Analogues of the $p^n$th Hilbert symbol in characteristic $p$ (updated)

Abstract

The $p$th degree Hilbert symbol $(\cdot ,\cdot {)}_p:K^{\times }/K^{\times p}\times K^{\times }/K^{\times p}\to {}_p\mathrm{Br}(K)$ from characteristic $\ne p$ has two analogues in characteristic $p$, $$[\cdot ,\cdot {)}_p:K/\wp (K)\times K^{\times }/K^{\times p}\to {}_p\mathrm{Br}(K),$$ where $\wp$ is the Artin-Schreier map $x\mapsto x^p-x$, and $$((\cdot ,\cdot ){)}_p:K/K^p\times K/K^p\to {}_p\mathrm{Br}(K).$$ The symbol $[\cdot ,\cdot {)}_p$ generalizes to an analogue of $(\cdot ,\cdot {)}_{p^n}$ via the Witt vectors, $$[\cdot ,\cdot {)}_{p^n}:W_n(K)/\wp (W_n(K))\times K^{\times }/K^{\times p^n}\to {}_{p^n}\mathrm{Br}(K).$$ Here $W_n(K)$ is the truncation of length $n$ of the ring of $p$-typical Witt wectors, i.e. $W_{\{1,p,\dots ,p^{n-1}\}}(K)$. In this paper we construct similar generalizations for $((\cdot ,\cdot ){)}_p$. Our construction involves Witt vectors and Weyl algebras. In the process we obtain a new kind of Weyl algebras in characteristic $p$, with many interesting properties. The symbols we introduce, $((\cdot ,\cdot ){)}_{p^n}$ and, more generally, $((\cdot ,\cdot ){)}_{p^m,p^n}$, which here are defined in terms of central simple algebras, coincide with the homonymous symbols we introduced in [arXiv:1711.00980] in terms of the symbols $[\cdot ,\cdot {)}_{p^n}$. This will be proved in a future paper. In the present paper we only introduce the symbols and we prove that they have the same properties with the symbols from [arXiv:1711.00980]. These properies are enough to obtain the representation theorem for ${}_{p^n}\mathrm{Br}(K)$ from [arXiv:1711.00980], Theorem 4.10.

Some changes from the previous version: The base field changed from $F$ to$K$. The notation for the Frobenius morphism changed from $\Phi$ to $F$. The old $Wn-1$ is now $W_n$. The old $B_{m-1,n-1}$ is now $B_{m,n}$. The new $B$ algebra is the opposite of the old one. I added the new section 5, "The adjoint property of Frobenius and Verschiebung"