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For a proper smooth variety of even dimension over a field of characteristic different from 2 or ℓ, the second Stiefel–Whitney class of the ℓ-adic cohomology and the second Hasse–Witt class of the de Rham cohomology are both defined in the second Galois cohomology. We state a conjecture on their relation and give several evidences.

The second Stiefel–Whitney class of ℓ-adic cohomology