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

If $K$ is a field of characteristic $p$ then the $p$-torsion of the Brauer group, ${}_p\mathrm{B}\mathrm{r}(K)$, is represented by a quotient of the group of $1$-forms, ${\mathrm{\Omega }}^1(K)$. Namely, we have a group isomorphism $${\alpha }_p:{\mathrm{\Omega }}^1(K)\mathrm{/}⟨\mathrm{d}a,(a^p-a)\mathrm{d}\mathrm{l}\mathrm{o}\mathrm{g}b:a,b\in K,b\mathrm{\ne }0⟩\to {}_p\mathrm{B}\mathrm{r}(K),$$ given by $a\mathrm{d}b\mapsto [ab,b{)}_p$ $\mathrm{\forall }a,b\in K$, $b\mathrm{\ne }0$. Here $[\cdot,\cdot )_p:K/\wp (K)\times K^\times/K^{\times p}\to{}_p{\rm Br,}(K)$ 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 ${}_{p^n}\mathrm{B}\mathrm{r}(K)$ by a quotient of ${\mathrm{\Omega }}^1(W_n(K))$, 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 $${\alpha }_{p^n}:{\mathrm{\Omega }}^1(W_{p^n}(K))\mathrm{/}⟨Fa\mathrm{d}b-a\mathrm{d}Vb:a,b\in W_n(K),([a^p]-[a])\mathrm{d}\mathrm{l}\mathrm{o}\mathrm{g}[b]:a,b\in K,b\mathrm{\ne }0⟩\to {}_{p^n}\mathrm{B}\mathrm{r}(K)\mathrm{.}$$ Here $F$ is the Frobenius isomorphism, $V$ is the Verschiebung map and $[a]$ is the Teichm"uller representative of $a\in K$,$[a]=(a,0,0,\ldots )$.