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 PP over a Hopf algebra HH with bijective antipode is called principal if the coaction of HH is Galois and PP is HH-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra PcoHP^{\mathrm{co}H}. We prove that principality is a piecewise property: given NN comodule-algebra surjections PPiP \rightarrow P_i whose kernels intersect to zero, PP is principal if and only if all PiP_i's are principal. Furthermore, assuming the principality of PP, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with PcoHP^{\mathrm{co}H}. 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 NN-families of surjections PPiP \rightarrow P_i and such that the comodule algebra of global sections is PP.

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