  Let $u_\zeta(g)$ denote the small quantum group associated to the simple
complex Lie algebra $g$, with parameter $q$ specialized to a primitive
$\ell$-th root of unity $\zeta$ in the field $k$. Generalizing a result of
Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional
$u_\zeta(g)$-module admitting a compatible torus action, then the injectivity
of $M$ as a module for $u_\zeta(g)$ can be detected by the restriction of $M$
to certain root subalgebras of $u_\zeta(g)$. If the characteristic of $k$ is
positive, then this injectivity criterion also holds for the higher
Frobenius--Lusztig kernels $U_\zeta(G_r)$ of the quantized enveloping algebra
$U_\zeta(g)$. Now suppose that $M$ lifts to a $U_\zeta(g)$-module. Using a new
rank variety type result for the support varieties of $u_\zeta(g)$, we prove
that the injectivity of $M$ for $u_\zeta(g)$ can be detected by the restriction
of $M$ to a single root subalgebra.
