The Stacks project

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21.43 Pseudo-coherent modules

In this section we discuss pseudo-coherent complexes.

Definition 21.43.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}$-modules. Let $m \in \mathbf{Z}$.

  1. We say $\mathcal{E}^\bullet $ is $m$-pseudo-coherent if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ 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 a strictly perfect complex of $\mathcal{O}_{U_ i}$-modules 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 $D(\mathcal{O})$ 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 $\mathcal{O}$-modules.

If $\mathcal{C}$ has a final object $X$ which is quasi-compact (for example if every covering of $X$ can be refined by a finite covering), then an $m$-pseudo-coherent object of $D(\mathcal{O})$ is in $D^-(\mathcal{O})$. But this need not be the case in general.

Lemma 21.43.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$.

  1. If $\mathcal{C}$ has a final object $X$ and if there exist a covering $\{ U_ i \to X\} $, strictly perfect complexes $\mathcal{E}_ i^\bullet $ of $\mathcal{O}_{U_ i}$-modules, 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 of $\mathcal{O}$-modules representing $E$ is $m$-pseudo-coherent.

  3. If for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that $E|_{U_ i}$ is $m$-pseudo-coherent, then $E$ is $m$-pseudo-coherent.

Proof. Let $\mathcal{F}^\bullet $ be any complex representing $E$ and let $X$, $\{ U_ i \to X\} $, and $\alpha _ i : \mathcal{E}_ i \to E|_{U_ i}$ be as in (1). We will show that $\mathcal{F}^\bullet $ is $m$-pseudo-coherent as a complex, which will prove (1) and (2) in case $\mathcal{C}$ has a final object. By Lemma 21.42.8 we can after refining the covering $\{ U_ i \to X\} $ 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.

Proof of (2). By the above we see that $\mathcal{F}^\bullet |_ U$ is $m$-pseudo-coherent as a complex of $\mathcal{O}_ U$-modules for all objects $U$ of $\mathcal{C}$. It is a formal consequence of the definitions that $\mathcal{F}^\bullet $ is $m$-pseudo-coherent.

Proof of (3). Follows from the definitions and Sites, Definition 7.6.2 part (2). $\square$

Lemma 21.43.3. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal {C})$. If $E$ is $m$-pseudo-coherent, then $Lf^*E$ is $m$-pseudo-coherent.

Proof. Say $f$ is given by the functor $u : \mathcal{D} \to \mathcal{C}$. Let $U$ be an object of $\mathcal{C}$. By Sites, Lemma 7.14.10 we can find a covering $\{ U_ i \to U\} $ and for each $i$ a morphism $U_ i \to u(V_ i)$ for some object $V_ i$ of $\mathcal{D}$. By Lemma 21.43.2 it suffices to show that $Lf^*E|_{U_ i}$ is $m$-pseudo-coherent. To do this it is enough to show that $Lf^*E|_{u(V_ i)}$ is $m$-pseudo-coherent, since $Lf^*E|_{U_ i}$ is the restriction of $Lf^*E|_{u(V_ i)}$ to $\mathcal{C}/U_ i$ (via Modules on Sites, Lemma 18.19.5). By the commutative diagram of Modules on Sites, Lemma 18.20.1 it suffices to prove the lemma for the morphism of ringed sites $(\mathcal{C}/u(V_ i), \mathcal{O}_{u(V_ i)}) \to (\mathcal{D}/V_ i, \mathcal{O}_{V_ i})$. Thus we may assume $\mathcal{D}$ has a final object $Y$ such that $X = u(Y)$ is a final object of $\mathcal{C}$.

Let $\{ V_ i \to Y\} $ be a covering such that for each $i$ there exists a strictly perfect complex $\mathcal{F}_ i^\bullet $ of $\mathcal{O}_{V_ i}$-modules and a morphism $\alpha _ i : \mathcal{F}_ i^\bullet \to E|_{V_ i}$ of $D(\mathcal{O}_{V_ i})$ such that $H^ j(\alpha _ i)$ is an isomorphism for $j > m$ and $H^ m(\alpha _ i)$ is surjective. Arguing as above it suffices to prove the result for $(\mathcal{C}/u(V_ i), \mathcal{O}_{u(V_ i)}) \to (\mathcal{D}/V_ i, \mathcal{O}_{V_ i})$. Hence we may assume that there exists a strictly perfect complex $\mathcal{F}^\bullet $ of $\mathcal{O}_\mathcal {D}$-modules and a morphism $\alpha : \mathcal{F}^\bullet \to E$ of $D(\mathcal{O}_\mathcal {D})$ such that $H^ j(\alpha )$ is an isomorphism for $j > m$ and $H^ m(\alpha )$ is surjective. In this case, choose a distinguished triangle

\[ \mathcal{F}^\bullet \to E \to C \to \mathcal{F}^\bullet [1] \]

The assumption on $\alpha $ means exactly that the cohomology sheaves $H^ j(C)$ are zero for all $j \geq m$. Applying $Lf^*$ we obtain the distinguished triangle

\[ Lf^*\mathcal{F}^\bullet \to Lf^*E \to Lf^*C \to Lf^*\mathcal{F}^\bullet [1] \]

By the construction of $Lf^*$ as a left derived functor we see that $H^ j(Lf^*C) = 0$ for $j \geq m$ (by the dual of Derived Categories, Lemma 13.17.1). Hence $H^ j(Lf^*\alpha )$ is an isomorphism for $j > m$ and $H^ m(Lf^*\alpha )$ is surjective. On the other hand, since $\mathcal{F}^\bullet $ is a bounded above complex of flat $\mathcal{O}_\mathcal {D}$-modules we see that $Lf^*\mathcal{F}^\bullet = f^*\mathcal{F}^\bullet $. Applying Lemma 21.42.4 we conclude. $\square$

Lemma 21.43.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site and $m \in \mathbf{Z}$. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$.

  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). Let $U$ be an object of $\mathcal{C}$. Choose a covering $\{ U_ i \to U\} $ 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 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 21.42.7 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 21.42.2). 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 21.43.2.

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

Lemma 21.43.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$.

  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 $K \otimes _\mathcal {O}^\mathbf {L} L$ is $t$-pseudo-coherent with $t = \max (m + a, n + b)$.

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

Proof. Proof of (1). Let $U$ be an object of $\mathcal{C}$. By replacing $U$ by the members of a covering and replacing $\mathcal{C}$ by the localization $\mathcal{C}/U$ 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

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

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). Let $U$ be an object of $\mathcal{C}$. We may first replace $U$ by the members of a covering and $\mathcal{C}$ by the localization $\mathcal{C}/U$ to reduce to the case that $K$ and $L$ are bounded above. Then the statement follows immediately from case (1). $\square$

Lemma 21.43.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $m \in \mathbf{Z}$. If $K \oplus L$ is $m$-pseudo-coherent (resp. pseudo-coherent) in $D(\mathcal{O})$ so are $K$ and $L$.

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

\[ (K \oplus L, K \oplus L, L \oplus L[1]) = (K, K, 0) \oplus (L, L, L \oplus L[1]) \]

see Derived Categories, Lemma 13.4.9. By Lemma 21.43.4 we see that $L \oplus L[1]$ is $m$-pseudo-coherent. Hence also $L[1] \oplus L[2]$ 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 21.43.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O})$. 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 $\mathcal{O}$-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 $\mathcal{O}$-module.

Proof. Proof of (1). Let $U$ be an object of $\mathcal{C}$. We have to show that $H^ m(K)$ is can be generated by finitely many sections over the members of a covering of $U$ (see Modules on Sites, Definition 18.23.1). Thus during the proof we may (finitely often) choose a covering $\{ U_ i \to U\} $ and replace $\mathcal{C}$ by $\mathcal{C}/U_ i$ and $U$ by $U_ i$. In particular, by our definitions 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 21.42.5 we can (after replacing $U$ by the members of a covering) 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). Pick an object $U$ of $\mathcal{C}$. As in the proof of (1) we may work locally on $U$. 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. $\square$


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