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All sources that I know that study formal schemes seem to assume that they are locally noetherian. For instance, in Hartshorne "Algebraic Geometry", the author states: "For technical reasons we will limit our discussion to noetherian schemes".

What are these technical reasons? what basic facts about formal schemes fail if they are not noetherian (or not locally noetherian)?

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    $\begingroup$ I would have a look at EGA I, where general formal schemes are discussed but noetherian assumptions show up in many theorems. For instance, a locally noetherian formal scheme is a sequential colimit of schemes. $\endgroup$
    – Marc Hoyois
    Jul 13, 2016 at 16:48
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    $\begingroup$ Perhaps one reason why many references restrict attention to locally Noetherian formal schemes is that adic completion is more well-behaved for Noetherian rings than it is for non-Noetherian ones. $\endgroup$
    – Keenan Kidwell
    Jul 13, 2016 at 17:16
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    $\begingroup$ The discussion of formal schemes in Hartshorne's book is too limited to give a sense of the purpose of formal schemes. Look at Illusie's chapter in "FGA Explained"; inspection of the proofs there will make clear the huge range of issues that come up if one tries to avoid noetherian hypotheses or some replacement for that condition (no Artin-Rees, no flatness of completion, no coherence, failure of finitely generated ideals to be closed, etc.). There are examples of real interest beyond the noetherian case (as arise in non-archimedean geometry), but this lies quite beyond basic references. $\endgroup$
    – nfdc23
    Jul 14, 2016 at 0:30

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