The Stacks project

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 15.64.7 $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 15.64.9. The pseudo-coherent case of (1) $\Rightarrow $ (2) follows from this and Lemma 15.64.5.

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 15.64.7. 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

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 15.62.4. 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 15.64.9) are so, see Lemma 15.64.2. 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 15.64.2 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 15.64.5. $\square$