This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the underlying ring, and we show how this generalises to modules over schemes. We also relate these notions to stratifications on modules, as introduced by A. Grothendieck in order to extend integrable (ordinary) connections to finite characteristic. Over smooth rings, we obtain an equivalence of stratifications and integrable iterative connections. Furthermore, over a regular ring in positive characteristic, we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by D. Gieseker. In the second part of this article, we set up a Picard-Vessiot theory for fields of solutions. For such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes into account even nonreduced closed subgroup schemes of the Galois group scheme on one hand and inseparable intermediate extensions of the Picard-Vessiot extension on the other hand. Finally, we compare our Galois theory with the Galois theory for purely inseparable field extensions.
For characteristic zero, N. Katz described in [Kat87] a general setting of modules with connection to describe partial linear differential equations, and established a Galois theory from an abstract point of view: He showed that -under some assumptions on the ring -the category of modules with connection (and also that of modules with integrable connection) forms a neutral Tannakian category over the field of constants and neutral Tannakian categories are known to be equivalent to categories of finite dimensional representations of proalgebraic groups (see [DM89]). However, this theory works only in characteristic zero. This is mainly caused by the fact that in positive characteristic p, every p-th power of an element in a ring is differentially constant. A. Grothendieck gave a notion of stratifications (cf. [BO78]) which generalises the notion of integrable connections to arbitrary characteristic, and which turns out to be a "good" category. In positive characteristic, a theorem of Katz (see [Gie75]) shows that over smooth schemes, modules with stratifications are equivalent to flat bundles (or F-divided sheaves as they are called in [San07]), which enables Gieseker and Dos Santos to obtain further properties of the fundamental group scheme resp. the Tannakian group scheme.
In the first part of this article, we set up a theory over rings of arbitrary characteristic, which generalises the characteristic zero setting not only in the integrable case, but also in the nonintegrable case, using so called iterative connections. The integrable version, however, (so called modules with integrable iterative connection) is again equivalent to flat bundles over a regular ring in positive characteristic (cf. Section 8).
For obtaining this theory, differentials will be replaced by a family of higher differentials, similar to the step from derivations to higher/iterative derivations in positive characteristic (see for example [Mat01] and [MvdP03]). In getting the right setting, the main idea is the following: For an algebra R over a perfect field K, regard an iterative derivation on R over K (or more generally, a higher derivation) not as a sequence of K-linear maps ∂ (k) : R → R k∈N (as it is done in [HS37], [Mat01] etc.) but as a homomorphism of K-algebras ψ : R → R[[T ]] by summing up, in detail ψ(r) := ∞ k=0 ∂ (k) (r)T k (ψ is often called the Taylor series), and moreover regard the ring of power series R[[T ]] as a completion of the graded R-algebra R[T ]. This leads to the notion of “cgas” (completions of graded algebras; cf. Section 2), which allows to generalise the definition of a higher derivation and to obtain a universal object ΩR/K with a universal higher derivation d R : R → ΩR/K , replacing the module of differentials Ω R/K used in the classical theory (cf. Theorem 3.10).
In Section 4, we introduce the definition of a higher connection on an R-module. Furthermore, we show that a finitely generated R-module that admits such a higher connection is locally free, if R is regular and a finitely generated K-algebra (Corollary 4.5). At least in positive characteristic, this is an improvement to the literature, since no integrability condition is needed. Although modules with higher connection might be interesting on their own, our main concern are modules with so called iterative connection and modules with integrable iterative connection (cf. Section 5), which are obtained by requiring additional properties on the higher connection. One of the main results of the first part is given in Section 6, namely Theorem 6.10. Let R be a regular ring over a perfect field K and the localisation of a finitely generated K-algebra, such that Spec(R) has a K-rational point. Then the categories HCon(R/K), ICon(R/K) and ICon int (R/K) of R-modules with higher connection resp. iterative connection resp. integrable iterative connection are neutral Tannakian categories over K.
The reason for considering iterative and integrable iterative connections becomes clear in the next two sections. In Section 7, we have a look at characteristic zero. Here we show that iterative connections on modules are in one-to-one correspondence to ordinary connections, if R is regular, and that the integrability conditions coincide via this correspondence. Hence the theory of modules with (integrable) iterative connection really is a generalisation of modules with (integrable) connection in characteristic zero.
Section 8 is dedicated to the case of positive characteristic. The main result here is the equivalence between the category ICon int (R/K) and the category of Frobenius compatible projective systems (Fc-projective systems) over the ring R. (Again under the assumption that R is regular.) Essentially, Fc-projective systems over R can be identified with flat bundles over Spec(R) resp. F-divided sheaves on Spec(R). Using the equivalence above, we can deduce from Corollary 4.5 that for an Fc-projective system {M i } i∈N , the R-module M 0 is loca
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