Arithmetic Analogues of Derivations

Jet operators on fields

“…It is implicit in [1] and shown in [2]. The point is that this map is a homomorphism on the graph of δ and that a jet operator can be extended to a jet operator (corresponding to the same algebraic ring) on a bigger field such that the graph of the new jet operator is Zariski dense in the affine plane.…”

“…It is worth to notice that a jet operator δ does not necessarily determine P and Q, e.g., the 0-map is both a derivation and a difference operator. Also for a map δ : K → K satisfying the "(P , Q)-rule" from the above definition, the ring condition is equivalent to the existence of a "(P , Q)-extension" of δ to a generic map on an extension of K. For the proof see [1,Lemma 2] …”

“…In [1], Buium introduced jet operators on rings; they are natural generalizations of derivations and endomorphisms (more precisely, difference operators). In the same paper, Buium classifies jet operators on local domains of characteristic 0.…”

“…In the remaining of this introduction, we recall definitions from [1]. Throughout this paper K denotes an infinite field of characteristic p 0.…”

An embedding theorem for tangent categories

“…Another direction we intend to explore in future research involves modifying the category E to capture generalised forms of differential structure. One possibility involves non-linear or arithmetic differential geometry in the sense of [4], which should involve enrichments over a suitable category of k-k-birings in the sense of [22,3]. Another possibility would be to explore "two-dimensional Lie theory" by replacing the cartesian closed category E with a suitable cartesian closed bicategory of k-linear categories, and considering generalised enrichments over this in the sense of [12].…”