Suppose $R=Z_p[t]$ , and $\hat{R}$ its p-adic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And $R_{\infty}=dir.lim\hat{R}$, the direct limit is over copies of $\hat{R}$ indexed by the natrual numbers, and the transition maps are given by $\Phi$ mapping from one copy to the next. If we denote $\hat{R_{\infty}}$ the p-adic completion of $R_{\infty}$. Then is it true that $\hat{R_{\infty}}$ isomorphic to $W(\hat{R_{\infty}}/p\hat{R_{\infty}})$, the Witt-vectors over $\hat{R_{\infty}}/p\hat{R_{\infty}}$ and why it is true? Thank you!
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3$\begingroup$ Here's a (probably overkill) explanation using the cotangent complex. Assume k is a perfect F_p-algebra. If A and B are two flat Z_p-algebras with A/p =~ k =~ B/p, then A/p^n =~ B/p^n compatibly for all n, so the completions are isomorphic. For this, view A/p^2 as a square-zero extension of k by k. Hence, it's classified by a map L_{k/F_p} ---> k[1]. The perfectness of k implies that L_{k/F_p} = 0, so all such extensions are canonically isomorphic. This shows A/p^2 =~ B/p^2, and similarly A/p^n =~ B/p^n by induction on n. In your question, this applies with A = R_\infty, and B = W(R_\infty/p). $\endgroup$– BhargavCommented Aug 15, 2012 at 11:49
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$\begingroup$ Thank you!So by the same arguement, if we hava a $Z_p$ algebra $\bar{R}$, such that $\bar{R}/p\bar{R}$ is perfect, then we will have $\hat{\bar{R}}=W(\bar{R}/p\bar{R})$, is that right ? $\endgroup$– TOMCommented Aug 15, 2012 at 13:13
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$\begingroup$ One more question , if we have a $Z_p$ linear action of a group G over R , then is the isomporthism equivariant under the group action ? $\endgroup$– TOMCommented Aug 15, 2012 at 13:17
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2$\begingroup$ The above argument shows that the category of p-adically complete flat Z_p-algebras R with R/pR perfect is equivalent to the category of perfect F_p-algebras via reduction modulo p. In particular, Aut(R/pR) =~ Aut(R) for such R, so group actions lift. $\endgroup$– BhargavCommented Aug 15, 2012 at 13:57
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