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In their book "Almost Ring Theory" (http://arxiv.org/abs/math/0201175), Ofer Gabber and Lorenzo Ramero define a valued field $K$ to be "deeply ramified" if the module of Kähler differentials $\Omega_{K^{s+}/K^+}$ is zero. Here, $K^{s}$ denotes a separable-algebraic closure of $K$, and $K^+$ and $K^{s+}$ are the valuation rings.

By Proposition 6.6.2 that's equivalent to $\Omega_{K^{s+}/K^+}$ being almost zero, which means that $\Omega_{K^{s+}/K^+}$ is killed by the maximal ideal of $K^+$.

This implies that the valuation on $K$ is not discrete, and that's what I don't understand. Can anyone give me a hint? Thank you!

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  • $\begingroup$ I don't understand what you don't understand. Is "Yes the valuation is not discrete for a deeply ramified field" the answer? $\endgroup$
    – eric
    Dec 30, 2015 at 12:26
  • $\begingroup$ This "step 0" in the proof of 6.6.2 rests on 6.3.20. Let $K={\rm{Frac}}(A)$ for a dvr $A$. Extend the valuation to $K^s$ to define $K^{s+}$. Assume $\Omega^1_{K^{s+}/A}=0$. We seek a contradiction. WLOG, $A$ is henselian. By 6.3.20 and the distinguished triangle for cotangent complexes (for valuation rings), if $E/K$ is a subextension of $K^s/K$ then $\Omega^1_{E^+/A}\otimes_A K^{s+}\rightarrow\Omega^1_{K^{s+}/A}=0$ is injective, so $\Omega^1_{E^+/A}=0$. For each $K$-finite $E$ the flat local $A\rightarrow E^+$ is module-finite (as $A$ is henselian!) and so is finite etale. That is absurd. QED $\endgroup$
    – nfdc23
    Dec 30, 2015 at 15:04
  • $\begingroup$ The use of cotangent complexes seems to be essential. Indeed, the development of the link between ramification theory and $\Omega^1$ for valuation rings as in Serre's book on local fields has pervasive separability hypotheses on residue field extensions (or perfectness hypotheses on residue fields), which is too restrictive for our purposes since $K^{s+}$ always has an algebraically closed residue field and we definitely must avoid assuming $K$ has perfect residue field. The proof of 6.3.20 involves hands-on analysis of ramification theory to replace the methods in Serre's book. $\endgroup$
    – nfdc23
    Dec 30, 2015 at 15:10

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