Ideles in higher dimension

We propose a notion of idele class groups of finitely generated fields using the concept of Parshin chains. This new class group allows us to give an idelic interpretation of the higher class field theory of Kato and Saito.

💡 Research Summary

The paper introduces a new definition of idele class groups for finitely generated fields of arbitrary dimension, using Parshin chains as the underlying geometric data. A Parshin chain is a flag of points (P=(x_0\subset x_1\subset\cdots\subset x_n)) on an integral scheme (X); by iteratively completing and normalizing along the chain one obtains a higher‑dimensional local field (K_P). These local fields carry Milnor (K)-theory groups (K_n^M(K_P)) and natural discrete valuations, providing the building blocks for a global object.

The authors define the higher adèle ring (\mathbb{A}_X) as a restricted product over all Parshin chains, imposing the condition that almost all components lie in the integral subrings (\mathcal{O}_P). A topology is introduced by taking the product topology on each factor and then the restricted product topology, making (\mathbb{A}_X) a complete topological ring. The group of units (\mathbb{A}_X^\times) contains the diagonal image of the global unit group (\mathcal{O}_X^\times); the quotient
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