A representation theorem for the $p^n$ torsion of the Brauer group in characteristic $p$

Abstract

If $K$ is a field of characteristic $p$ then the $p$-torsion of the Brauer group, ${}_p\mathrm{Br}(K)$, is represented by a quotient of the group of $1$-forms, ${\Omega }^1(K)$. Namely, we have a group isomorphism $${\alpha }_p:{\Omega }^1(K)/⟨\mathrm{d}a,(a^p-a)\mathrm{dlog}b:a,b\in K,b\ne 0⟩\to {}_p\mathrm{Br}(K),$$ given by $a\mathrm{d}b\mapsto [ab,b{)}_p$ $\forall a,b\in K$, $b\ne 0$. Here $[\cdot ,\cdot {)}_p:K/\wp (K)\times K^{\times }/K^{\times p}\to {}_p\mathrm{Br}(K)$ denotes the Artin-Schreier symbol. In this paper we generalize this result. Namely, we prove that for every $n\ge 1$ we have a representation of ${}_{p^n}\mathrm{Br}(K)$ by a quotient of ${\Omega }^1(W_n(K))$, where $W_n(K)$ is the truncation of length $n$ of the ring of $p$-typical Witt vectors, i.e. $W_{\{1,p,\dots ,p^{n-1}\}}(K)$. Explicitly, we have a group isomorphism $${\alpha }_{p^n}:{\Omega }^1(W_{p^n}(K))/⟨Fa\mathrm{d}b-a\mathrm{d}Vb:a,b\in W_n(K),([a^p]-[a])\mathrm{dlog}[b]:a,b\in K,b\ne 0⟩\to {}_{p^n}\mathrm{Br}(K).$$ Here $F$ is the Frobenius isomorphism, $V$ is the Verschiebung map and $[a]$ is the Teichmüller representative of $a\in K$, $[a]=(a,0,0,\dots )$.