Corrigendum to "On injective modules and support varieties for the small quantum group"

TL;DR

This paper corrects a previous proof regarding injective modules and support varieties for the small quantum group, addressing an algebraic oversight related to Hopf subalgebras.

Contribution

It provides a revised proof for Theorem 5.12, clarifying the conditions under which certain subalgebras are Hopf subalgebras in the context of small quantum groups.

Findings

Corrected proof of Theorem 5.12

Clarification on algebraic structures of subalgebras

Ensures validity of support variety results

Abstract

The proof of Theorem 5.12 in [C.M. Drupieski, On injective modules and support varieties for the small quantum group, Int. Math. Res. Not. 2011 (2011), 2263-2294] does not make sense as written because the algebra $u_\zeta(\mathfrak{b}_\alpha^+)$ need not be a Hopf subalgebra of $u_\zeta(\mathfrak{b}^+)$ unless $\alpha$ is a simple root. This note describes how the proof should be modified to work around this fact.