Differential Galois cohomology and parameterized Picard–Vessiot extensions

Abstract

Assuming that the differential field [Formula: see text] is differentially large, in the sense of [León Sánchez and Tressl, Differentially large fields, preprint (2020); arXiv:2005.00888 ], and “bounded” as a field, we prove that for any linear differential algebraic group [Formula: see text] over [Formula: see text], the differential Galois (or constrained) cohomology set [Formula: see text] is finite. This applies, among other things, to closed ordered differential fields in the sense of [Singer, The model theory of ordered differential fields, J. Symb. Logic 43(1) (1978) 82–91], and to closed[Formula: see text]-adic differential fields in the sense of [Tressl, The uniform companion for large differential fields of characteristic [Formula: see text], Trans. Amer. Math. Soc. 357(10) (2005) 3933–3951]. As an application, we prove a general existence result for parameterized Picard–Vessiot (PPV) extensions within certain families of fields; if [Formula: see text] is a field with two commuting derivations, and [Formula: see text] is a parameterized linear differential equation over [Formula: see text], and [Formula: see text] is “differentially large” and [Formula: see text] is bounded, and [Formula: see text] is existentially closed in [Formula: see text], then there is a PPV extension [Formula: see text] of [Formula: see text] for the equation such that [Formula: see text] is existentially closed in [Formula: see text]. For instance, it follows that if the [Formula: see text]-constants of a formally real differential field [Formula: see text] is a closed ordered[Formula: see text]-field, then for any homogeneous linear [Formula: see text]-equation over [Formula: see text] there exists a PPV extension that is formally real. Similar observations apply to [Formula: see text]-adic fields.

More on Galois Cohomology, Definability, and Differential Algebraic Groups

“…By Lemma 5.5 (and 5.4), H 1 ∂ (K, ker( ) ∩ A)) is countable, and the same holds also for any K-form of ker( ) ∩ A (as it also has finite Morley rank). By Theorem 4.1 of [3]…”

“…Alternatively, note that the finitely many points of G are in K alg so really This is an adaptation of Case 1 of the proof of Theorem 4.1 of [3]. We give some details.…”

“…Differential Galois cohomology. In this section we complete the results on differential constrained cohomology of differential algebraic groups over bounded, differentially large, fields from [3]. We also clarify a proof from [1] about the algebraic Galois cohomology of algebraic groups over bounded fields.…”

“…For model-theoretic notions, differential algebraic notions and their interrelations, see the introductions to [3] and [7], and the various references there (such as Poizat's [9]). But we recall some key definitions: The field K (which we are assuming to be of characteristic 0) is bounded or equivalently has Serre's property (F), if K has only finitely many extensions of degree n, for each n. A field K is said to be large if whenever W is a K-irreducible variety with a smooth K-point, then V has a Zariski-dense set of K-points.…”

“…is finite as well. The corollary above, in the special case of Example 1.3(ii), is used in [3,Lemma 2.6]. A sketch of the argument is pointed out there, in the differential case, with details left to the reader.…”