  The kernel $\mathcal{K}^{\operatorname{st}}$ of a descent statistic
$\operatorname{st}$, introduced by Grinberg, is a subspace of the algebra
$\operatorname{QSym}$ of quasisymmetric functions defined in terms of
$\operatorname{st}$-equivalent compositions, and is an ideal of
$\operatorname{QSym}$ if and only if $\operatorname{st}$ is shuffle-compatible.
This paper continues the study of kernels of descent statistics, with emphasis
on the peak set $\operatorname{Pk}$ and the peak number $\operatorname{pk}$.
The kernel $\mathcal{K}^{\operatorname{Pk}}$ in particular is precisely the
kernel of the canonical projection from $\operatorname{QSym}$ to Stembridge's
algebra of peak quasisymmetric functions, and is the orthogonal complement of
Nyman's peak algebra. We prove necessary and sufficient conditions for
obtaining spanning sets and linear bases for the kernel
$\mathcal{K}^{\operatorname{st}}$ of any descent statistic $\operatorname{st}$
in terms of fundamental quasisymmetric functions, and give characterizations of
$\mathcal{K}^{\operatorname{Pk}}$ and $\mathcal{K}^{\operatorname{pk}}$ in
terms of the fundamental basis and the monomial basis of $\operatorname{QSym}$.
Our results imply that the peak set and peak number statistics are
$M$-binomial, confirming a conjecture of Grinberg.
