Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of (coming, in fact, from relations in the Chow ring of ). One immediate consequence of these relations is that the classes generate the tautological ring of , which was conjectured by Faber in [F] and recently proven at the level of cohomology by Morita in [M].
"Relations in the tautological ring of ." Duke Math. J. 129 (1) 157 - 186, 15 July 2005. https://doi.org/10.1215/S0012-7094-04-12916-1