In order to further relate the arc-topology with the arc

\(_p\)-topology, we study the effect of tilting on valuation rings. This can be thought of as a continuation of our discussion of perfectoid fields (

Subsection 8.3).

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Lemma 22.4.2.

Let \(V\) be a lens. Then \(V\) is a valuation ring if and only if \(V^\flat\) is. In this case, the value groups of \(V\) and \(V^\flat\) are isomorphic; in particular, \(V\) is eudoxian if and only if \(V^\flat\) is.

Let \(\sharp\colon V^\flat \to V\) be the multiplicative map obtained by composing the constant lift \([\bullet]: V^\flat \to W(V^\flat)\) with the quotient map \(W(V^\flat) \to V\text{.}\) It is customary to write the image of \(x\) under \(\sharp\) as \(x^\sharp\) rather than \(\sharp(x)\text{.}\)

Suppose that \(V\) is a valuation ring. If \(\Frac V\) has characteristic \(p\text{,}\) then \(V = V^\flat\) and there is nothing more to check; we may thus assume that \(\Frac V\) has characteristic 0. Since \(V\) is an integral domain, the \(p\)-power map on \(V\) is injective; hence for \(x \in V^\flat\text{,}\) \(x^\sharp = 0\) if and only if \(x = 0\text{.}\) This in turn implies that \(V^\flat\) is an integral domain (if \(xy = 0\) then \(x^\sharp y^\sharp = 0\)) and that the principal ideals of \(V^\flat\) are totally ordered with respect to inclusion (if \(x^\sharp\) is divisible by \(y^\sharp\text{,}\) then the ratio admits a coherent sequence of \(p\)-power roots and so is itself in the image of \(\sharp\)). Hence \(V^\flat\) is a valuation ring.

Conversely, suppose that

\(V^\flat\) is a valuation ring. Again, we may assume that

\(p \neq 0\) in

\(V\text{;}\) since

\(V^\flat\) is an integral domain, we may apply

Exercise 8.5.3 to deduce that

\(V\) is

\(p\)-torsion-free. Since

\(V^\flat\) is a local ring, so are

\(W(V^\flat)\) and its quotient

\(V\text{.}\) Choose

\(\varpi \in V\) as per

Lemma 8.2.3.

We need to show that given any two nonzero elements \(x,y \in V\text{,}\) one is a multiple of the other. By dividing by powers of \(\varpi\) as needed, we may reduce to the case where \(x\) and \(y\) have nonzero images in \(V/\varpi\) and hence in \(V/p = V^\flat/d\text{.}\) Since \(V^\flat\) is a valuation ring, after possibly swapping terms we can write \(x = yz + pu\) for some \(z,u \in V\text{.}\) Similarly, we can write \(\varpi \equiv yw + pv\) for some \(w,v \in V\text{.}\) Since \(V\) is classically \(\varpi\)-complete, \(1 - (p/\varpi)v\) is a unit in \(V\text{;}\) hence \(\varpi\) is divisible by \(y\text{,}\) as then is \(p\text{.}\) Consequently, \(x = yz + pu\) is also divisible by \(y\text{,}\) as desired.

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Lemma 22.4.3.

Let \(V\) be a lens which is a valuation ring. Then \(V\) and \(V^\flat\) have residue fields isomorphic to each other (and to that of \(W(V^\flat)\)) and \(\sharp: V^\flat \to V\) induces an isomorphism between the value groups of \(V\) and \(V^\flat\text{.}\) In particular, \(V\) is eudoxian if and only if \(V'\) is.

By

Lemma 22.4.2,

\(V\) is a valuation ring if and only if

\(V^\flat\) is; it thus remains to show that if

\(V^\flat\) is not AIC, then neither is

\(V\text{.}\) We may assume that

\(V\) has characteristic 0, as otherwise there is nothing to check.

Let

\(R\) be the integral closure of

\(V^\flat\) in a nontrivial finite Galois extension of its fraction field with Galois group

\(G\text{.}\) By

Theorem 7.3.5,

\(V' = V \otimes_{W(V^\flat)} W(R)\) is a lens with

\(V^{\prime \flat} \cong R\) and, by

Lemma 22.4.2, again a valuation ring.

By construction, \(G\) acts on \(V^{\prime \flat}\) with fixed subring \(V\text{.}\) By functoriality, \(G\) also acts on \(V'\text{;}\) since \(V'\) is of characteristic \(0\text{,}\) we can we can see by averaging over the group action that the fixed subring \(V'^{G}\) is equal to \(V\text{.}\)

By the Artin and Dedekind lemmas in Galois theory, we see that \(\Frac V'\) is a finite Galois extension of \(\Frac V\) of degree \(\#G \gt 1\text{.}\) This proves that \(V\) is not AIC.