Piecewise principal comodule algebras

  • Piotr M. Hajac

    IMPAN, Warsaw, Poland
  • Ulrich Krähmer

    University of Glasgow, UK
  • Rainer Matthes

    University of Warsaw. Poland
  • Bartosz Zieliński

    University of Lodz, Poland


A comodule algebra over a Hopf algebra with bijective antipode is called principal if the coaction of is Galois and is -equivariantly projective (faithfully flat) over the coaction-invariant subalgebra . We prove that principality is a piecewise property: given comodule-algebra surjections whose kernels intersect to zero, is principal if and only if all 's are principal. Furthermore, assuming the principality of , we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with . Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such -families of surjections and such that the comodule algebra of global sections is .

Cite this article

Piotr M. Hajac, Ulrich Krähmer, Rainer Matthes, Bartosz Zieliński, Piecewise principal comodule algebras. J. Noncommut. Geom. 5 (2011), no. 4, pp. 591–614

DOI 10.4171/JNCG/88