On l-adic cohomology of Artin stacks: L-functions, weights, and the decomposition theorem

We develop the notion of stratifiability in the context of derived categories and the six operations for stacks in [26, 27]. Then we reprove Behrend's Lefschetz trace formula for stacks, and give the meromorphic continuation of the L-series of Fq-stacks. We give an upper bound for the weights of the cohomology groups of stacks, and as an application, prove the decomposition theorem for perverse sheaves on stacks with affine diagonal, both over finite fields and over the complex numbers. Along the way, we generalize the structure theorem of iota-mixed sheaves and the generic base change theorem to stacks. We also give a short exposition on the lisse-analytic topoi of complex analytic stacks, and give a comparison between the lisse-etale topos of a complex algebraic stack and the lisse-analytic topos of its analytification.