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Total \(p\)-differentials on schemes over \(\mathbb Z/p^{2}\). (English) Zbl 1408.13054

Summary: For a scheme \(X\) defined over the length \(2p\)-typical Witt vectors \(W_2(\mathbf{k})\) of a characteristic \(p\) field, we introduce total \(p\)-differentials which interpolate between Frobenius-twisted differentials and Buium’s \(p\)-differentials. They form a sheaf over the reduction \(X_0\), and behave as if they were the sheaf of differentials of \(X\) over a deeper base below \(W_2(\mathbf{k})\). This allows us to construct the analogues of Gauss-Manin connections and Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to Frobenius lifts, Borger-Weiland’s biring formalism, and Deligne-Illusie classes.

MSC:

13F35 Witt vectors and related rings
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
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References:

[1] Borger, James, \(λ\)-rings and the field with one element (2009), arXiv preprint
[2] Buium, Alexandru, Differential characters of abelian varieties over \(p\)-adic fields, Invent. Math., 122, 1, 309-340 (1995) · Zbl 0841.14037
[3] Buium, Alexandru, Geometry of \(p\)-jets, Duke Math. J., 82, 2, 349-367 (1996) · Zbl 0882.14007
[4] Buium, Alexandru, Arithmetic analogues of derivations, J. Algebra, 198, 1, 290-299 (1997) · Zbl 0892.13008
[5] Buium, Alexandru, Arithmetic Differential Equations, vol. 118 (2005), American Mathematical Soc. · Zbl 1088.14001
[6] Borger, James; Wieland, Ben, Plethystic algebra, Adv. Math., 194, 2, 246-283 (2005) · Zbl 1098.13033
[7] Dupuy, Taylor; Freitag, James; Royer, Aaron, Order one differential equations on nonisotrivial algebraic curves (2017), arXiv preprint
[8] Deligne, Pierre; Illusie, Luc, Relèvements modulo \(p^2\) et décomposition du complexe de de Rham, Invent. Math., 89, 2, 247-270 (1987) · Zbl 0632.14017
[9] Dupuy, Taylor, Lifted torsors of lifts of the Frobenius for curves, preprint, available at · Zbl 1364.11130
[10] Dupuy, Taylor; Zureick-Brown, David, Deligne-Illusie classes as arithmetic Kodaira-Spencer classes (2018), arXiv preprint · Zbl 1471.14057
[11] Hartshorne, Robin, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0367.14001
[12] Katz, Nicholas M.; Oda, Tadao, On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ., 8, 199-213 (1968) · Zbl 0165.54802
[13] Serre, J.-P., Corps locaux, Publications de l’Université de Nancago, vol. VIII (1968), Hermann: Hermann Paris
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