The Formalism of ℓ-adic Sheaves

Let k be an algebraically closed field, let X be an algebraic curve over k, and let G be a smooth affine group scheme over X. For each closed point x ∈ X, let Gx denote the fiber of G at x and let BGx denote its classifying stack. One of our principal aims in this book is to make sense of the idea that (under mild hypotheses) the cohomology of Bun G (X) should admit a “continuous” Künneth decomposition \[\underset{x\in X}{\mathop{\otimes }}\,{{\text{H}}^{*}}(\text{B}{{\text{G}}_{x}})\simeq {{\text{H}}^{*}}(\text{Bu}{{\text{n}}_{G}}(X))\]. (2.1) In §1.5, we gave a precise formulation of this heuristic (see Example 1.5.4.15) in the case where k =