Linear Algebra over a division ring

Cédric Milliet (1)

1 ICJ - Institut Camille Jordan

We consider an analogue of the Zariski topology over a division ring~$\R$ equipped with a ring morphism $\sigma:\R\rightarrow\R$. A basic closed subset of $\R^n$ is given by the zero set of a (finite) family of linear combinations of $\left\{\sigma^{i_1}(x_1),\dots,\sigma^{i_n}(x_n):(i_1,\dots,i_n)\in\mathbb N^n\right\}$ having left coefficients in~$\R$. This enables us to define elementary notions of algebraic geometry: algebraic sets, $\sigma$-morphisms and comorphisms, a notion of Zariski dimension, a notion of radical component of an algebraic set. We classify the algebraic sets over $\R$ up to $\sigma$-isomorphisms when $\sigma$ is onto $\R$ and $[\R:{\rm Fix}(\sigma)]$ infinite (and as a by-product, the additive algebraic groups over a perfect field), and show that any division ring with infinite $[\R:{\rm Fix}(\sigma)]$ has an extension in which each affine polynomial $r+r_0x+r_1\sigma(x)+\dots+r_n\sigma^n(x)$ has a root. In such an extension, Chevalley's projection Theorem for constructible sets holds, as well as affine Nullstellensätze. These results are intended to be applied in a further paper to division rings that do not have Shelah's independence property.

Mots clés: théorie des modèles; géométrie agébrique; corps non commutatif; propriété d'indépendance

Domaines: Logique [math.LO]; Géométrie algébrique [math.AG]

Cédric Milliet. Linear Algebra over a division ring. 2016. ⟨hal-01283071v4⟩