Formally proper Artin stack conjecture on integral étale cohomology

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Let KK be a finite extension of Qp\mathbb Q_p, let OK\mathcal O_K be its ring of integers, and let \mathpzcX\mathpzc X be a smooth formally proper Artin stack over OK\mathcal O_K. Let \mathpzcX^\widehat{\mathpzc X} denote its formal completion, and let Υ\mathpzcX\Upsilon_{\mathpzc X} be the comparison map. Formally proper Artin stack conjecture.

Υ\mathpzcX ⁣:RΓeˊt(\mathpzcXC,Fℓ)⟶RΓeˊt(\mathpzcX^C,Fℓ)\Upsilon_{\mathpzc X}\colon R\Gamma_{\mathrm{\acute et}}(\mathpzc X_C,\mathbb F_\ell)\longrightarrow R\Gamma_{\mathrm{\acute et}}(\widehat{\mathpzc X}_C,\mathbb F_\ell)

is an equivalence for any prime ℓ\ell. This is a stronger comparison statement than the rational Hodge-proper prediction and is intended to connect algebraic and formal étale cohomology under formal GAGA. It remains conjectural in the supplied text.

References

Primary source

Dmitry Kubrak and Artem Prikhodko, “p-adic Hodge theory for Artin stacks”, arXiv:2105.05319 (2021).

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