Cohomology with integral coefficients of stacks of shtukas

Abstract

We construct the cohomology groups with compact support of stacks of shtukas with $\mathbb Z_{\ell}$-coefficients. We construct the cuspidal cohomology groups and prove that they are $\mathbb Z_{\ell}$-modules of finite type. We prove that the cohomology groups are modules of finite type over a Hecke algebra with $\mathbb Z_{\ell}$-coefficients. As an application, we prove that the cuspidal cohomology groups with $\mathbb Q_{\ell}$-coefficients are equal to the Hecke-finite cohomology groups with $\mathbb Q_{\ell}$-coefficients defined by V. Lafforgue. We also state the Eichler-Shimura relations for cohomology groups with $\mathbb Z_{\ell}$-coefficients and prove the compatibility of the excursion operators and the constant term morphisms.