The Stacks project

21.45 Pseudo-coherent modules

In this section we discuss pseudo-coherent complexes.

Definition 21.45.1. 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^\bullet $ 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 21.45.2. 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 21.44.8 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 21.45.3. 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 21.45.2 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 13.16.1). 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 21.44.4 we conclude. $\square$

Lemma 21.45.4. 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 21.44.7 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 21.44.2). 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 21.45.2.

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

Lemma 21.45.5. 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 21.45.6. 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 13.4.10. By Lemma 21.45.4 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 21.45.7. 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 21.44.5 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08FS. Beware of the difference between the letter 'O' and the digit '0'.