  Looijenga recently proved that the tautological ring of M_g vanishes in
degree d>g-2 and is at most one-dimensional in degree g-2, generated by the
class of the hyperelliptic locus. Here we show that K_{g-2} is non-zero on M_g.
The proof uses the Witten conjecture, proven by Kontsevich. With similar
methods, we expect to be able to prove some (possibly all) of the identities in
degree g-2 in the tautological ring that are part of the author's conjectural
explicit description of the ring.
