Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials
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- by Michiel Hazewinkel PDF
- Trans. Amer. Math. Soc. 259 (1980), 47-63 Request permission
Abstract:
For any ring $R$ let $\Lambda (R)$ denote the multiplicative group of power series of the form $1 + {a_1}t + \cdots$ with coefficients in $R$. The Artin-Hasse exponential mappings are homomorphisms $W_{p, \infty } (k) \to \Lambda ({W_{p, \infty }}(k))$, which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism $E: {W_{p, \infty }}( - ) \to {W_{p,\infty }}({W_{p,\infty }}( - ))$, where ${W_{p,\infty }}$ is the functor of infinite-length Witt vectors associated to the prime $p$. In this paper we present ramified versions of both ${W_{p,\infty }}( - )$ and $E$, with ${W_{p,\infty }}( - )$ replaced by a functor $W_{q,\infty }^F( - )$, which is essentially the functor of $q$-typical curves in a (twisted) Lubin-Tate formal group law over $A$, where $A$ is a discrete valuation ring that admits a Frobenius-like endomorphism $\sigma$ (we require $\sigma (a) \equiv {a^q} \bmod \mathfrak {m}$ for all $a \in A$, where $\mathfrak {m}$ is the maximal idea of $A$). These ramified-Witt-vector functors $W_{q,\infty }^F( - )$ do indeed have the property that, if $k = A/\mathfrak {m}$ is perfect, $A$ is complete, and $l/k$ is a finite extension of $k$, then $W_{q,\infty }^F(l)$ is the ring of integers of the unique unramified extension $L/K$ covering $l/k$.References
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Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 259 (1980), 47-63
- MSC: Primary 14L05; Secondary 13K05
- DOI: https://doi.org/10.1090/S0002-9947-1980-0561822-1
- MathSciNet review: 561822