Paolo Saracco
ULB - Université Libre de Bruxelles, Bruxelles, Belgium

An ubiquitous result relates coideal subalgebras of a Hopf algebra $H$ with one-sided ideal coideals in $H$. In some favourable cases, this relation becomes a one-to-one correspondence. For instance, in a celebrated paper from 1972, Takeuchi exhibited a bijection between normal Hopf ideals and sub-Hopf algebras of a commutative Hopf algebra, providing also a purely algebraic proof of the fact that affine commutative $\Bbbk$-group schemes form an abelian category. In this talk, I will report on our results concerning a similar correspondence between coideal subrings and left ideal coideals in an arbitrary bialgebroid and its relationship with the structure theorem for relative Hopf modules. In particular, we will discuss a one-to-one correspondence between certain normal Hopf ideals (satisfying a purity condition) and certain sub-Hopf algebroids of a commutative Hopf algebroid and how this relates with the quotient theory of affine groupoid schemes. Based on a joint work with A. Ghobadi, L. El Kaoutit, J. Vercruysse