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In this paper we prove that the cohomology groups with compact support ofstacks of shtukas are modules of finite type over a Hecke algebra. As anapplication, we extend the construction of excursion operators, defined by V.Lafforgue on the space of cuspidal automorphic forms, to the space ofautomorphic forms with compact support. This gives the Langlandsparametrization for some quotient spaces of the latter, which is compatiblewith the constant term morphism.

Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications