Homomorphisms and involutions of unramified henselian division algebras

S. V. Tikhonov ; V. I. Yanchevskii

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 26, Tome 423 (2014), pp. 264-275

Résumé

Let $K$ be a henselian field with the residue field $\overline K$, and let $\mathcal A_1$, $\mathcal A_2$ be finite dimensional division unramified $K$-algebras with residue algebras $\overline{\mathcal A}_1$ and $\overline{\mathcal A}_2$. Further, let $\mathrm{Hom}_K(\mathcal A_1,\mathcal A_2)$ be the set of nonzero $K$-homomorphisms from $\mathcal A_1$ to $\mathcal A_2$. We prove that there is a natural bijection between the set of nonzero $\overline K$-homomorphisms from $\overline{\mathcal A}_1$ to $\overline{\mathcal A}_2$ and the factor set of $\mathrm{Hom}_K(\mathcal A_1,\mathcal A_2)$ under the equivalence relation: $\phi_1\sim\phi_2$ $\Leftrightarrow$ there exists $m\in1+M_{\mathcal A_2}$ such that $\phi_2=\phi_1i_m$, where $i_m$ is the inner automorphism of $\mathcal A_2$ induced by $m$. A similar result is obtained for unramified algebras with involutions.