Timeline for maximal unramified extension of Breuil ring in $A_{cris}$

8 events

when toggle format	what		by	license	comment
Jan 22, 2021 at 15:50	comment	added	MAS		@SashaP, Thanks a lot. Second, ofcourse this argument is not used in your answer but somehow I thought about it and so asked. Thanks again
Jan 22, 2021 at 15:49	comment	added	SashaP		@M.A.SARKAR An element of a $p$-adically complete ring $A$ is invertible if and only if its reduction in $A/p$ is invertible. In this case the reduction is a field so the claim is true because the mod $p$ reduction of $E(u)$ is non-zero,
Jan 22, 2021 at 15:46	comment	added	MAS		@SashaP, sorry I meant to say $E(u)$ is unit in $\widehat{W(k)[[u]][\frac{1}{u}]}$. How to show it ?
Jan 22, 2021 at 15:34	comment	added	SashaP		@M.A.SARKAR This is not true: if $E(u)$ was invertible in $W(k)[[u]][\frac{1}{u}]$ there would be an element $f(u)\in W(k)[[u]]$ that is not divisible by $u$ such that $f(u)E(u)=u^n$ for some $n'\geq 0$. This gives a contradiction by plugging in $u=0$. Do you think this fact is used in the above argument?
Jan 22, 2021 at 14:02	comment	added	MAS		How to show $E(u)$ is invertible in $W(k)[[u]][\frac{1}{u}]$ ?
Jul 27, 2019 at 22:23	comment	added	quasi-mathematician		Thanks so much both for correcting my wrong question and giving the right answer!
Jul 27, 2019 at 22:18	vote	accept	quasi-mathematician
Jul 26, 2019 at 18:31	history	answered	SashaP	CC BY-SA 4.0