Model theory of fields with finite group scheme actions
aa r X i v : . [ m a t h . L O ] J un MODEL THEORY OF FIELDS WITH FINITE GROUP SCHEMEACTIONS
DANIEL MAX HOFFMANN † AND PIOTR KOWALSKI ♠ Abstract.
We study model theory of fields with actions of a fixed finite groupscheme. We prove the existence and simplicity of a model companion of thetheory of such actions, which generalizes our previous results about truncatediterative Hasse-Schmidt derivations ([13]) and about Galois actions ([14]). Asan application of our methods, we obtain a new model complete theory ofactions of a finite group on fields of finite imperfection degree. Introduction
In this paper, we deal with model theory of actions of finite group schemes onfields. Our results here provide a generalization of both the results from [13] (modeltheory of truncated iterative Hasse-Schmidt derivations on fields) and the resultsfrom [14] (model theory of actions of finite groups on fields). We explain below thenature of this generalization.Let us fix a field k . By a finite group scheme over k , we mean the dual object to afinite-dimensional Hopf algebra over k (see Section 2). Let us fix such a finite groupscheme g . It is a classical result ([29, Section 6.7]) that g fits into the followingshort exact sequence: 1 −→ g −→ g −→ g −→ , where the finite group scheme g is infinitesimal (that is: its underlying scheme isconnected) and the finite group scheme g is ´etale (that is: it becomes a constantfinite group scheme after a finite Galois base extension). As explained in [13], theactions of infinitesimal finite group schemes on fields correspond to truncated iter-ative Hasse-Schmidt derivations. It is also clear that the actions of constant finitegroup schemes on fields correspond to actions of finite groups, and the model theoryof such actions was analyzed in [14]. It should be clear now that the model theoryof actions of finite group schemes, which is the topic of this paper, encompasses thesituations from [13] and [14].Before describing the main results of this paper, let us compare the structureswe consider here with a certain kind of operators on fields. Moosa and Scanlon de-veloped in [22] a general theory of rings with iterative operators. They analyzed themodel theory of fields with such operators in [24] under two additional assumptions:the base field is of characteristic 0 and the operators are free (every free operatoris inter-definable in a natural way with a certain iterative operator as explained in † SDG. The first author is supported by the Polish National Agency for Academic Exchange. ♠ Supported by the Narodowe Centrum Nauki grant no. 2018/31/B/ST1/00357 and by theT¨ubitak 1001 grant no. 119F397.2010
Mathematics Subject Classification
Primary 03C60, 14L15 Secondary 11S20.
Key words and phrases . Finite group scheme action, Model companion, Simple theory. [24]). The results from [24] were extended to an arbitrary characteristic case (stillin the free context) in [1]. The actions of finite group schemes, which are the topicof this paper, are the same as iterative operators in the sense of Moosa-Scanlon for constant iterative systems (as we explain in Section 2), and they should be thoughtof as “very non-free” iterative operators.We list here the main results of this paper.(1) The theory of actions of g on fields has a model companion, which we denoteby g -DCF (Theorem 3.6).(2) The theory g -DCF is simple (Theorem 4.12).(3) The theory of actions of a finite group G on fields of characteristic p > e has a model companion, which is bi-interpretable (after adding finitely many constants) with the theory ( G a [1] × G F p )-DCF, where G a [1] is the kernel of the Frobenius endomorphism onthe additive group scheme over the prime field F p (Theorem 5.6).Coming back to the situation from [24], we should mention that one can not ex-pect a generalization of the above results to the case of general iterative operatorsconsidered in [24], since the theory of such operators on fields may be not compan-ionable as it is shown in [24] and [1] (we comment more on this issue in Remark4.13(5)).This paper is organized as follows. In Section 2, we give the main definitions,provide some examples, and we also discuss the notion of a prolongation withrespect to a fixed action of g . In Section 3, we show the existence of a modelcompanion of the theory of g -actions on fields using the notion of a prolongationfrom Section 2. In Section 4, we analyze the model-theoretic properties of the theoryobtained in Section 3. In particular, we show by the Galois-theoretic methods thatthis theory is simple. As an application of the results from Sections 3 and 4, we givein Section 5 a new example of a theory fitting to the set-up considered by the firstauthor in [11], that is we prove the existence of a model companion of the theoryof actions of a finite group on fields of a fixed finite imperfection degree.2. Finite group scheme actions: first-order set-up and prolongations
In this section, we describe our set-up of finite group scheme actions both in thealgebraic as well as in the model-theoretic context. Then, we define the notion of aprolongation and collect some technical tools, which will be needed in the sequel.We fix for the rest of this paper a base field k .2.1. Finite group scheme actions: definitions and examples.
For the nec-essary background about affine group schemes, their actions, and Hopf algebras,we refer the reader to [29]. All the group schemes considered in this paper will beaffine, so we will skip the adjective “affine” from now on. Let us fix a finite groupscheme g over k . Then g = Spec( H ), where H is a finite dimensional Hopf algebraover k . We denote the comultiplication in H by µ : H → H ⊗ k H , and we denotethe counit map in H by π : H → k . We also fix the number e := dim k ( H ), whichis called the order of the finite group scheme g . Definition 2.1.
We say that R is a g -ring , if R is a k -algebra together with agroup scheme action of g on Spec( R ).Similarly, we define g -fields , g -morphisms , g -extensions , etc. Remark 2.2.
Let R be a k -algebra. ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 3 (1) A g -ring structure on R is the same as a k -algebra map ∂ : R → R ⊗ k H such that π ◦ ∂ = id R (the counit condition) and the following diagramcommutes (the coassociativity condition): R ⊗ k H id ⊗ µ / / ( R ⊗ k H ) ⊗ k HR ∂ O O ∂ / / R ⊗ k H. ∂ ⊗ id O O If R is a g -ring and we want to emphasize the group scheme action on R ,then we say: “( R, ∂ ) is a g -ring”.(2) A k -algebra map ∂ : R → R ⊗ k H satisfying only the counit condition (fromItem (1) above) was called an H -operator in [1]. This notion originated from[23, 24] and should be thought of as a “free operator” in the sense that it isfree from the coassociativity condition (or the iterativity condition ) givenby the multiplication in g . Example 2.3.
We give below several examples of finite group schemes and weinterpret their actions.(1) For a finite group G , we denote by G k the finite group scheme over k corresponding to G , that is: G k := Spec(Func( G, k )) , where the comultiplication in Func( G, k ) comes from the group operationin G . Then, G k -rings coincide with k -algebras with actions of the group G by k -algebra automorphisms.(2) Let us assume that char( k ) = p >
0. A truncated group scheme (see [4])over k is a finite group scheme whose underlying scheme is isomorphic toSpec (cid:16) k [ X , ..., X f ] / (cid:16) X p m , . . . , X p m f (cid:17)(cid:17) for some f, m >
0. If the base field k is perfect, then each infinitesimalgroup scheme over k is truncated (see e.g. [9, Corollary 6.3, p. 347]).If g is a truncated group scheme, then ( R, ∂ ) is a g -ring if and only if ∂ is a g -derivation on R in the sense of [12, Definition 3.8] (see also [12,Definition 3.9] for an interpretation in terms of m -truncated f -dimensionalHasse-Schmidt derivations on R over k satisfying a “ g -iterativity” rule).The simplest example of a truncated group scheme is the kernel of theFrobenius morphisms on the additive group scheme over k and we denotethis kernel by G a [1]. Then, a G a [1]-ring is the same as a k -algebra R together with a k -derivation d such that d ( p ) = 0 (the composition of d with itself p times).(3) Let us assume that g = g × g . Then, an action of g on a k -scheme X maybe understood as an action of g on X together an action of g on X suchthat these two actions commute with each other.In the case when char( k ) = p > g = G a [1] × G k for a finite group G , a g -ring R is the same as a derivation d on R such that d ( p ) = 0 (seeItem (2) above) together with an action of G on R by k -algebra differentialautomorphisms. D. M. HOFFMANN AND P. KOWALSKI (4) Assume that g is a finite ´etale group scheme. Then, there is a finite Galoisextension k ⊆ k ′ such that the group scheme g k ′ := g × Spec( k ) Spec( k ′ )over k ′ is constant, that is: there is finite group G such that g k ′ ∼ = G k ′ (seee.g. [29, Section 6]).As an explicit non-constant example, let us take g as the finite groupscheme of third roots of unity over Q . That is: g = Spec( Q [ ε ]) , Q [ ε ] := Q [ X ] / ( X − , and the counit and the comultiplication in Q [ ε ] are given by: π ( ε ) = 1 , µ ( ε ) = ε ⊗ ε. We also have that: g Q ( ζ ) ∼ = ( Z / Z ) Q ( ζ ) , where ζ = exp(2 πi/
3) is the third primitive root of unity in C .The following Q -algebra map: ∂ : Q ( √ −→ Q ( √ ⊗ Q Q [ ε ] , ∂ ( √
2) := √ ⊗ ε gives Q ( √
2) a g -field structure. This action has an obvious functorialdescription on R -rational points ( R is a Q -algebra): the group of thirdroots of unity in R naturally acts by the ring multiplication on the set ofthird roots of 2 in R .As mentioned in the Introduction, we will provide now a comparison betweenthe notion of a g -ring considered in this paper and the notion of a D -ring for ageneralized Hasse-Schmidt system D from [22]. In short, the notion considered in[22] is much more general than ours. A generalized Hasse-Schmidt system D is aninverse system of schemes with scheme morphisms between them, which may bethough of as “partial group scheme structures”. Such an idea is also consideredin [17], where D is replaced with a basically equivalent notion of a formal groupscheme , which is a direct system of finite schemes together with some morphismsbetween them such that they become a group operation on the level of the formallimit. In our case, we consider just a constant system consisting of one finite groupscheme g (and the identity maps).2.2. First-order set-up for finite group scheme actions.
In this subsection,we will describe a first-order language L g and an L g -theory g -DF such that modelsof g -DF are exactly g -fields. If we skip the “iterativity condition” (see Remark2.2(2)), then we are exactly in the set-up from [1]. Let us recall this set-up briefly.We fix { b , . . . , b e − } , which is a k -basis of H such that π ( b ) = 1 and π ( b i ) = 0for i >
0. Then, for a k -algebra R , any map ∂ : R → R ⊗ k H can be identified witha sequence ∂ , ∂ , . . . , ∂ e − : R → R. The map ∂ satisfies the counit condition (see Remark 2.2(1)) if and only if ∂ = id R .To express the coassociativity condition, we need to use the structure constants for ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 5 the comultiplication on H , which for i, j, l < e are the elements c i,jl ∈ k such thatthe following equality holds: µ ( b l ) = X i,j R, ∂ ) satisfies the coassociativity condition if and only if for any r ∈ R , wehave: ∂ i ∂ j ( r ) = e − X l =0 c i,jl ∂ l ( r ) . Let L g be the language of rings expanded by e − k . It is clear by what is written above,that there is an L g -theory, which we denote by g -DF, whose models are exactly g -fields. We are aiming towards showing that this theory has a model companion(Theorem 3.6) and that this model companion is simple (Corollary 4.12). Example 2.4. We discuss here model-theoretic properties of g -fields from Example2.3. (1) If g = G k for a finite group G , then the theory g -DF coincides with thetheory of G -transformal fields from [14]. It is proved in [14] that the theoryof G -transformal fields has a model companion (denoted G -TCF), which issupersimple of finite rank coinciding with the order of G .(2) If char( k ) = p > g is a truncated group scheme, then the theory g -DF was already introduced (with the same name in this truncated context)in [13]. It is shown in [13] that the theory g -DF has a model companion(denoted g -DCF), which is strictly stable.(3) If g = g × g , where g is non-trivial truncated and g is non-trivial, then,as for as we know, the model theory of g -fields has not been consideredbefore. If the finite group scheme g is constant, then this theory fits tothe set-up considered by the first author in [11], which will be explained inSection 5.(4) If g is a finite ´etale group scheme which is not constant, then, as far aswe know, the iterativity rules (and the “Leibniz rules”) coming from suchactions have not been studied yet. We consider the action from Example2.3(4) and we use the notation from there. We have to fix a “good basis” b , b , b first such that π ( b ) = 1 and π ( b ) = π ( b ) = 0. This choice isquite arbitrary, we have chosen the following: b := 1 + ε + ε , b := ε − , b := ε − . (a) We have the following multiplication table: b = b , b = − b + b ,b b = 0 , b = b − b ,b b = 0 , b b = − b − b . D. M. HOFFMANN AND P. KOWALSKI Then, the multiplicative rule for g -operators (with respect to this basis)is the following one: ∂ ( xy ) = − ∂ ( x ) ∂ ( y ) − ∂ ( x ) ∂ ( y ) − ∂ ( x ) ∂ ( y ) + ∂ ( x ) ∂ ( y )3 ,∂ ( xy ) = ∂ ( x ) ∂ ( y ) − ∂ ( x ) ∂ ( y ) − ∂ ( x ) ∂ ( y ) − ∂ ( x ) ∂ ( y )3 . (b) We have the following comultiplication table: µ ( b ) = b ⊗ b + 2 b ⊗ b − b ⊗ b − b ⊗ b + 2 b ⊗ b ,µ ( b ) = b ⊗ b + b ⊗ b + b ⊗ b − b ⊗ b − b ⊗ b ,µ ( b ) = b ⊗ b + b ⊗ b + b ⊗ b − b ⊗ b − b ⊗ b , which gives the following g -iterativity rules ( ∂ = id): ∂ ◦ ∂ = 2 ∂ + ∂ ,∂ ◦ ∂ = 2 ∂ + ∂ ,∂ ◦ ∂ = − ∂ − ∂ − ∂ = ∂ ◦ ∂ . (c) It is interesting to see how the rules from Item (b) above transforminto the rules for the usual action of Z / Z in the case when our g -fieldcontains the primitive third root of unity ζ in the constants of theaction. In such a case, we set the following new basis: b ′ := b = 1 + ε + ε , b ′ = 1 + ζ ε + ζε , b ′ = 1 + ζε + ζ ε . It is easy to see that it is an “orthogonal basis”, that is b ′ i b ′ j = δ ij (theKronecker delta) and that the comultiplication on b ′ , b ′ , b ′ gives theconstant group scheme coming from the group Z / Z . Then, we havethe following transformation rules explaining how a g -action given by ∂ , ∂ becomes an action of Z / Z given by ∂ ′ , ∂ ′ : ∂ ′ = ζ ∂ − ζ∂ ζ − ζ , ∂ ′ = − ζ∂ + ζ ∂ ζ − ζ . Prolongations with respect to finite group scheme actions. We fix inthis subsection a g -field ( K, ∂ ). We will recall the notion of a prolongation from[23], which is closely related to the Weil restriction of scalars. We need the followingdefinition first. Definition 2.5. For a K -algebra R , we consider a “ ∂ -twisted” K -algebra structureon R ⊗ k H , which is the K -algebra structure given by the following composition: K ∂ / / K ⊗ k H / / R ⊗ k H. We denote the ring R ⊗ k H with the above ∂ -twisted K -algebra structure by R ⊗ ∂ k H . Remark 2.6. The functor R R ⊗ ∂ k H controls the g -ring extensions of K . Moreprecisely, a map ∂ ′ : R −→ R ⊗ ∂ k H is a K -algebra homomorphism if and only if ( R, ∂ ′ ) is a g -ring extension of ( K, ∂ ).By a K -variety , we always mean a reduced K -irreducible algebraic subvariety of A nK for some n > ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 7 Definition 2.7. Let V be a K -variety.(1) A K -variety ∇ ( V ) is called a ∂ -prolongation of V , if for each K -algebra R ,there is a functorial bijection: ∇ ( V )( R ) ←→ V (cid:0) R ⊗ ∂ k H (cid:1) . (2) For any K -algebra R , the counit map µ : H → k induces a natural K -algebra map R ⊗ ∂ k H → R , which yields the following functorial morphism: π V : ∇ ( V ) −→ V. From now on, we will regard the natural bijection from Definition 2.7(1) as theidentity map. Remark 2.8. We collect here several observations regarding the notion of a ∂ -prolongation.(1) As explained in [23], ∂ -prolongations exist and they coincide with the Weilrestriction to K of the base extension from K to K ⊗ ∂ k H .(2) On the level of rings, the ∂ -prolongation functor is a left-adjoint functor tothe following functor:Alg K ∋ R R ⊗ ∂ k H ∈ Alg K . (3) It is good to point out that the notion of a ∂ -prolongation does not dependon the group scheme structure on g . In particular, ∂ -prolongations alsoexist if ∂ is a “non-iterative” H -operator on K in the sense of Remark2.2(2). However, we notice below that if ( K, ∂ ) is a g -field (which is ourassumption here), then the ∂ -prolongation functor still has some in-builtiterativity properties (see Definition 2.13 and Remark 2.18). Example 2.9. For finite group schemes considered in Example 2.3(1) and (2), the ∂ -prolongation functor has a natural description. Let V be a K -variety.(1) If g = G k for a finite group G , then G acts on K by field automorphisms.Let us recall that for any σ ∈ Aut( K ), we have the “ σ -twisted” variety V σ defined as: V σ := V × Spec( K ) (Spec( K ) , Spec( σ )) . If G = { g , . . . , g e } , then we have: ∇ ( V ) = V g × . . . × V g e (see [14, Remark 2.7(4)]).(2) If char( k ) = p > g is a finite truncated group scheme, then ∇ ( V ) is atorsor of a higher tangent bundle of V . For a more explicit description, letus consider the simplest case when g = G a [1] and p = 2. Then, we knowthat ∂ corresponds to a derivation d on K such that d ◦ d = 0. If V isdefined over the kernel of d , then ∇ ( V ) coincides with the tangent bundleof V . Definition 2.10. Let V be a K -variety and ( R, ∂ ′ ) be a g -ring extension of ( K, ∂ ).By Remark 2.6, the map ∂ ′ : R −→ R ⊗ ∂ k H is a K -algebra homomorphism, hence it induces the following map, which is naturalin V (but it is not a morphism!): ∂ ′ V := V ( ∂ ′ ) : V ( R ) −→ V (cid:0) R ⊗ ∂ k H (cid:1) = ∇ ( V )( R ) . D. M. HOFFMANN AND P. KOWALSKI It is easy to observe the following. Remark 2.11. Let us assume that we are in the situation from Definition 2.10.(1) The following diagram is commutative: V ( R ) ∂ ′ V / / ∇ ( V )( R ) V ( K ) ⊆ O O ∂ V / / ∇ ( V )( K ) . ⊆ O O (2) Let as assume that V ⊆ A n and let ∂ , . . . , ∂ e − (see Section 2.2) actcoordinate-wise on A n ( R ) = R n . Then, the map ∂ ′ V : V ( R ) → ∇ ( V )( R ) isgiven by: ∂ ′ V ( r ) = (cid:0) r, ∂ ′ ( r ) , . . . , ∂ ′ e − ( r ) (cid:1) , and the map π V corresponds to the projection on the first ( n -tuple) coor-dinate.We will describe now a morphism ∇ ( V ) −→ ∇ ( ∇ ( V )), which reflects the “itera-tivity” of ∂ on the level of the ∂ -prolongation (it was mentioned in Remark 2.8(3)).We need a lemma first. Lemma 2.12. The map id ⊗ µ : R ⊗ ∂ k H −→ (cid:0) R ⊗ ∂ k H (cid:1) ⊗ ∂ k H is a morphism of K -algebras.Proof. Let ι : K → R be the K -algebra structure map. By Remark 2.2(1), thefollowing diagram commutes: R ⊗ k H id ⊗ µ / / R ⊗ k H ⊗ k HK ⊗ k H id ⊗ µ / / ι ⊗ id O O K ⊗ k H ⊗ k H ι ⊗ id ⊗ µ O O K ∂ / / ∂ O O K ⊗ k H. ∂ ⊗ id O O Since the ring homomorphism ( ι ⊗ id) ◦ ∂ gives the K -algebra structure of R ⊗ ∂ k H and the ring homomorphism ( ι ⊗ id ⊗ id) ◦ ( ∂ ⊗ id) ◦ ∂ gives the K -algebra structureon ( R ⊗ ∂ k H ) ⊗ ∂ k H , the result follows. (cid:3) Definition 2.13. Let V be an affine K -scheme. We define the following schememorphism (natural in V ) c V : ∇ ( V ) −→ ∇ ( ∇ ( V ))using for any K -algebra R the following map (well-defined by Lemma 2.12): c V := V (id ⊗ µ ) : V (cid:0) R ⊗ ∂ k H (cid:1) −→ V (cid:0)(cid:0) R ⊗ ∂ k H (cid:1) ⊗ ∂ k H (cid:1) . ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 9 Remark 2.14. (1) In the set-up of Remark 2.11(2), the morphism c V aboveis given on rational points by the following formula: c V ( x , . . . , x e − ) = e − X l =0 c i,jl x l ! i,j We have the following: ∂ ∇ ( V ) ◦ ∂ V = c V ◦ ∂ V . Proof. By Definitions 2.10 and 2.13, we have: ∂ V = V ( ∂ ) , c V := V (id ⊗ µ ) . Therefore, the statement we are showing follows from applying V (regarded as thefunctor of rational points) to the commutative diagram from Remark 2.2(1). (cid:3) By the definition of ∇ ( V ), the morphisms V → ∇ ( V ) correspond to the K -algebra maps K [ V ] → K [ V ] ⊗ ∂ k H . The following result explains when such mor-phisms correspond to g -ring structures on K [ V ] extending ∂ on K (equivalently: g -actions on V ). Lemma 2.16. Let us fix a morphism ϕ : V → ∇ ( V ) . Then, the following areequivalent. (1) The corresponding map K [ V ] → K [ V ] ⊗ ∂ k H makes K [ V ] a g -ring extensionof K . (2) We have π V ◦ ϕ = id V and the following diagram is commutative: ∇ ( V ) c V / / ∇ ( ∇ ( V )) V ϕ O O ϕ / / ∇ ( V ) . ∇ ( ϕ ) O O Proof. It follows directly from the adjointness property defining the functor ∇ andfrom Remark 2.2(1). (cid:3) Example 2.17. By Lemma 2.16, we obtain that the morphism c V : ∇ ( V ) −→ ∇ ( ∇ ( V ))gives the ring K [ ∇ ( V )] a canonical g -ring structure e ∂ = ( e ∂ , . . . , e ∂ e − ). This canon-ical structure is dual (or rather “adjoint”) to the one considered in [12, Proposition3.4].Let us give an explicit description of this structure for V = A , so K [ ∇ ( V )] = K [ X , . . . , X e − ] =: K [ X 0, we have that ∇ ( A n ) = A ne , by Remark 2.14 we get thefollowing interpretation (using the fixed basis b , . . . , b e − of H over k ): c A : ∇ ( A )( R ) = R e −→ ∇ ( ∇ ( A ))( R ) = R e , c A ( r , . . . , r e − ) = e − X l =0 c i,jl r l ! i,j We would like to say few words about an explicit constructionof the ∂ -prolongation functor understood as the left-adjoint functor from Remark2.8(2). If R = K [ X ] /I is an arbitrary K -algebra, then there is a “non-iterative H -operator ∂ from K [ X ] to K { X } := K [ X, X , . . . , X e − ]” defined by ∂ i ( X ) := X i ,and the ∂ -prolongation of R coincides with K { X } / ( ∂ ( I )).In our “iterative” situation, the operator ∂ extends to the g -ring structure e ∂ asexplained in Example 2.17. Then, we have that ∂ ( I ) = e ∂ ( I ) is an “ g -ideal”, hence ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 11 the quotient K { X } / ( ∂ ( I )) gets a natural g -ring structure, which coincides with theone described in Example 2.17.3. Model companion In this section, we prove the existence of a model companion of the theory of g -fields for an arbitrary finite group scheme g over the field k . The proofs in thissection follow the lines of the proofs from [19] and [13], however, we made someimprovements here in order to make these proofs simpler and “coordinate-free”.For the next result, we need to introduce the notion of constants of a g -ring(these are the same constants as in [1, Definition 2.2(3)]). Definition 3.1. Let ( R, ∂ ) be a g -ring. We denote by R g the ring of constants of( R, ∂ ), that is: R g := { x ∈ R | ∂ ( x ) = x } , where ∂ is understood as a map R → R ⊗ k H and R is considered as a subring of R ⊗ k H . Remark 3.2. We note here several properties of the ring extension R g ⊆ R .(1) By [21, Theorem 4.2.1], the ring extension R g ⊆ R is integral (see also theproof of [25, Section 12, Theorem 1]).(2) The extension R g ⊆ R need not be finite, that is R is not necessarily afinitely generated R g -module (by Item (1) above, it is equivalent to sayingthat R is not necessarily a finitely generated R g -algebra). Finite generationfails in the simplest non-trivial case, that is for the constant group g =( Z / Z ) k as demonstrated in [20, Example 5.5] (a Noetherian domain R with an action of G = Z / Z such that R G is not Noetherian and R is nota finite R G -module).Finite generation holds in the case when R is a finitely generated k -algebra (it is quite easy to show knowing that the ring extension R g ⊆ R is integral, see e.g. [27, Lemma 10, page 49]), but this is not a goodassumption for us.(3) We will show in Section 4 (Theorem 4.4) that if K is a field, then the fieldextension K g ⊆ K is finite and of degree bounded by e , as in the case ofgroup actions. It is hard to believe that this result is new, but we wereunable to find it in the literature. The closest result we could find is [21,Theorem 8.3.7], but there is an extra assumption there saying that theextension K g ⊆ K should be Hopf Galois (see [21, Definition 8.1.1]) and,as [21, Example 8.1.3] shows, the field extension K g ⊆ K need not be HopfGalois in general. Lemma 3.3. Suppose that R is a g -ring and R → S, R → T are homomorphismsof g -rings. Then, we have the following. (1) There is a unique g -ring structure on S ⊗ R T such that the natural maps S → S ⊗ R T, T → S ⊗ R T are homomorphisms of g -rings. (2) If R is a domain, then the g -ring structure on R extends uniquely to itsfield of fractions, which we denote by Frac( R ) .Proof. Item (1) is an instance of the following very general fact: if a group G in acategory acts on the objects X and Y , then there is a unique G -action on X × Y such that the projection maps X × Y → X, X × Y → Y commute with the actionof G .For Item (2), we notice first that by Remark 3.2(1) the extension of domains R g ⊆ R is integral, which implies that:Frac ( R g ) [ R ] = Frac( R ) . Since tensor products commute with localizations, we get that:Frac ( R g ) ⊗ R g R ∼ = R Frac( R ) . Therefore, using Item (1), we obtain that there is an g -action on Frac( R ) extendinguniquely the trivial action on Frac ( R g ) and the given action on R . (cid:3) Remark 3.4. We do not know whether Lemma 3.3 holds without the finitenessassumption on the group scheme.For the rest of this section we fix a g -field K and a K -variety V . The followinglemma is an extended version of [19, Lemma 1.1(ii)]. Proposition 3.5. Assume that W is a K -variety such that W ⊆ ∇ ( V ) and c V ( W ) ⊆ ∇ ( W ) . Then, we have the following. (1) There is a g -field structure ∂ ′ on K ( W ) such that K ⊆ K ( W ) is a g -fieldextension. (2) Let b : W → V denote the composition of the inclusion W ⊆ ∇ ( V ) andthe projection morphism π V : ∇ ( V ) → V . We regard b as an element of V ( K ( W )) . Then, we have: ∂ ′ V ( b ) ∈ W ( K ( W )) , where ∂ ′ comes from Item (1) above.Proof. For the proof of Item (1), we consider the morphism: ϕ := c V | W : W −→ ∇ ( W ) . By Example 2.17, the morphism ϕ satisfies the second condition from Lemma 2.16,hence K ⊆ K [ W ] has a natural structure of a g -ring extension. By Lemma 3.3, the g -ring structure on K [ W ] extends to a g -field structure, which we denote by ∂ ′ , on K ( W ).For Item (2), we notice first that the rational point ∂ ′ V ( b ) ∈ ∇ ( V )( K ( W )) cor-responds to the following morphism: ∇ ( b ) ◦ ϕ : W −→ ∇ ( V ) . It is enough to show that this morphism coincides with the inclusion W ⊆ ∇ ( V ),which follows from the commutativity of the following diagram: W ⊆ / / ϕ (cid:15) (cid:15) ∇ ( V ) c V (cid:15) (cid:15) ∇ ( W ) ⊆ / / ∇ ( b ) ( ( PPPPPPPPPPPPP ∇ ( ∇ ( V )) ∇ ( π V ) (cid:15) (cid:15) ∇ ( V ) ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 13 and the facts that ∇ ( π V ) = π ∇ ( V ) and π ∇ ( V ) ◦ c V = id ∇ ( V ) (since c V satisfies thecounit condition from Remark 2.2(1)). (cid:3) We can formulate now: Geometric axioms for g -DCF For each positive integer n , suppose that V ⊆ A n and W ⊆ ∇ ( V ) are K -varieties.If c V ( W ) ⊆ ∇ ( W ), then there is a ∈ V ( K ) such that ∂ V ( a ) ∈ W ( K ).It is standard to see that these axioms are first-order, see e.g. the discussion in [14,Remark 2.7]. Theorem 3.6. The g -field ( K, ∂ ) is existentially closed if and only if ( K, ∂ ) satisfiesthe geometric axioms above.Proof. For the left-to-right implication, let us assume that the g -field K is exis-tentially closed and take V, W as in the assumptions of the geometric axioms for g -DCF. As in the statement of Proposition 3.5, we regard the projection morphism W → V as a rational point b ∈ V ( K ( W )). By Proposition 3.5, ( K, ∂ ) ⊆ ( K ( W ) , ∂ ′ )is a g -field extension and ∂ ′ V ( b ) ∈ W ( K ( W )). By Lemma 2.11 and our assump-tion saying that the g -field ( K, ∂ ) is existentially closed, we obtain that there is a ∈ V ( K ) such that ∂ V ( a ) ∈ W ( K ).For the right-to-left implication, let us assume that ( K, ∂ ) satisfies the geometricaxioms for g -DCF. We take a g -field extension ( K, ∂ ) ⊆ ( L, ∂ ′ ) and a quantifier-free L g -formula φ ( x ) over K having a realisation v in ( L, ∂ ′ ). By the usual trickof introducing extra variables (to get rid of any negations of equalities), we canassume that for V := locus K ( v ) , W := locus K ( ∂ ′ V ( v ))the formula φ ( x ) is implied by the formula saying that “ x ∈ V and ∂ V ( x ) ∈ W ”(the latter is clearly a first-order formula by Remark 2.11(2)). By Lemma 2.15, weget that c V ( W ) ⊆ ∇ ( W ), thus by the geometric axioms for g -DCF, the formula φ ( x ) has a realisation in ( K, ∂ ). (cid:3) Example 3.7. The geometric axioms for g -DCF generalize both the axiomati-zations given in [13] and in [14]. We explain below how the axioms for g -DCFspecialize to these two cases.(1) In [13], the case of a truncated (see Example 2.3(2)) finite group scheme g is considered. The axioms for the theory of existentially closed g -fieldsfrom [13, Section 5.2] have basically the same form as the axioms above.However, the idea that similar axioms as in [13, Section 5.2] may work foran arbitrary finite group scheme g only came to us at the final moments ofwriting the article [14] (see [14, Remark 5.1]).(2) In [14], we studied model theory of actions of finite groups on fields, whichcorrespond to actions of finite constant group schemes (see Example 2.4(1)).The geometric version of the corresponding axioms is given in [14, Remark2.7(2)(b)] and the comparison to axioms from Item (1) above (so, also tothe axioms from Theorem 3.6) is discussed in [14, Remark 2.7(4)].4. Fields of constants and simplicity The main result of this section is Theorem 4.12, which says that the theory g -DCF (see Theorem 3.6) is simple. This result follows from a Galois-theoreticanalysis of the field of constants of an existentially closed g -field: we show that this field is pseudo-algebraically closed and bounded. We also show that any g -field isa finite extension of its field of constants (as a pure field).Our first aim is to show the finite generation result mentioned above. The nextlemma holds in any category in which quotients of group actions exist (as in thecase of standard group actions). We recall that quotients exist in the category offinite group schemes (see e.g. [29, Section 16.3]). Lemma 4.1. Assume that K is a g -field and n is a normal subgroup scheme of g .Then, we have the following. (1) K is an n -field and K n is a g / n -field. (2) The following two subfields of invariants coincide: ( K n ) g / n = K g . Thanks to Lemma 4.1, we can reduce the proof of our first aim to the case ofa reasonably simple finite group scheme g . The next two results deal with someparticular cases of such group schemes. We recall that e is the order of g , that is e = dim k ( H ), where g = Spec( H ). Lemma 4.2. Assume that K is a g -field, char( k ) = p > and g = ker(Fr g ) . Then,we have: [ K : K g ] e. Proof. The proof of [13, Corollary 3.21] works in the case considered here as well(our e here plays the role of pe from [13]). (cid:3) Lemma 4.3. Assume that K is a g -field and g is ´etale. Then we have: [ K : K g ] e. Proof. Let k ⊆ l be a finite Galois extension such that g l ∼ = G l , where G is a finitegroup of order e (see Example 2.3(4)). Since K is a g -field, K ⊗ k l becomes a g l -ring,which means that the group G acts on K ⊗ k l by l -algebra automorphisms. By [1,Theorem 4.4], it is enough to show that K ⊗ k l can be generated by e elements asa ( K ⊗ k l ) G -module.Since the extension k ⊆ l is finite separable, the ring K ⊗ k l is a reduced Artinian K -algebra, so it is semisimple. By [20, Theorem 2.19] and the comments about thefinite number of generators after its proof (since a commutative semisimple ring isnecessarily self-injective), our inequality follows. (cid:3) We are ready now to show our main finite generation result. As already pointedout in Remark 3.2(3), we doubt whether this result is new, but we could not findany reference to it. Theorem 4.4. Assume that K is a g -field. Then we have: [ K : K g ] e. Proof. By the short exact sequence of finite group schemes from the Introduction(see [29, Section 6.7]) and Lemma 4.1, we can assume that g is connected or g is´etale. The ´etale case holds by Lemma 4.3. For the connected case, by Lemma 4.1again we can assume that ker(Fr g ) = g (see [13, Remark 2.3]), and then the proofis concluded by Lemma 4.2. (cid:3) ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 15 We recall (the reader may consult e.g. [10, Section 11]) that a field M is pseudo-algebraically closed , abbreviated PAC , if each absolutely irreducible variety over M has an M -rational point. Proposition 4.5. If K is an existentially closed g -field, then K g is PAC.Proof. By [10, Proposition 11.3.5], we need to show that K g is existentially closed ineach regular field extension K g ⊆ L . We can assume that K and L are algebraicallydisjoint over K g . Let us define: M := K ⊗ K g L. By Lemma 3.3(1), there is a g -ring structure ∂ ′ on M such that L ⊆ R ∂ ′ . Since thefield extension K g ⊆ L is regular and the field extension K g ⊆ K is finite algebraic(by Theorem 4.4), the ring M is a field. Then we have the following: • ( K, ∂ ) ⊆ ( M, ∂ ′ ) is a g -field extension; • the g -field ( K, ∂ ) is existentially closed; • L ⊆ M ∂ ′ .The conditions above imply that for any quantifier-free formula ϕ ( x ) over K g in thelanguage of fields, ϕ ( K g ) = ∅ if and only if ϕ ( L ) = ∅ . Therefore, K g is existentiallyclosed in L , what we needed to show. (cid:3) Corollary 4.6. If K is an existentially closed field g -field, then K is PAC.Proof. Since an algebraic extension of a PAC field is a PAC field (see [10, Corollary11.2.5]), it is enough to use Proposition 4.5 and Theorem 4.4. (cid:3) For the boundedness part, we would like to single out first the following Galois-theoretic result, which appeared in some form in [28], [14, Lemma 3.7], and [2,Lemma 2.8]. We recall that for a field M , Gal( M ) denotes the absolute Galoisgroup of M (consider as a topological profinite group), that is:Gal( M ) := Gal ( M sep /M ) . For the necessary background about Frattini covers, we refer the reader to [10,Chapter 22]. Proposition 4.7. Suppose that M ⊆ N is a finite Galois field extension. Thenthe following conditions are equivalent. (1) The restriction map res : Gal( M ) −→ Gal( N/M ) is a Frattini cover. (2) For any separable field extension M ⊆ M ′ , M ′ is not linearly disjoint from N over M .Proof. We will use [2, Lemma 2.8], which says that for any closed subgroup H Gal( M ) and for M ′ := (cid:0) M alg (cid:1) H , the following conditions are equivalent:(A) res( H ) = Gal( N/M ), (B) the restriction mapGal( N M ′ /M ′ ) −→ Gal( N/M )is an isomorphism.It is easy to see that Condition (B) above is equivalent to saying that M ′ is linearlydisjoint from N over M . Since the restriction mapres : Gal( M ) −→ Gal( N/M )is a Frattini cover if and only if for each proper closed subgroup H as above, wehave res( H ) = Gal( N/M ), the result follows. (cid:3) Our next aim is to show that if K is an existentially closed g -field, then K g isbounded, that is the absolute Galois group Gal( K g ) is small. As in the case ofactions of finite groups, we will achieve this aim by identifying Gal( K g ) with theuniversal Frattini cover of a certain finite group. Similarly as in the group case,the whole strength of the existential closedness assumption is not necessary here,we will only need the property expressed in the following definition. Definition 4.8. A g -field K is g -closed , if there are no non-trivial g -field extensions K ⊂ L such that the pure field extension K ⊂ L is algebraic.The following result is crucial for our proof of the simplicity of the theory g -DCF. Theorem 4.9. Assume that a g -field K is g -closed and C = K g . Let C ⊆ K nc bethe normal closure of the field extension C ⊆ K . Then, the restriction map: Gal( C ) −→ Gal( K nc /C ) is a Frattini cover.Proof. Let us define: C ins := ( K nc ) Gal( K nc /C ) . By [16, Theorem 19.18], the extension C ⊆ C ins is purely inseparable, the extension C ins ⊆ K nc is Galois, and the following diagram commutes:Gal( C ) res / / res ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Gal( K nc /C ins ) = (cid:15) (cid:15) Gal( K nc /C ) . Therefore, by Proposition 4.7, it is enough to show that for any proper Galois fieldextension C ins ⊂ C ins ′ , we have that K nc is not linearly disjoint from C ins ′ over C ins . Let us assume that there is a proper Galois field extension C ins ⊂ C ins ′ suchthat K nc is linearly disjoint from C ins ′ over C ins . We aim to reach a contradiction.Let C ⊆ C ′ be the maximal separable subextension of the extension C ⊂ C ins ′ .The situation is summarized in the following commutative diagram, where all the ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 17 arrows are inclusions of fields: K nc K = = ④④④④④④④④ C ins / / O O C ins ′ C a a ❈❈❈❈❈❈❈❈❈ / / O O C ′ . O O We have the following:[ C ins ′ : C ins ] = [ C ins ′ : C ] sep = [ C ′ : C ] , [ C ins : C ] = [ C ins ′ : C ] ins = [ C ins ′ : C ′ ] . Therefore, C ⊂ C ′ is a proper Galois extension and we have:Gal( C ′ /C ) ∼ = Gal( C ins ′ /C ins ) , K nc ∩ C ′ = C. Since the extension C ⊂ C ′ is Galois and K ∩ C ′ = C , we obtain that K is linearlydisjoint from C ′ over C . Hence, we have: KC ′ ∼ = K ⊗ C C ′ and by Lemma 3.3, the field extension K ⊂ KC ′ has a g -field extension structure.Since the field extension K ⊂ KC ′ is proper algebraic, we get a contradiction withthe assumption that K is g -closed. (cid:3) One could wonder whether the field extension C ⊆ K appearing in the statementof Theorem 4.9 is a Galois extension (as it is the case when we consider the actions ofgroups) or perhaps it is only a normal extension or it is only a separable extension.The example below shows that “anything may happen”, hence we really need toconsider the fields K nc and C ins in the proof of Theorem 4.9 above. Example 4.10. (1) If char( k ) = p > g = G a [1], and K is a g -field such thatthe g -action is non-trivial, then the extension K g ⊂ K is purely inseparableand non-trivial (see [13, Definition 3.5 and Lemma 3.16]).(2) To see that the field extension K g ⊆ K need not be normal (even in thecase when g is ´etale), one can consider the g -field Q ( √ 2) from the end ofExample 2.3(4), where g is the group scheme of the third roots of unityover Q . Then, we have: (cid:16) Q ( √ (cid:17) g = Q and the field extension Q ⊂ Q ( √ 2) is clearly not normal. Corollary 4.11. If K is an existentially closed g -field, then the field K g is PACand bounded.Proof. The PAC part is exactly Proposition 4.5.We proceed to show the boundedness part. Since an existentially closed g -fieldis g -closed, we can apply Theorem 4.9 and obtain that the restriction mapGal( K g ) −→ Gal( K nc /K g )is a Frattini cover. Since a Frattini cover of a topologically finitely generated profi-nite group (in our case, it just the finite group Gal( K nc /K g )) is finitely generated again (see [10, Corollary 22.5.3]), we get that the profinite group Gal( K g ) is topolog-ically finitely generated. By [26, Proposition 2.5.1(a)], the profinite group Gal( K g )is small, therefore the field K g is bounded. (cid:3) Theorem 4.12. The theory g − DCF is simple.Proof. For any model ( K, g ) of g -DCF, by Theorem 4.4, the theory of ( K, g ) isbi-interpretable with the theory of the pure field K g (the description of this bi-interpretation given in [14, Remark 2.3] works here as well). By Corollary 4.11, thefield K g is PAC and bounded. Therefore, by [18, Fact 2.6.7], the theory of the field K g is simple. (cid:3) Remark 4.13. We describe here some special cases and comment on possibleimprovements and generalizations.(1) If the group scheme g is infinitesimal (see Example 2.3(2)), then the theory g -DCF is stable and it coincides with one of the theories considered in [13].(2) If the group scheme g = G k is constant ( G is a finite group), then we have g − DCF = G − TCF , where G -TCF is the theory considered in [14]. In particular, this theory issupersimple of finite rank e coinciding with the order of G .(3) Suppose that the group scheme g is neither infinitesimal nor ´etale. It meansthat g = g and g = g , where g and g come from the exact sequenceconsidered in the beginning of the Introduction. In this case, using [18,Fact 2.6.7] again, the theory g -DCF is strictly simple (that is: simple, notstable, and not supersimple), since the PAC field K g is neither separablyclosed nor perfect.(4) We consider in Section 5 finite group schemes g of some special kind, whichare of the type described in Item (3) above. For such finite group schemes,the theory g − DCF considered in a certain expanded language has a finerdescription, which is obtained using the results from [11] (see Corollary5.8).(5) As we pointed out in Section 2, the g -rings considered in this paper area very special case of the rings with iterative operators considered in [23].One may wonder whether our main results (Theorems 3.6 and 4.12) couldbe generalized to the context considered in [23]. Unfortunately, it looks im-possible at this moment, since, for example, actions of locally finite groupson rings constitute still quite a special case of the situation from [23]. Butthe theory of actions of such groups on fields need not be companionable,as a recent work of Beyarslan and the second author shows [2]. More pre-cisely, the case of Abelian torsion groups is considered in [2] and a ratherunexpected algebraic condition on such groups is given in [2, Theorem 1.1],which is equivalent to the companionability the theory mentioned above.For the general case of locally finite groups, it is even unclear what shouldbe conjecturally taken for this algebraic condition.5. Group actions on fields of finite imperfection degree In this section, we use the results from the previous sections to show the existenceand to describe some properties of a model complete theory of actions of a fixedfinite group on fields of finite imperfection degree. Let us describe first the general ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 19 setting of the model-theoretic dynamics from [11], which will be used here. Westart from an L -theory T having a model companion T mc and we assume that T mc has elimination of quantifiers. For a fixed group G , we expand the language L tothe language L G by adding unary function symbols for elements of G . Then, T G is the obvious L G -theory of G -actions by automorphisms on models of T , and it isan interesting question whether a model companion of T G exists (if it exists, it isdenoted by T mc G ). In the case of G = Z , the above question is about the existenceof an axiomatization of the theory of generic automorphism of models of T (see[5]). It is well-known (see e.g. [7]) that if T is the theory of fields, then the theory T mc Z (=ACFA) exists. In [6], Chatzidakis shows the existence of the theory T mc Z ,where T is the theory of separably closed fields of a fixed imperfection degree inthe language of fields expanded by symbols for p -basis and λ -functions.If G is a finite group and T is the theory of fields, then the theory T mc G exists(see [28] and [14]); this theory was denoted by G -TCF in [14]. It is a good momentnow to warn the reader that the theory T mc G usually does not imply the theory T mc : for example the models of G -TCF are not algebraically (or even separably)closed for a non-trivial finite group G (see [14]). The main aim of this sectionis to show a variant of Chatzidakis’ existence result from [6] for a finite group G replacing the infinite cyclic group, which also gives one more example of the classof theories satisfying the conditions from [11]. To achieve this aim, we do not followthe methods from [6], but we provide a new argument using Theorem 3.6.We start with describing our version of the general set-up from [11]. Let us fixa positive integer e (it will not play the role of the number e from the previoussections!) and a prime number p . For the notion of a p -basis, the reader mayconsult [6, Section 1.5]. The imperfection degree of a field K of characteristic p isthe cardinality of its p -basis. We define the language L as the language of fields withthe extra constant symbols b = { b , . . . , b e } and the extra unary function symbols λ = { λ i | i ∈ [ p ] e } , where [ p ] := { , . . . , p − } . We set as T the L -theory SF p,e ,which is such that its models ( K ; b, λ ) satisfy the following: • K is a field of characteristic p ; • the set { b , . . . , b e } is a p -basis of K ; • for each x ∈ K , we have: x = X i ∈ [ p ] e λ i ( x ) p b i . It is well-known that the model companion T mc is the theory of separably closedfields which are models of SF p,e ; this theory is denoted by SCF p,e . The theorySCF p,e is strictly stable and has quantifier elimination (Theorem 2.3 in [8]), so weare in the general situation from [11].We describe now how to add dynamics to the theory T . Let G be a finite groupof order e ′ . As explained above, L G denotes the language L expanded by unaryfunction symbols { σ g | g ∈ G } and T G = (SF p,e ) G is the theory of group actionsof G on models of SF p,e by L -automorphisms. To show the existence of the theory T mc G , we will use the theory g − DCF (see Theorem 3.6) for k = F p and the followingfinite group scheme: g := G e a [1] × G F p . For the rest of this section, g denotes the finite group scheme defined above, so wediverge again from the notation of the previous sections, since e is not anymore theorder of g (which is now p e + e ′ ).By Example 2.3(3), a g -field ( K, ∂ ) is the same as a field K of characteristic p withderivations D , . . . , D e and with an action of the group G by field automorphismson K such that the following conditions hold: • the derivations D , . . . , D e commute with each other; • for each i e , we have D ( p ) i = 0 (the composition of D i with itself p times); • the action of G commutes with the derivations D , . . . , D e .Let us define: C K := K G e a [1] = ker ( D ) ∩ . . . ∩ ker ( D e ) . We clearly have K p ⊆ C K and the next result shows that there is a g -field extensionof K where the equality holds, which will be crucial in the proof of Theorem 5.6(our aim here). Lemma 5.1. For any g -field K , there is a g -field extension K ⊆ M such that C M = M p .Proof. If C K = K p , then there is nothing to do, so let us take t ∈ C K \ K p . Then,we have: G · t ⊂ C K \ K p . We take a p -basis { t = t, . . . , t k } of K p ( G · t ) over K p and we define: K := K (cid:16) t /p , . . . , t /pk (cid:17) . By setting for any i e and j k : D i (cid:16) t /pj (cid:17) := 0 ,K becomes a G e a [1]-field extension of K . For any g ∈ G and any j k , we define: g · t /pj := ( g · t j ) /p , which gives a unique extension of the G -action from K to K . This makes K a g -field extension of K and by repeating this process, we can define M as the unionof a tower of g -field extensions of K . (cid:3) For a g -field K , it will be crucial to find a p -basis of K in the subfield of G -invariants K G , which is allowed by the next result. Lemma 5.2. Suppose that the group G acts on a field K of characteristic p andof imperfection degree e . Then, there is a p -basis of K in K G .Proof. Let us denote K := K G and consider the following commutative diagramof field extensions: KK p < < ③③③③③③③③ K b b ❉❉❉❉❉❉❉❉ K p . a a ❈❈❈❈❈❈❈❈ = = ④④④④④④④④ ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 21 By applying the Frobenius map, we have [ K : K ] = [ K p : K p ]. Therefore, weobtain: [ K : K p ] = [ K : K p ] , so the imperfection degree of K is e as well. Since the field extension K p ⊆ K is purely inseparable and the extension K p ⊆ K p is Galois (because the extension K ⊆ K is Galois), we obtain that K is linearly disjoint from K p over K p . Thelinear disjointness and the fact that the imperfection degree of K is finite andcoincides with the imperfection degree of K imply that any p -basis of K is a p -basis of K as well. (cid:3) Proposition 5.3. If K is a model of g − DCF , then C K = K p and the imperfectiondegree of K is e .Proof. Since K is an existentially closed g -field, Lemma 5.1 implies that C K = K p .By [14, Corollary 3.21], we get that [ K : C K ] p e . Therefore (by the existentialclosedness again), it is enough to find a g -field extension K ⊆ N such that theimperfection degree of N is e .Let us consider the field of rational functions F p ( X , . . . , X e ) with the standard G e a [1]-field structure given by D i ( X j ) = δ ji (the Kronecker delta) and with thetrivial action of G . Then, F p ( X , . . . , X e ) becomes a g -field of imperfection degree e and such that: C F p ( X ,...,X e ) = F p ( X p , . . . , X pe ) = ( F p ( X , . . . , X e )) p . We consider now the following field: M := Frac (cid:0) K ⊗ F p F p ( X , . . . , X e ) (cid:1) . By Lemma 3.3 (both of the items), M is a g -field and it is naturally a g -fieldextension of K and F p ( X , . . . , X e ). Let N be a g -field extension of M pro-vided by Lemma 5.1, so N p = C N . By [14, Lemma 3.22], the field extension F p ( X , . . . , X e ) ⊆ N is separable, hence the imperfection degree of N is at least e .Since N p = C N , we obtain (using [14, Corollary 3.21] again) that the imperfectiondegree of N is exactly e . (cid:3) For our main bi-interpretability result, we need to consider the theory g -DCF inan expanded language. We set the following.(1) Let L g ,b be the language L g (see Section 2.2) expanded by the constantsymbols b , . . . , b e .(2) Let g − DF b be the L g ,b -theory of g -fields such that:(a) { b , . . . , b e } is a p -basis;(b) { b , . . . , b e } is contained in the invariants of the action of G ;(c) for all i, j e , we have: D i ( b j ) = δ ji (the Kronecker delta).(3) We define the L g ,b -theory g − DCF b as the theory g − DCF with the sameextra conditions on the fixed p -basis and the G -invariants as in Item (2)above.The next lemma shows that the theory g − DF b has the expected properties. Lemma 5.4. We have the following. (1) If K is a model of g − DF b , then the imperfection degree of K is e and C K = K p . (2) The theory g − DF b is bi-interpretable with the theory T G = (SF p,e ) G .Proof. For the proof of Item (1), the statement about the imperfection degree of K is clear from the axiomatization of the theory g − DF b . To show that C K = K p ,we consider for each i e the following subfield of K : C i := ker( D ) ∩ . . . ∩ ker( D i ) . Then, we have the following tower of fields: K p ⊆ C K = C e C e − . . . C K, where the properness of the inclusions is witnessed by b i ∈ C i − \ C i (we set C := K ). Therefore, we get: p e = [ K : K p ] > [ K : C K ] > p e , which implies that C K = K p .For the proof of Item (2), if ( K ; b, D , . . . D e , σ g ) g ∈ G is a model of g − DF b , then( K ; b, σ g ) g ∈ G obviously interprets a model of T G , since λ -functions are definable ina field with a p -basis. If ( K ; b, λ, σ g ) g ∈ G is a model of T G , then we forget about the λ -functions and define the derivations by the following formula (as in the proof ofProposition 5.3): D j ( b i ) = δ ji . Then, it is standard that ( K ; b, D , . . . D e , σ g ) g ∈ G is a model of g − DF b . It is alsoclear that the two interpretations above are mutually inversive. (cid:3) The next result is crucial for the proof of Theorem 5.6. Proposition 5.5. The theory g − DCF b is a model companion of the theory g − DF b .Proof. Clearly, any model of g − DCF b is a model of g − DF b . We need to showthat each model of g − DF b embeds into a model of g − DCF b and that the theory g − DCF b is model complete.It is easy to see the model completeness part, since the theory g − DCF b is anexpansion by constants of the model complete (by Theorem 3.6) theory g − DCF.Let ( K ; b, ∂ ) be a model of the theory g − DF b . By Theorem 3.6, the g -field ( K, ∂ )has a g -field extension ( K ′ , ∂ ′ ) which is a model of the theory g − DCF. By Lemma5.4, we have C K = K p . Therefore, by [14, Lemma 3.22], the field extension K ⊆ K ′ is separable. Since b is a p -basis of K and the extension K ⊆ K ′ is separable, b is p -independent in K ′ . By Proposition 5.3, the imperfection invariant of K ′ is e , so b is a p -basis of K ′ as well. Hence ( K ′ ; b, ∂ ′ ) is a model of g − DCF b and ( K ′ ; b, ∂ ′ )is also an extension of ( K ; b, ∂ ), which finishes the proof. (cid:3) Before stating the main result of this section, we advice the reader to recall ournotational set-up regarding the theories T, T G , T mc , T mc G from the beginning of thissection. Theorem 5.6. The theory T mc G exists and it is bi-interpretable with the theory g − DCF b . In particular, the theory T mc G is strictly simple.Proof. The existence part follows from the bi-interpretability part, which is clear byLemma 5.4(2) and Proposition 5.5. The “in particular” part follows from Remark4.13 and the bi-interpretability part above. (cid:3) ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 23 Remark 5.7. As we have pointed out before, the theory T mc G need not imply thetheory T mc and indeed: if G is non-trivial, then the underlying fields of models of T mc G are not separably closed, since the are no non-trivial actions of a finite groupon a separably closed field of positive characteristic (the Artin-Schreier theorem).We can apply now the techniques from [11] to obtain several model-theoreticproperties of the theory T mc G . We assume that C is a sufficiently saturated model of T mc G and that C is embedded in a sufficiently saturated model D of T mc = SCF p,e . Inother words, C is an existentially closed field with a p -basis, λ -functions and a groupaction of G , and D is an existentially closed field with a p -basis and λ -functions.We will consider the model-theoretic algebraic closure inside C , denoted by acl G ( · ),and the model-theoretic algebraic closure inside D , denoted by acl sep ( · ). By [15,Fact 2.7], if A ⊆ D then acl sep ( A ) coincides with the field theoretic algebraic closurein D of the subfield generated by the p -basis and the values of the λ -functions onthe elements of A . Corollary 5.8. Let A, B, C be small subsets of C such that C ⊆ A ∩ B . (1) By [11, Corollary 4.12] , we have: acl G ( A ) = acl sep ( G · A ) ∩ C . (2) By [11, Corollary 4.28] , we have: A T mc G | ⌣ C B ⇐⇒ acl sep ( G · A ) and acl sep ( G · B ) are linearly disjoint over acl sep ( G · C ) , i.e. the forking independence in T mc G is given by the G -action and theforking independence in T mc = SCF p,e . (3) By [11, Remark 4.13] , T mc G has “almost quantifier elimination” (similarlyas ACFA). (4) By [11, Remark 4.36] , T mc G has geometric elimination of imaginaries. It was shown in [3] that the theory of actions of a fixed virtually free group onfields has a model companion. An analogous result holds in the case of actionsof a finite group (Theorem 5.6) or the infinite cyclic group ([6]) on fields of finiteimperfection degree. The following natural question arises. Question 5.9. Does the theory of actions of a fixed virtually free group on fieldsof finite imperfection degree have a model companion? More generally, what is theclass of groups such that the model companion above exists?Let us point out that the “more generally” part of Question 5.9 is wide openeven in the case of group actions on arbitrary fields. References [1] ¨Ozlem Beyarslan, Daniel Max Hoffmann, Moshe Kamensky, and Piotr Kowalski. Model theoryof fields with free operators in positive characteristic. Transactions AMS , 372(8):5991–6016,2019.[2] ¨Ozlem Beyarslan and Piotr Kowalski. Model theory of Galois actions of torsion abelian groups.Submitted, available on https://arxiv.org/pdf/2003.02329.pdf .[3] ¨Ozlem Beyarslan and Piotr Kowalski. Model theory of fields with virtually free group actions. Proc. London Math. Soc. , 118(2):221–256, 2019. [4] Stephen U. Chase. Infinitesimal group scheme actions on finite field extensions. AmericanJournal of Mathematics , 98(2):441–480, 1976.[5] Z. Chatzidakis and A. Pillay. Generic structures and simple theories. Annals of Pure andApplied Logic , 95:71–92, 1998.[6] Zoe Chatzidakis. Generic automorphisms of separably closed fields. Illinois Journal of Math-ematics , 45(3):693–733, 2002.[7] Zo´e Chatzidakis and Ehud Hrushovski. Model theory of difference fields. Trans. AMS ,351(8):2997–3071, 2000.[8] Fran¸coise Delon. Separably closed fields. In Elisabeth Bouscaren, editor, Model theory, alge-bra, and geometry: An Introduction to E. Hrushovskis proof of the geometric Mordell-Langconjecture , volume 1696 of Lecture Notes in Mathematics , pages 143–176. Springer, Berlin,1998.[9] M. Demazure and P. Gabriel. Groupes alg´ebriques, Vol. 1: G´eom´etrie alg´ebrique, g´en´eralit´es,groupes commutatifs . Masson, Paris, 1970.[10] M.D. Fried and M. Jarden. Field Arithmetic . Ergebnisse der Mathematik und ihrer Grenzge-biete. 3. Folge / A Series of Modern Surveys in Mathematics. 3rd Edition, Springer, 2008.[11] Daniel Max Hoffmann. Model theoretic dynamics in Galois fashion. Annals of Pure andApplied Logic , 170(7):755–804, 2019.[12] Daniel Max Hoffmann and Piotr Kowalski. Integrating Hasse-Schmidt derivations. J. PureAppl. Algebra , 219(4):875–896, 2015.[13] Daniel Max Hoffmann and Piotr Kowalski. Existentially closed fields with G -derivations. Journal of London Mathematical Society , 93(3):590–618, 2016.[14] Daniel Max Hoffmann and Piotr Kowalski. Existentially closed fields with finite group actions. Journal of Mathematical Logic , 18(1):1850003, 2018.[15] Ehud Hrushovski. The Mordell-Lang conjecture for function fields. J. Amer. Math. Soc. ,9(3):667–690, 1996.[16] Isaacs I.M. Algebra. A graduate course . Graduate Studies in Mathematics 100. AMS, 1994.[17] Moshe Kamensky. Tannakian formalism over fields with operators. International MathematicsResearch Notices , 24:5571–5622, 2013.[18] B. Kim. Simplicity Theory . Oxford Logic Guides. OUP Oxford, 2013.[19] Piotr Kowalski. Geometric axioms for existentially closed Hasse fields. Annals of Pure andApplied Logic , 135:286–302, 2005.[20] S. Montgomery. Fixed Rings of Finite Automorphism Groups of Associative Rings . LectureNotes in Mathematics. Springer, 1980.[21] S. Montgomery. Hopf Algebras and Their Actions on Rings . Number 82 in Regional conferenceseries in mathematics. American Mathematical Society, 1993.[22] Rahim Moosa and Thomas Scanlon. Jet and prolongation spaces. Journal of the Inst. ofMath. Jussieu , 9(2):391–430, 2010.[23] Rahim Moosa and Thomas Scanlon. Generalized Hasse-Schmidt varieties and their jet spaces. Proc. Lond. Math. Soc. , 103(2):197–234, 2011.[24] Rahim Moosa and Thomas Scanlon. Model theory of fields with free operators in characteristiczero. Journal of Mathematical Logic , 14(02):1450009, 2014.[25] D. Mumford. Abelian varieties . Tata Institute of fundamental research studies in mathemat-ics. Published for the Tata Institute of Fundamental Research, Bombay [by] Oxford UniversityPress, 1974.[26] Luis Ribes and Pavel Zalesskii. Profinite groups . Springer New York, 2000.[27] J.P. Serre. Algebraic Groups and Class Fields: Translation of the French Edition . GraduateTexts in Mathematics Series. Springer-Verlag New York Incorporated, 1988.[28] Nils Sj¨ogren. The Model Theory of Fields with a Group Action. Research Reportsin Mathematics, Department of Mathematics Stockholm University , 2005. Available on .[29] William C. Waterhouse. Introduction to Affine Group Schemes . Springer-Verlag, 1979. ODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS 25 † Instytut Matematyki, Uniwersytet Warszawski, Warszawa, Poland and Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA E-mail address : [email protected] URL : https://sites.google.com/site/danielmaxhoffmann/home ♠ Instytut Matematyczny, Uniwersytet Wroc lawski, Wroc law, Poland E-mail address : [email protected] URL ::