, b]$. Then $K^\bullet $ is isomorphic in $D(A)$ to a complex $K^\bullet $ of flat $A$-modules with $K^ i = 0$ for $i \not\in [a, b]$, see More on Algebra, Lemma 15.65.3. Then $E$ is isomorphic to $\widetilde{K^\bullet }$. Since each $\widetilde{K^ i}$ is a flat $\mathcal{O}_ X$-module, we see that $E$ has tor amplitude in $[a, b]$ by Cohomology, Lemma 20.45.3.

Assume that $E$ has tor amplitude in $[a, b]$. Then $E$ is bounded whence $M^\bullet $ is in $K^-(A)$. Thus we may replace $M^\bullet $ by a bounded above complex of $A$-modules. We may even choose a projective resolution and assume that $M^\bullet $ is a bounded above complex of free $A$-modules. Then for any $A$-module $N$ we have

\[ E \otimes _{\mathcal{O}_ X}^\mathbf {L} \widetilde{N} \cong \widetilde{M^\bullet } \otimes _{\mathcal{O}_ X}^\mathbf {L} \widetilde{N} \cong \widetilde{M^\bullet \otimes _ A N} \]

in $D(\mathcal{O}_ X)$. Thus the vanishing of cohomology sheaves of the left hand side implies $M^\bullet $ has tor amplitude in $[a, b]$. $\square$

Lemma 36.10.5. Let $f : X \to S$ be a morphism of affine schemes corresponding to the ring map $R \to A$. Let $M^\bullet $ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_ X)$. Then

  1. $E$ as an object of $D(f^{-1}\mathcal{O}_ S)$ has tor amplitude in $[a, b]$ if and only if $M^\bullet $ has tor amplitude in $[a, b]$ as an object of $D(R)$.

  2. $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ S)$ if and only if $M^\bullet $ has finite tor dimension as an object of $D(R)$.

Proof. Consider a prime $\mathfrak q \subset A$ lying over $\mathfrak p \subset R$. Let $x \in X$ and $s = f(x) \in S$ be the corresponding points. Then $(f^{-1}\mathcal{O}_ S)_ x = \mathcal{O}_{S, s} = R_\mathfrak p$ and $E_ x = M^\bullet _\mathfrak q$. Keeping this in mind we can see the equivalence as follows.

If $M^\bullet $ has tor amplitude in $[a, b]$ as a complex of $R$-modules, then the same is true for the localization of $M^\bullet $ at any prime of $A$. Then we conclude by Cohomology, Lemma 20.45.5 that $E$ has tor amplitude in $[a, b]$ as a complex of sheaves of $f^{-1}\mathcal{O}_ S$-modules. Conversely, assume that $E$ has tor amplitude in $[a, b]$ as an object of $D(f^{-1}\mathcal{O}_ S)$. We conclude (using the last cited lemma) that $M^\bullet _\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $R_\mathfrak p$-modules for every prime $\mathfrak q \subset A$ lying over $\mathfrak p \subset R$. By More on Algebra, Lemma 15.65.15 we find that $M^\bullet $ has tor amplitude in $[a, b]$ as a complex of $R$-modules. This finishes the proof of (1).

Since $X$ is quasi-compact, if $E$ locally has finite tor dimension as a complex of $f^{-1}\mathcal{O}_ S$-modules, then actually $E$ has tor amplitude in $[a, b]$ for some $a, b$ as a complex of $f^{-1}\mathcal{O}_ S$-modules. Thus (2) follows from (1). $\square$

Lemma 36.10.6. Let $X$ be a quasi-separated scheme. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $a \leq b$. The following are equivalent

  1. $E$ has tor amplitude in $[a, b]$, and

  2. for all $\mathcal{F}$ in $\mathit{QCoh}(\mathcal{O}_ X)$ we have $H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F}) = 0$ for $i \not\in [a, b]$.

Proof. It is clear that (1) implies (2). Assume (2). Let $U \subset X$ be an affine open. As $X$ is quasi-separated the morphism $j : U \to X$ is quasi-compact and separated, hence $j_*$ transforms quasi-coherent modules into quasi-coherent modules (Schemes, Lemma 26.24.1). Thus the functor $\mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ U)$ is essentially surjective. It follows that condition (2) implies the vanishing of $H^ i(E|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \mathcal{G})$ for $i \not\in [a, b]$ for all quasi-coherent $\mathcal{O}_ U$-modules $\mathcal{G}$. Write $U = \mathop{\mathrm{Spec}}(A)$ and let $M^\bullet $ be the complex of $A$-modules corresponding to $E|_ U$ by Lemma 36.3.5. We have just shown that $M^\bullet \otimes _ A^\mathbf {L} N$ has vanishing cohomology groups outside the range $[a, b]$, in other words $M^\bullet $ has tor amplitude in $[a, b]$. By Lemma 36.10.4 we conclude that $E|_ U$ has tor amplitude in $[a, b]$. This proves the lemma. $\square$

Lemma 36.10.7. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M^\bullet $ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_ X)$. Then $E$ is a perfect object of $D(\mathcal{O}_ X)$ if and only if $M^\bullet $ is perfect as an object of $D(A)$.

Proof. This is a logical consequence of Lemmas 36.10.2 and 36.10.4, Cohomology, Lemma 20.46.5, and More on Algebra, Lemma 15.73.2. $\square$

As a consequence of our description of pseudo-coherent complexes on schemes we can prove certain internal homs are quasi-coherent.

Lemma 36.10.8. Let $X$ be a scheme.

  1. If $L$ is in $D^+_\mathit{QCoh}(\mathcal{O}_ X)$ and $K$ in $D(\mathcal{O}_ X)$ is pseudo-coherent, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and locally bounded below.

  2. If $L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $K$ in $D(\mathcal{O}_ X)$ is perfect, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

  3. If $X = \mathop{\mathrm{Spec}}(A)$ is affine and $K, L \in D(A)$ then

    \[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\widetilde{K}, \widetilde{L}) = \widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} \]

    in the following two cases

    1. $K$ is pseudo-coherent and $L$ is bounded below,

    2. $K$ is perfect and $L$ arbitrary.

  4. If $X = \mathop{\mathrm{Spec}}(A)$ and $K, L$ are in $D(A)$, then the $n$th cohomology sheaf of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\widetilde{K}, \widetilde{L})$ is the sheaf associated to the presheaf

    \[ X \supset D(f) \longmapsto \mathop{\mathrm{Ext}}\nolimits ^ n_{A_ f}(K \otimes _ A A_ f, L \otimes _ A A_ f) \]

    for $f \in A$.

Proof. The construction of the internal hom in the derived category of $\mathcal{O}_ X$ commutes with localization (see Cohomology, Section 20.39). Hence to prove (1) and (2) we may replace $X$ by an affine open. By Lemmas 36.3.5, 36.10.2, and 36.10.7 in order to prove (1) and (2) it suffices to prove (3).

Part (3) follows from the computation of the internal hom of Cohomology, Lemma 20.43.11 by representing $K$ by a bounded above (resp. finite) complex of finite projective $A$-modules and $L$ by a bounded below (resp. arbitrary) complex of $A$-modules.

To prove (4) recall that on any ringed space the $n$th cohomology sheaf of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (A, B)$ is the sheaf associated to the presheaf

\[ U \mapsto \mathop{\mathrm{Hom}}\nolimits _{D(U)}(A|_ U, B|_ U[n]) = \mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ U)}(A|_ U, B|_ U) \]

See Cohomology, Section 20.39. On the other hand, the restriction of $\widetilde{K}$ to a principal open $D(f)$ is the image of $K \otimes _ A A_ f$ and similarly for $L$. Hence (4) follows from the equivalence of categories of Lemma 36.3.5. $\square$

Lemma 36.10.9. Let $X$ be a scheme. Let $K, L, M$ be objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The map

\[ K \otimes _{\mathcal{O}_ X}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

of Cohomology, Lemma 20.39.6 is an isomorphism in the following cases

  1. $M$ perfect, or

  2. $K$ is perfect, or

  3. $M$ is pseudo-coherent, $L \in D^+(\mathcal{O}_ X)$, and $K$ has finite tor dimension.

Proof. Lemma 36.10.8 reduces cases (1) and (3) to the affine case which is treated in More on Algebra, Lemma 15.97.3. (You also have to use Lemmas 36.10.2, 36.10.7, and 36.10.4 to do the translation into algebra.) If $K$ is perfect but no other assumptions are made, then we do not know that either side of the arrow is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ but the result is still true because we can work locally and reduce to the case that $K$ is a finite complex of finite free modules in which case it is clear. $\square$


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