Mathematical Research Letters

Volume 24 (2017)

Number 4

A module structure and a vanishing theorem for cycles with modulus

Pages: 1147 – 1176



Amalendu Krishna (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Jinhyun Park (Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea)


We show that the higher Chow groups with modulus of Binda–Kerz–Saito for a smooth quasi-projective scheme $X$ is a module over the Chow ring of $X$. From this, we deduce certain pull-backs, the projective bundle formula, and the blow-up formula for higher Chow groups with modulus.

We prove vanishing of $0$-cycles of higher Chow groups with modulus on various affine varieties of dimension at least two. This shows in particular that the multivariate analogue of Bloch–Esnault–Rülling computations of additive higher Chow groups of $0$-cycles vanishes.


algebraic cycle, $K$-theory

2010 Mathematics Subject Classification

Primary 14C25. Secondary 13F35, 19E15.

Jinhyun Park was partially supported by the National Research Foundation of Korea (NRF) grant (No. 2013042157) and Korea Institute for Advanced Study (KIAS) grant, both funded by the Korean government (MSIP), and TJ Park Junior Faculty Fellowship funded by POSCO TJ Park Foundation.

Received 6 January 2015

Accepted 10 May 2016

Published 9 November 2017