Pre-Nichols algebras and generation of pointed Hopf algebras in positive characteristic

We study when a finite-dimensional pre-Nichols algebra $R$ of $V$ over an algebraically closed field of characteristic $p>0$, with $p\nmid \dim R$, is necessarily isomorphic to the Nichols algebra $\text{B}(V)$. A trace identity gives a rigidity criterion involving the twist on minimal defining relations. Using integral forms and Hochschild cohomology, we establish uniform rigidity for Yetter–Drinfeld modules of bounded dimension over fixed finite abelian groups, and obtain related results for certain modules over nonabelian groups. For every finite group $G$ of odd order, outside finitely many characteristics depending only on $G$, every finite-dimensional pointed Hopf algebra $H$ with $G(H)\simeq G$ and dimension prime to the characteristic is generated by group-like and skew-primitive elements. If $G$ is abelian, then, outside a possibly larger finite set of characteristics, every such $H$ is a Hopf $2$-cocycle deformation of its associated graded Hopf algebra.