Partial Actions In Algebraic Geometry

We introduce geometrically partial comodules over coalgebras in monoidal categories,as an alternative notion to the notion of partial action and coaction of Hopf algebras introduced by Caenepeel and Janssen.We show that our new notion suits better if one wants to describe phenomena of partial actions in algebraic geometry.We show that under mild conditions,the category of geometric partial comodules is complete and cocomplete and the category of partial comodules over a Hopf algebra is lax monoidal.We develop a Hopf-Galois theory for geometric partial coactions to illustrate that our new notion might be a useful additional tool in Hopf algebra theory.