Generators for the tautological algebra of the moduli space of curves

In this paper, we prove that the tautological algebra in cohomology of the moduli space Mg of smooth projective curves of genus g is generated by the first [g/3] Mumford–Morita–Miller classes. This solves a part of Faberʹs conjecture (Moduli of Curves and Abelian Varieties Vieweg, Braunschweig, 1999) concerning the structure of the tautological algebra affirmatively. More precisely, for any k we express the kth Mumford–Morita–Miller class ek as an explicit polynomial in the lower classes for all genera g=3k−1, 3k−2,…,2.