#### Vol. 8, No. 5, 2014

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

### Alexandru Chirvasitu

Vol. 8 (2014), No. 5, 1179–1199
##### Abstract

The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting “exact sequence” $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $H$ is a CQG algebra.

##### Keywords
cosemisimple Hopf algebra, CQG algebra, faithfully flat, right coideal subalgebra, quotient left module coalgebra, expectation
##### Mathematical Subject Classification 2010
Primary: 16T20
Secondary: 16T15, 16T05, 20G42