In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

Keywords:

threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsion threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsion

MSC Classifications:

14F30, 14J30 show english descriptions $p$-adic cohomology, crystalline cohomology
$3$-folds [See also 32Q25] 14F30 - $p$-adic cohomology, crystalline cohomology
14J30 - $3$-folds [See also 32Q25]

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