Remark 38.6.3. Note that the $R$-algebras $B_ i$ for all $i$ and $A_ i$ for $i \geq 2$ are of finite presentation over $R$. If $S$ is of finite presentation over $R$, then it is also the case that $A_1$ is of finite presentation over $R$. In this case all the ring maps in the complete dÃ©vissage are of finite presentation. See Algebra, Lemma 10.6.2. Still assuming $S$ of finite presentation over $R$ the following are equivalent

$M$ is of finite presentation over $S$,

$M_1$ is of finite presentation over $A_1$,

$M_1$ is of finite presentation over $B_1$,

each $M_ i$ is of finite presentation both as an $A_ i$-module and as a $B_ i$-module.

The equivalences (1) $\Leftrightarrow $ (2) and (2) $\Leftrightarrow $ (3) follow from Algebra, Lemma 10.36.23. If $M_1$ is finitely presented, so is $\mathop{\mathrm{Coker}}(\alpha _1)$ (see Algebra, Lemma 10.5.3) and hence $M_2$, etc.

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