Pseudo-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 $(D,\sigma,\delta)$ equipped with a ring morphism $\sigma$, a $\sigma$-derivation $\delta$, and a pseudo-linear transformation $\theta$ as introduced by Ore and Jacobson. A basic closed subset of $D^n$, which we call $\theta$-affine, is the zero set of a (finite) family of linear combinations of $\left\{\theta^{i_1}(x_1),\dots,\theta^{i_n}(x_n), 1\colon(i_1,\dots,i_n)\in\mathbf N^n\right\}$ having left coefficients in $D$. This enables to define elementary notions of algebraic geometry: $\theta$-affine sets, $\theta$-morphisms, a Zariski dimension, and a notion of comorphism that witnesses a duality between the category of $\theta$-affine sets and the category of $D[t;\sigma,\delta]$-modules. Using results of P. Cohn, we show that when $\sigma$ and $\delta$ commute, $(D,\sigma,\delta)$ has an extension in which each nonzero polynomial $a_0x+a_1{\theta}(x)+\dots+a_n{\theta}^n(x)$ is surjective. In such an extension, using Baur-Monk's quantifier elimination, we show that Chevalley's projection Theorem holds, as well as a Nullstellensatz that provides an equivalence between the category of $\theta$-affine sets having no proper $\theta$-affine subset of the same Zariski dimension, and the category of torsion-free finitely generated $D[t;\sigma,\delta]$-modules. These results are applied in a further paper to division rings that do not have Shelah's independence property.

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

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

Cédric Milliet. Pseudo-linear algebra over a division ring. 2018. ⟨hal-01283071v5⟩