The Stacks project

Lemma 15.62.12. Let $A \to B$ be a ring map. Assume that $B$ is pseudo-coherent as an $A$-module. Let $K^\bullet $ be a complex of $B$-modules. The following are equivalent

  1. $K^\bullet $ is $m$-pseudo-coherent as a complex of $B$-modules, and

  2. $K^\bullet $ is $m$-pseudo-coherent as a complex of $A$-modules.

The same equivalence holds for pseudo-coherence.

Proof. Assume (1). Choose a bounded complex of finite free $B$-modules $E^\bullet $ and a map $\alpha : E^\bullet \to K^\bullet $ which is an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$. Consider the distinguished triangle $(E^\bullet , K^\bullet , C(\alpha )^\bullet )$. By Lemma 15.62.8 $C(\alpha )^\bullet $ is $m$-pseudo-coherent as a complex of $A$-modules. Hence it suffices to prove that $E^\bullet $ is pseudo-coherent as a complex of $A$-modules, which follows from Lemma 15.62.10. The pseudo-coherent case of (1) $\Rightarrow $ (2) follows from this and Lemma 15.62.5.

Assume (2). Let $n$ be the largest integer such that $H^ n(K^\bullet ) \not= 0$. We will prove that $K^\bullet $ is $m$-pseudo-coherent as a complex of $B$-modules by induction on $n - m$. The case $n < m$ follows from Lemma 15.62.8. Choose a bounded complex of finite free $A$-modules $E^\bullet $ and a map $\alpha : E^\bullet \to K^\bullet $ which is an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$. Consider the induced map of complexes

\[ \alpha \otimes 1 : E^\bullet \otimes _ A B \to K^\bullet . \]

Note that $C(\alpha \otimes 1)^\bullet $ is acyclic in degrees $\geq n$ as $H^ n(E) \to H^ n(E^\bullet \otimes _ A B) \to H^ n(K^\bullet )$ is surjective by construction and since $H^ i(E^\bullet \otimes _ A B) = 0$ for $i > n$ by the spectral sequence of Example 15.60.4. On the other hand, $C(\alpha \otimes 1)^\bullet $ is $m$-pseudo-coherent as a complex of $A$-modules because both $K^\bullet $ and $E^\bullet \otimes _ A B$ (see Lemma 15.62.10) are so, see Lemma 15.62.2. Hence by induction we see that $C(\alpha \otimes 1)^\bullet $ is $m$-pseudo-coherent as a complex of $B$-modules. Finally another application of Lemma 15.62.2 shows that $K^\bullet $ is $m$-pseudo-coherent as a complex of $B$-modules (as clearly $E^\bullet \otimes _ A B$ is pseudo-coherent as a complex of $B$-modules). The pseudo-coherent case of (2) $\Rightarrow $ (1) follows from this and Lemma 15.62.5. $\square$


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