 
Summary: Hopf Pairings and (Co)induction Functors over
Commutative Rings
Jawad Y. Abuhlail
Mathematics Department
Birzeit University
P.O.Box 14, Birzeit  Palestine
Abstract
(Co)induction functors appear in several areas of Algebra in different forms. In
teresting examples are the so called induction functors in the Theory of Affine Al
gebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and
use them to study (co)induction functors for affine group schemes over arbitrary
commutative ground rings. We present also a special type of Hopf pairings (bialge
bra pairings) satisfying the so called condition. For those pairings the coinduction
functor is studied and nice descriptions of it are obtained. Along the paper several
interesting results are generalized from the case of base fields to the case of arbitrary
commutative (Noetherian) ground rings.
Introduction
Hopf pairings (respectively bialgebra pairings) were presented by M. Takeuchi [35, Page
15] (respectively S. Majid [26, 1.4]). With the help of these, several authors studied affine
group schemes and quantum groups over arbitrary commutative ground rings (e.g. [16],
