Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with respect to the maximal ideal. We have the containments $R_\mathfrak{m} \subset \hat{R}^h_\mathfrak{m} \subset \hat{R}_\mathfrak{m}$. The Henslization sits in between and enjoy some of the properties of each of the other rings: it satisfies Hensel's lemma, but it can be defined in a completely algebraic way and its cardinality is closer to that of the localization. The localization and Henselization are both local rings in a Grothendieck topology, which makes them more cooperative with algebraic geometry, and the completion is not (to my knowledge), but it empowers you with many analytic techniques.

What is a good way to think about the relation between these rings? I'm trying to gain a better understanding of their similarities and differences, particularly for the roles they play in (higher dimensional) class field theory.

In Moritz Kerz' paper Ideles in Higher Dimension, he starts off by defining the Henselian finite idele group of a number field $F$ with ring of integers $\mathcal{O}_F$ as the direct limit $$I_{cl}(F) = \lim_{S\subset |X|}\Big(\prod_{v\in S}F_v^\times \times \prod_{v\not\in S}\mathcal{O}_{F_v}^\times \Big)$$ Where the limit is taken over finite subsets of the closed points $|X|$ of $X = Spec(\mathcal{O}_{F_v})$, $F_v$ the Henselization of $F$ at $v$ and $\mathcal{O}_{F_v}$ its ring of integers. Or we could use a restricted product over all valuations and obtain the same object. A similar construction can be performed with the full completions at each valuation (which seems to be more common in class field theory), or with simply the field $F$ and its localization $(\mathcal{O}_F)_v$ at each place. The last case is the idele version of the ring of repartitions or the pre-adeles.

What differences are there between this product when we use either the completion, the Henselization, or just the localization? The ring of repartitions can already be used to prove a version of Riemann-Roch, as done in Lang's Introduction to Algebraic and Abelian Functions. But the topological ring/group structure on the adeles/ideles is vitally important in their study. The Henselian adeles/ideles sit somewhere inbetween. What advantages or disadvantages do they have, and what is a good way to think of them?

The other construction I'm picking apart, a repeated Henselization process, is described at the bottom of page 4 of Kerz' paper. Given a Parshin chain/flag $\mathfrak{p}_0 \subset ... \subset \mathfrak{p}_n$ of prime ideals of a ring $R$, we repeatedly localize and Henselize with respect to the primes, from the primes of largest height to the smallest height. If this process is performed with completion instead of Henselization, we get a higher dimensional local field. If we just use localization, it forgets the chain of primes, and we obtain the localization at the smallest primes, often the fraction field. Again, the Henselization sits somewhere inbetween.

What parts of completion's ability to store information about the entire chain is retained by the Henselization, and what similarities/differences does it have?

I would very much appreciate any of: precise technical explanations and lists of important properties, vague intuition, or references to chase.


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