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For a commutative ring $R$, consider $R[[t]]$-modules $$t^k R[[t]]^n \subset M \subset t^{-k} R[[t]]^n \subset R((t))^n.$$ It is known that if $t^{-k} R[[t]]^n / M$ is finitely generated projective $R$-module, then $M$ is finitely generated projective $R[[t]]$-module.

One line of proof, as found in http://web.stanford.edu/~tonyfeng/Zhu.pdf and http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Oct13(AffGr).pdf, reduces to a Noetherian $R$, and then to the global situation (basically, replacing power series by polynomials).

Another line of proof is in the chapter "Affine Springer Fibers and Affine Deligne–Lusztig Varieties" by Ulrich Gortz (lemam 2.11).

The problem with the second line is that it is sketchy (for example, I think that it might actually use Noetherness of $R$ when formalized). So, my question is, do you know, or have a reference, to a proof of that fact, which is not along the first line above?

Thank you, Sasha

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    $\begingroup$ I think you can find more details in Martin Kreidl's thesis, available at uni-due.de/~hx0051/Dissertation.pdf. Best, $\endgroup$
    – Matthieu Romagny
    Commented Jan 30, 2016 at 20:00
  • $\begingroup$ @Matthieu Romagny: Thank you very much! This looks very helpful. It's pity I can't mark your comment as an answer. $\endgroup$
    – Sasha
    Commented Jan 31, 2016 at 10:25

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I am being told it is better not to leave answers in comments. So:

I think you can find more details in Martin Kreidl's thesis, available at uni-due.de/~hx0051/Dissertation.pdf.

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  • $\begingroup$ Precisely, see Theorem 1.9. $\endgroup$
    – ACL
    Commented Jan 31, 2016 at 21:45

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