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## Frontiers of Operator Theory / Frontières de la théorie des opérateurs

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## Upper ramification groups of local fields with imperfect residue fields (1/3)

Appears in collection : Takeshi Saito - Upper ramification groups of local fields with imperfect residue fields

Upper ramification groups of local fields with imperfect residue fields were introduced by two of the organizers, Abbes and myself in 2000. Recently the graded quotients are shown to be F_p-vector spaces and related to Frobenius-Witt differentials. In three lectures, we outline the definition and recent developments. After short heuristic observation using rigid geometry leading to the definition, we will sketch:

1. Definition purely in the language of schemes.
2. Proof of the property that the graded quotients are F_p-vector spaces using the cotangent complexes.
3. Construction of the characteristic form relating the graded quotients to Frobenius-Witt differentials. The third topic may be omitted depending on the time.

### Bibliography

• Abbes, Saito, Ramification of local fields with imperfect residue fields, American Journal of Mathematics, 124.5 (2002), 879-920.
• Saito, Graded quotients of ramification groups of local fields with imperfect residue fields, preprint arXiv:2004.03768.

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