Timeline for Length of a module and Frobenius map

9 events

when toggle format	what		by	license	comment
Mar 28, 2021 at 6:51	history	edited	Martin Sleziak		edited tags
Nov 7, 2017 at 17:55	vote	accept	Cusp
Oct 29, 2017 at 14:20	history	edited	Martin Sleziak	CC BY-SA 3.0	Replaced the newly created (and ambiguous) tag by already existing (local-rings)
Oct 23, 2017 at 12:56	comment	added	Wille Liu		Let $\hat R$ denote the completion of $R$. Then for all $n\in \mathbf{N}$ we have $R/\mathfrak{m}^n\cong \hat R/\hat{\mathfrak{m}}^n$. We know that $\mathfrak{m}^{dp^e}\subseteq \mathfrak{m}^{[p^e]}$ and $\hat{\mathfrak m}^{dp^e}\subseteq \hat{\mathfrak{m}}^{[p^e]}$. The image of $\mathfrak{m}^{[p^e]}$ in $R/\mathfrak{m}^{dp^e}$ is identified with the image of $\hat{\mathfrak{m}}^{[p^e]}$ in $\hat R/\hat{\mathfrak{m}}^{dp^e}$. It amounts thus to determine the length for $(\hat R, \hat{\mathfrak{m}})$. Finally, the Cohen structure theorem implies $\hat R\cong (R/\mathfrak{m})[[x_1, ..., x_d]]$.
Oct 23, 2017 at 12:17	comment	added	Cusp		@WilleLiou It would be helpful if you kindly explain it more.
Oct 23, 2017 at 11:16	comment	added	Wille Liu		Take the completion w.r.t. the maximal ideal and use the Cohen structure theorem, which reduces the task to formal power series rings, for which the result is evident.
Oct 23, 2017 at 11:00	history	edited	Cusp	CC BY-SA 3.0	deleted 3 characters in body
Oct 23, 2017 at 10:47	answer	added	js21		timeline score: 3
Oct 23, 2017 at 8:20	history	asked	Cusp	CC BY-SA 3.0