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 dévissage 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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