English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups  G  of the typesD  4{D_{4}}   ,E  6{E_{6}}   ,E  7{E_{7}}   ,E  8{E_{8}}   ,G  2{G_{2}}    on an affine variety defined over a local function field  k , andv  ∈   V  ⁢   (  k  ){v\in V(k)}    such that the geometric orbitG  .  v{G.v}    is Zariski closed although the corresponding relative orbitG  ⁢   (  k  )    .  v{G(k).v}    is not closed in the topology induced from  k . Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.

On the topology of geometric and rational orbits for algebraic group actions over valued fields