# Multiplicativity of the Double Ramification Cycle

### David Holmes

Mathematisch Instituut, Universiteit Leiden, Postbus 9512 2300 RA, Leiden, Netherlands### Aaron Pixton

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA### Johannes Schmitt

Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland

## Abstract

The double ramification cycle satisfies a basic multiplicative relation $\mathrm{DRC}_a \cdot \mathrm{DRC}_b = \mathrm{DRC}_a \cdot \mathrm{DRC}_{a + b}$ over the locus of compact-type curves, but this relation fails in the Chow ring of the moduli space of stable curves. We restore this relation over the moduli space of stable curves by introducing an extension of the double ramification cycle to the small $\mathrm{b}$-Chow ring (the colimit of the Chow rings of all smooth blowups of the moduli space). We use this to give evidence for the conjectured equality between the (twisted) double ramification cycle and a cycle $P_g^{d,k}(A)$ described by the second author in [*F. Janda* et al., Publ. Math., Inst. Hautes Étud. Sci. 125, 221--266 (2017; Zbl 1370.14029)].

## Cite this article

David Holmes, Aaron Pixton, Johannes Schmitt, Multiplicativity of the Double Ramification Cycle. Doc. Math. 24 (2019), pp. 545–562

DOI 10.4171/DM/688