Jet operators on fields

Abstract

We classify jet operators in the sense of Buium [J. Algebra 198 (1997) 290-299] on a field of an arbitrary characteristic. In the positive characteristic we obtain a class of "new" operators: derivations of the Frobenius map.

Operators coming from ring schemes

“…In this section, we define certain functors "governing" the class of operators, which will be introduced in the next section. We also prove a classification result (Theorem 2.19) extending the main theorem of [11].…”

“…(i) A classification of k-algebra schemes for a perfect field k (Theorem 2. 19) extending the two-dimensional classification from [11]. This result yields a comparison (given by a sequence of Frobenius maps) between arbitrary k-algebra schemes and the k-algebra schemes related to D-rings from [15].…”

“…For example, if R = K is a field of is infinite imperfection degree, then dim K B(K) is infinite as well. (3) Similarly as in Item (2) above, we get a ring scheme structure B on A 2 k associated to any jet operator on k (see [5] and [11,Definition 1.1]). In this case, there is still a morphism π : B → S k , but B need not be a k-algebra scheme (see [5,Example(d)]).…”

“…However, if k is a field, then by a classification result of the second author [11,Theorem 2.1], we get that B has a k-algebra scheme structure ι compatible with π, so B becomes a coordinate k-algebra scheme.…”

“…(4) Hardouin's notion of iterative q-difference operators (see [8]). The situations considered in Items (2) and (3) above jointly generalize the set-up from Item (1) (actually, by [11], the "intersection" of (2) and (3) is exactly (1)). We introduce here a certain class of operators, which was invented in an attempt of bringing into a common framework the set-ups from Items (2) and (3) above.…”