# 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

## Abstract

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

## 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