Drinfeld's lemma - Knowlpedia

Statement

Let $X$ be a connected scheme over a finite field $\mathbb F_q$ , let $I$ be a finite set, and consider $X^I$ . Drinfeld's lemma says, in one of its sheaf-theoretic forms, that a finite étale or lisse sheaf on $X^I$ equipped with compatible isomorphisms for the partial Frobenius morphisms carries the monodromy action expected from a product of copies of the fundamental or Weil group of $X$ , one copy for each element of $I$ .

In particular, the commuting partial Frobenius operators recover independent Frobenius elements in the different legs; using only the diagonal Frobenius would see just one copy.

Hypotheses and versions

Precise statements differ with the coefficient category. The original finite-étale form is a statement about fundamental groups. The lisse $\ell$ -adic form used for shtukas imposes a finiteness condition on geometric monodromy, or works through an appropriate Weil-group formulation. Modern versions for stacks and diamonds require corresponding smallness and continuity hypotheses.

Role in excursion operators

The cohomology of a multiple-leg $G$ -shtuka has partial Frobenius actions. Drinfeld's lemma upgrades them to an action of a product of global Galois groups. Together with coalescence of legs, this makes it possible to insert a tuple $(\gamma_i)_{i\in I}$ into an excursion operator.

References

Vladimir Drinfeld, “On a conjecture of Kashiwara,” Mathematical Research Letters 8 (2001), 713–728.
Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” Journal of the American Mathematical Society 31 (2018), 719–891, §8. arXiv.