  Let $\mathscr{R}_\alpha^\Lambda$ be the cyclotomic KLR algebra associated to
a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and polynomials
$\{Q_{ij}(u,v)\}_{i,j\in I}$. Shan, Varagnolo and Vasserot show that, when the
ground field $K$ has characteristic $0$, the degree $d$ component of the
cocenter $Tr(\mathscr{R}_\alpha^\Lambda)$ is nonzero only if $0\leq d\leq
d_{\Lambda,\alpha}$. In this paper we show that this holds true for arbitrary
ground field $K$, arbitrary $\mathfrak{g}$ and arbitrary polynomials
$\{Q_{ij}(u,v)\}_{i,j\in I}$. We generalize our earlier results on the
$K$-linear generators of $Tr(\mathscr{R}_\alpha^\Lambda),
Tr(\mathscr{R}_\alpha^\Lambda)_0,
Tr(\mathscr{R}_\alpha^\Lambda)_{d_{\Lambda,\alpha}}$ to arbitrary ground field
$K$. Moreover, we show that the dimension of the degree $0$ component
$Tr(\mathscr{R}_\alpha^\Lambda)_0$ is always equal to $\dim
V(\Lambda)_{\Lambda-\alpha}$, where $V(\Lambda)$ is the integrable highest
weight $U(\mathfrak{g})$-module with highest weight $\Lambda$, and we obtain a
basis for $Tr(\mathscr{R}_\alpha^\Lambda)_0$.
