Jets via Hasse-schmidt Derivations

This preprint is intended to provide a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA). The approach is rather concrete, using Hasse-Schmidt (divided) higher differentials. Discussion of projectivized jet spaces (as in Green and Griffiths [G-G]) is included. This paper contains a few brief notes on how to define jets using Hasse-Schmidt higher derivations. I wrote them in order to better understand jets, and also to generalize the situation more fully to the situation of arbitrary schemes. This includes allowing singularities, working in arbitrary characteristic (including mixed characteristic), and working in the relative situation of one scheme over another. Throughout this note, all rings (and algebras) are assumed to be commutative. Most of this note consists of straightforward generalizations of the theory of derivations of order 1 , as found in Grothendieck [EGA] or Matsumura [M]. Most of this is known already to the experts, but I am not aware of any general references, other than ([B-L-R], Sect. 9.6, proof of Lemma 2) and [L]. See also Green and Griffiths [G-G], in the context of complex manifolds. Section 1 gives the basic definition of higher-order divided derivations and differentials, leading up to the basic property of jets that they correspond to arcs in the scheme. The fundamental object is the algebra HSB/A of divided differentials, replacing ⊕ d≥0 S ΩB/A ; here m ∈ N ∪ {∞} . The algebra HSB/A is a graded algebra in which the higher differentials have varying degrees; therefore there is no obvious candidate for a module to replace ΩB/A . Section 2 extends the usual first and second fundamental exact sequences to the context of higher differentials (with some changes in the leftmost term in each case). Section 3 shows that higher differentials are preserved under passing to étale covers; this then implies that the definitions are preserved under localization (which can also Supported by NSF grant DMS-0200892. 1