  To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf
Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$,
the smallest multiple of the diagonal appearing in a cycle-theoretic
decomposition \`a la Bloch-Srinivas. We show that $\mathrm{Tor}(X)$ is the
exponent of the torsion in the N\'eron-Severi-group of $X$ when $X$ is a
surface over an algebraically closed field $k$, up to a power of the
exponential characteristic of $k$.
