## 20.44 Pseudo-coherent modules

In this section we discuss pseudo-coherent complexes.

Definition 20.44.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}_ X$-modules. Let $m \in \mathbf{Z}$.

1. We say $\mathcal{E}^\bullet$ is $m$-pseudo-coherent if there exists an open covering $X = \bigcup U_ i$ and for each $i$ a morphism of complexes $\alpha _ i : \mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{U_ i}$ where $\mathcal{E}_ i$ is strictly perfect on $U_ i$ and $H^ j(\alpha _ i)$ is an isomorphism for $j > m$ and $H^ m(\alpha _ i)$ is surjective.

2. We say $\mathcal{E}^\bullet$ is pseudo-coherent if it is $m$-pseudo-coherent for all $m$.

3. We say an object $E$ of $D(\mathcal{O}_ X)$ is $m$-pseudo-coherent (resp. pseudo-coherent) if and only if it can be represented by a $m$-pseudo-coherent (resp. pseudo-coherent) complex of $\mathcal{O}_ X$-modules.

If $X$ is quasi-compact, then an $m$-pseudo-coherent object of $D(\mathcal{O}_ X)$ is in $D^-(\mathcal{O}_ X)$. But this need not be the case if $X$ is not quasi-compact.

Lemma 20.44.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_ X)$.

1. If there exists an open covering $X = \bigcup U_ i$, strictly perfect complexes $\mathcal{E}_ i^\bullet$ on $U_ i$, and maps $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$ with $H^ j(\alpha _ i)$ an isomorphism for $j > m$ and $H^ m(\alpha _ i)$ surjective, then $E$ is $m$-pseudo-coherent.

2. If $E$ is $m$-pseudo-coherent, then any complex representing $E$ is $m$-pseudo-coherent.

Proof. Let $\mathcal{F}^\bullet$ be any complex representing $E$ and let $X = \bigcup U_ i$ and $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ be as in (1). We will show that $\mathcal{F}^\bullet$ is $m$-pseudo-coherent as a complex, which will prove (1) and (2) simultaneously. By Lemma 20.43.8 we can after refining the open covering $X = \bigcup U_ i$ represent the maps $\alpha _ i$ by maps of complexes $\alpha _ i : \mathcal{E}_ i^\bullet \to \mathcal{F}^\bullet |_{U_ i}$. By assumption $H^ j(\alpha _ i)$ are isomorphisms for $j > m$, and $H^ m(\alpha _ i)$ is surjective whence $\mathcal{F}^\bullet$ is $m$-pseudo-coherent. $\square$

Lemma 20.44.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\mathcal{O}_ Y)$. If $E$ is $m$-pseudo-coherent, then $Lf^*E$ is $m$-pseudo-coherent.

Proof. Represent $E$ by a complex $\mathcal{E}^\bullet$ of $\mathcal{O}_ Y$-modules and choose an open covering $Y = \bigcup V_ i$ and $\alpha _ i : \mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{V_ i}$ as in Definition 20.44.1. Set $U_ i = f^{-1}(V_ i)$. By Lemma 20.44.2 it suffices to show that $Lf^*\mathcal{E}^\bullet |_{U_ i}$ is $m$-pseudo-coherent. Choose a distinguished triangle

$\mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{V_ i} \to C \to \mathcal{E}_ i^\bullet $

The assumption on $\alpha _ i$ means exactly that the cohomology sheaves $H^ j(C)$ are zero for all $j \geq m$. Denote $f_ i : U_ i \to V_ i$ the restriction of $f$. Note that $Lf^*\mathcal{E}^\bullet |_{U_ i} = Lf_ i^*(\mathcal{E}|_{V_ i})$. Applying $Lf_ i^*$ we obtain the distinguished triangle

$Lf_ i^*\mathcal{E}_ i^\bullet \to Lf_ i^*\mathcal{E}|_{V_ i} \to Lf_ i^*C \to Lf_ i^*\mathcal{E}_ i^\bullet $

By the construction of $Lf_ i^*$ as a left derived functor we see that $H^ j(Lf_ i^*C) = 0$ for $j \geq m$ (by the dual of Derived Categories, Lemma 13.16.1). Hence $H^ j(Lf_ i^*\alpha _ i)$ is an isomorphism for $j > m$ and $H^ m(Lf^*\alpha _ i)$ is surjective. On the other hand, $Lf_ i^*\mathcal{E}_ i^\bullet = f_ i^*\mathcal{E}_ i^\bullet$. is strictly perfect by Lemma 20.43.4. Thus we conclude. $\square$

Lemma 20.44.4. Let $(X, \mathcal{O}_ X)$ be a ringed space and $m \in \mathbf{Z}$. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O}_ X)$.

1. If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent then $M$ is $m$-pseudo-coherent.

2. If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent.

3. If $L$ is $(m + 1)$-pseudo-coherent and $M$ is $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent.

Proof. Proof of (1). Choose an open covering $X = \bigcup U_ i$ and maps $\alpha _ i : \mathcal{K}_ i^\bullet \to K|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$ with $\mathcal{K}_ i^\bullet$ strictly perfect and $H^ j(\alpha _ i)$ isomorphisms for $j > m + 1$ and surjective for $j = m + 1$. We may replace $\mathcal{K}_ i^\bullet$ by $\sigma _{\geq m + 1}\mathcal{K}_ i^\bullet$ and hence we may assume that $\mathcal{K}_ i^ j = 0$ for $j < m + 1$. After refining the open covering we may choose maps $\beta _ i : \mathcal{L}_ i^\bullet \to L|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$ with $\mathcal{L}_ i^\bullet$ strictly perfect such that $H^ j(\beta )$ is an isomorphism for $j > m$ and surjective for $j = m$. By Lemma 20.43.7 we can, after refining the covering, find maps of complexes $\gamma _ i : \mathcal{K}^\bullet \to \mathcal{L}^\bullet$ such that the diagrams

$\xymatrix{ K|_{U_ i} \ar[r] & L|_{U_ i} \\ \mathcal{K}_ i^\bullet \ar[u]^{\alpha _ i} \ar[r]^{\gamma _ i} & \mathcal{L}_ i^\bullet \ar[u]_{\beta _ i} }$

are commutative in $D(\mathcal{O}_{U_ i})$ (this requires representing the maps $\alpha _ i$, $\beta _ i$ and $K|_{U_ i} \to L|_{U_ i}$ by actual maps of complexes; some details omitted). The cone $C(\gamma _ i)^\bullet$ is strictly perfect (Lemma 20.43.2). The commutativity of the diagram implies that there exists a morphism of distinguished triangles

$(\mathcal{K}_ i^\bullet , \mathcal{L}_ i^\bullet , C(\gamma _ i)^\bullet ) \longrightarrow (K|_{U_ i}, L|_{U_ i}, M|_{U_ i}).$

It follows from the induced map on long exact cohomology sequences and Homology, Lemmas 12.5.19 and 12.5.20 that $C(\gamma _ i)^\bullet \to M|_{U_ i}$ induces an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$. Hence $M$ is $m$-pseudo-coherent by Lemma 20.44.2.

Assertions (2) and (3) follow from (1) by rotating the distinguished triangle. $\square$

Lemma 20.44.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_ X)$.

1. If $K$ is $n$-pseudo-coherent and $H^ i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^ j(L) = 0$ for $j > b$, then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is $t$-pseudo-coherent with $t = \max (m + a, n + b)$.

2. If $K$ and $L$ are pseudo-coherent, then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is pseudo-coherent.

Proof. Proof of (1). By replacing $X$ by the members of an open covering we may assume there exist strictly perfect complexes $\mathcal{K}^\bullet$ and $\mathcal{L}^\bullet$ and maps $\alpha : \mathcal{K}^\bullet \to K$ and $\beta : \mathcal{L}^\bullet \to L$ with $H^ i(\alpha )$ and isomorphism for $i > n$ and surjective for $i = n$ and with $H^ i(\beta )$ and isomorphism for $i > m$ and surjective for $i = m$. Then the map

$\alpha \otimes ^\mathbf {L} \beta : \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ) \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$

induces isomorphisms on cohomology sheaves in degree $i$ for $i > t$ and a surjection for $i = t$. This follows from the spectral sequence of tors (details omitted).

Proof of (2). We may first replace $X$ by the members of an open covering to reduce to the case that $K$ and $L$ are bounded above. Then the statement follows immediately from case (1). $\square$

Lemma 20.44.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $m \in \mathbf{Z}$. If $K \oplus L$ is $m$-pseudo-coherent (resp. pseudo-coherent) in $D(\mathcal{O}_ X)$ so are $K$ and $L$.

Proof. Assume that $K \oplus L$ is $m$-pseudo-coherent. After replacing $X$ by the members of an open covering we may assume $K \oplus L \in D^-(\mathcal{O}_ X)$, hence $L \in D^-(\mathcal{O}_ X)$. Note that there is a distinguished triangle

$(K \oplus L, K \oplus L, L \oplus L) = (K, K, 0) \oplus (L, L, L \oplus L)$

see Derived Categories, Lemma 13.4.10. By Lemma 20.44.4 we see that $L \oplus L$ is $m$-pseudo-coherent. Hence also $L \oplus L$ is $m$-pseudo-coherent. By induction $L[n] \oplus L[n + 1]$ is $m$-pseudo-coherent. Since $L$ is bounded above we see that $L[n]$ is $m$-pseudo-coherent for large $n$. Hence working backwards, using the distinguished triangles

$(L[n], L[n] \oplus L[n - 1], L[n - 1])$

we conclude that $L[n - 1], L[n - 2], \ldots , L$ are $m$-pseudo-coherent as desired. $\square$

Lemma 20.44.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $m \in \mathbf{Z}$. Let $\mathcal{F}^\bullet$ be a (locally) bounded above complex of $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $(m - i)$-pseudo-coherent for all $i$. Then $\mathcal{F}^\bullet$ is $m$-pseudo-coherent.

Proof. Omitted. Hint: use Lemma 20.44.4 and truncations as in the proof of More on Algebra, Lemma 15.63.9. $\square$

Lemma 20.44.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ X)$. If $E$ is (locally) bounded above and $H^ i(E)$ is $(m - i)$-pseudo-coherent for all $i$, then $E$ is $m$-pseudo-coherent.

Proof. Omitted. Hint: use Lemma 20.44.4 and truncations as in the proof of More on Algebra, Lemma 15.63.10. $\square$

Lemma 20.44.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K$ be an object of $D(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$.

1. If $K$ is $m$-pseudo-coherent and $H^ i(K) = 0$ for $i > m$, then $H^ m(K)$ is a finite type $\mathcal{O}_ X$-module.

2. If $K$ is $m$-pseudo-coherent and $H^ i(K) = 0$ for $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented $\mathcal{O}_ X$-module.

Proof. Proof of (1). We may work locally on $X$. Hence we may assume there exists a strictly perfect complex $\mathcal{E}^\bullet$ and a map $\alpha : \mathcal{E}^\bullet \to K$ which induces an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$. It suffices to prove the result for $\mathcal{E}^\bullet$. Let $n$ be the largest integer such that $\mathcal{E}^ n \not= 0$. If $n = m$, then $H^ m(\mathcal{E}^\bullet )$ is a quotient of $\mathcal{E}^ n$ and the result is clear. If $n > m$, then $\mathcal{E}^{n - 1} \to \mathcal{E}^ n$ is surjective as $H^ n(E^\bullet ) = 0$. By Lemma 20.43.5 we can locally find a section of this surjection and write $\mathcal{E}^{n - 1} = \mathcal{E}' \oplus \mathcal{E}^ n$. Hence it suffices to prove the result for the complex $(\mathcal{E}')^\bullet$ which is the same as $\mathcal{E}^\bullet$ except has $\mathcal{E}'$ in degree $n - 1$ and $0$ in degree $n$. We win by induction on $n$.

Proof of (2). We may work locally on $X$. Hence we may assume there exists a strictly perfect complex $\mathcal{E}^\bullet$ and a map $\alpha : \mathcal{E}^\bullet \to K$ which induces an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$. As in the proof of (1) we can reduce to the case that $\mathcal{E}^ i = 0$ for $i > m + 1$. Then we see that $H^{m + 1}(K) \cong H^{m + 1}(\mathcal{E}^\bullet ) = \mathop{\mathrm{Coker}}(\mathcal{E}^ m \to \mathcal{E}^{m + 1})$ which is of finite presentation. $\square$

Lemma 20.44.10. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

1. $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is $0$-pseudo-coherent if and only if $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module, and

2. $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is $(-1)$-pseudo-coherent if and only if $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

Proof. Use Lemma 20.44.9 to prove the implications in one direction and Lemma 20.44.8 for the other. $\square$

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