Tangent lifting of deformations in mixed characteristic (Q2569426)

The authors introduce a new concept in deformation theory, the tangent lifting property (TLP), which is particularly useful in positive or mixed characteristics. Let \(\Lambda\) be a complete local noetherian ring, with residue field \(k\). A functor \(\bar{F}\) on local artinian \(\Lambda\)-algebras with residue field \(k\) is said to have the TLP if the map \(\bar{F} (A[\epsilon])\rightarrow\bar{F}(A)\times\bar{F}(k[\epsilon])\) is surjective, for all such algebras \(A\). This is related to the \(T^1\)-lifting property of \textit{Z. Ran} [J. Algebr. Geom. 1, 279--291 (1992; Zbl 0818.14003)] and \textit{Y. Kawamata} [J. Algebr. Geom. 1, 183--190 (1992; Zbl 0818.14004)], but the relation is not completely clear. The first main result is that if \(\bar{F}\) admits a hull \(R\) and has the TLP, then the following holds: If \(\Lambda\) contains the rational numbers, then \(R\) is formally smooth. If \(\Lambda\) has characteristic \(p>0\), then \(R\) is the quotient of a formal power series by an ideal generated by \(p\)-th powers. If \(\Lambda \) is of mixed characteristic, and the hull \(R\) has characteristic zero, then \(R\) is formally smooth over \(\Lambda\). The main application is to the deformation theory of smooth proper schemes \(X\) over a perfect field \(k\) whose dualizing sheaf is trivial. To apply the preceding general result in positive characteristic, one has to obtain some control over Hodge cohomology in infinitesimal deformations. The authors achieve this by working with divided power structures, as in [\textit{S. Schröer}, J. Algebr. Geom. 12, 699--714 (2003; Zbl 1079.14505)], and introducing the divided power tangent lifting property (DPTLP). This leads, among other interesting results, to the following theorem: Suppose that \(k\) is of characteristic \(p>0\), that \(b_n(X)=\sum_{i+j=n}h^{ij}(X)\) where \(n=\dim(X)\), and that \(X\) lifts to characteristic zero over some slightly ramified extension of the Witt vectors \(W(k)\). Then the versal deformation of \(X\) is formally smooth over \(W(k)\). The paper also contains some nice results on divided powers and smooth foliations on singular schemes.