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Differential étale extensions and differential modules over differential rings
Advances in Applied Mathematics 72 (2016) 195–214
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Differential étale extensions and differential modules over differential rings Andy R. Magid Department of Mathematics, University of Oklahoma, Norman, OK 73019, United States
a r t i c l e
i n f o
Article history: Received 12 December 2014 Received in revised form 7 September 2015 Accepted 7 September 2015 Available online 19 September 2015 MSC: 12H05 Keywords: Picard–Vessiot Differential Galois
a b s t r a c t This paper studies differential square zero extensions and differential modules of a commutative differential algebra R over a differential field F where the field of constants of F is algebraically closed and of characteristic 0. All elements of R and the differential R modules and algebras considered are assumed to satisfy linear homogeneous differential equations over F . For R differentially simple, we describe the invectives, and, using that all considered differential modules are R flat, provide a criterion for all square zero extensions to be differentially split. © 2015 Elsevier Inc. All rights reserved.
0. Introduction Let F be a differential field of characteristic zero whose field of constants C is algebraically closed. Suppose an order n linear monic homogeneous differential equation with coefficients in F is given. A Picard–Vessiot, or differential Galois, extension E of F for this equation is a minimal differential field extension E containing n solutions of the equation which are linearly independent over the field of constants of E, where the latter is required to be the constant field C of F . This is the exact analogue of the usual, E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aam.2015.09.006 0196-8858/© 2015 Elsevier Inc. All rights reserved.
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or polynomial, Galois extension: given a monic polynomial of degree n over a field K, a Galois extension of K for this polynomial is a minimal extension field L containing n distinct roots of the equation. In both cases there is a Galois group consisting of all automorphisms of L (respectively differential automorphisms of E) fixing K (respectively F ). This Galois group is finite for L/K and an algebraic group over C for E/F . Moreover, in algebraic geometry language, L is essentially the function field/coordinate ring of a principal homogeneous space (respectively E is essentially the function field of a principal homogeneous space) for the Galois group. In the differential case, the coordinate ring of the space is essentially the Picard–Vessiot ring R of E/F , which can be defined as the set of elements of E which satisfy a linear monic homogeneous differential equation with coefficients in F . The Picard–Vessiot ring R is a differential F algebra, and it has no proper differential ideals. For Picard–Vessiot theory, see [5] and [10]. The ordinary Galois extension L/K corresponds to a finite étale cover of schemes Spec(L) → Spec(K). In terms of the coordinate rings L and K the étale condition is the following: suppose T → S is a surjective homomorphism of commutative K algebras whose kernel I satisfies I 2 = 0. Then any K algebra homomorphism L → S lifts to a homomorphism L → T . (This is a way of saying that the K polynomial of which L is the splitting field is separable.) It is natural to inquire whether E/F , or more precisely R/F , shares a similar property. In previous work, we considered the related question for commutative rings on which a reductive algebraic group acted. It turned out that rings with reductive algebraic group H action with no H stable ideals have a sort of “equivariant étale” property: H ring homomorphisms from them to H equivariant square zero extensions lift [4, Thm. 1.2, p. 176]. Although this statement only involves rings with group action, the proof used required considering modules which were simultaneously modules for the group and the ring, in fact it occurs naturally in the context of a general investigation of such modules. To describe this context, and its differential analogue, we retain the meaning of the symbols F and C, but redefine R and G as needed. Let G be a C algebraic group. Consider those rational G modules which are simultaneously modules for a commutative C algebra R on which G acts rationally and such that the G action and R actions on the module are compatible. Such modules, which may be infinite dimensional, arise in the consideration of induced modules for quotients of G and were studied in [8]; in that paper they were called R · G modules, a convention since followed for the consideration of such modules in a general context. In the previous work [3], as well as the equivariant étale property, we considered in general injective modules and homology for R · G modules. In this paper, we consider similar questions for differential modules over a differential commutative F algebra R. The category of differential vector spaces over a differential field bears a close analogy to the category of rational modules for an algebraic group. Indeed, for the differential field F with field of constants C, and if the differential vector spaces are finite dimensional, their category is antiequivalent to the category of finite dimensional rational modules for a certain (pro)algebraic group over C [6].
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For the differential case, in addition to the obvious compatibility of differential and R actions, we require, analogously to what is built into the definition of rational module for an algebraic group, a local finiteness condition on R and the allowed differential modules, which we also call rational. Fortunately, for the main example of rational differential rings, namely Picard–Vessiot rings [10, Defn. 1.15, p. 12], this rationality condition obtains. Following the notation from the group case, we call this the category of R · D modules. Picard–Vessiot rings are differentially simple. For rings R with algebraic group G action, if the ring has no group stable ideals, finitely generated R · G modules are R projective [2, Prop. 3.5, p. 794] (and hence all are R flat), a fact which has a number of useful consequences. Yves André [1] has proved that a similar result obtains in the differential case as we recall below. This then allows us to establish the differential analogue of the lifting property for homomorphisms to square zero extensions, although the requirement is more stringent than differentially simple, and we show by example that not all Picard–Vessiot rings have this “differential étale” property. The requirement is met, however, for Picard–Vessiot rings of Picard–Vessiot closures. This paper is organized as follows: the initial section, Basics, gives the relevant definitions and standard properties of R · D modules. The next section F · D modules considers the special case that R is the base field F ; we look at the construction of injectives in this case. The following section, Injective, looks at the construction of injective R · D modules for general R. The next section, Simple, considers R · D modules when R has no proper differential ideals. We recall here André’s theorem that all R · D modules are R flat, and R projective if R finitely generated. The final section, Square zero, considers what we have called above the “differential étale” property and obtains a condition for when it holds. Most of the results in all but the final section of this paper were presented in a Special Session in Boston in January 2012 dedicated to the memory of Jerald Kovacic; the rest of the results were obtained as a consequence of questions raised at a BIRS workshop in June 2012. I am grateful to the organizers of both the session and the workshop for the opportunities provided, and I dedicate this work in its entirety to Jerry Kovacic, whose leadership in differential algebra was and is an inspiration. 1. Basics Let F be a differential field of characteristic zero with derivation DF whose field of constants C is algebraically closed. Let R be a commutative differential F algebra with derivation DR satisfying DR |F = DF . We assume that every element of R satisfies a monic linear homogeneous differential equation over F . We call such an R a differentially rational algebra over F . An R[D] module is an R module M furnished with a C endomorphism DM such that DM (rv) = DR (r)v + rDM (v) for all r ∈ R and v ∈ M . Thus R is an R[D] module. An R[D] module M such that every element of M satisfies
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a monic linear homogeneous differential equation over F is called an R · D module. Thus R is an R · D module. If M is an R[D] module and N is an R[D] submodule then M is an R · D module if and only if both N and M/N are R · D modules. For an R[D] module M , the kernel M D of DM is a vector space over C, and more generally a module over the ring RD . If V is an F finite dimensional F [D] module, it is an F · D module. Similarly, the symmetric algebra SF (V ), being a direct sum of F finite dimensional F [D] modules, is an F ·D module. This provides an example R = SF (V ) of a differentially rational F algebra. If S = {Vi |i ∈ I} is a set of F finite dimensional F [D] modules, then R = ⊗i∈I SF (Vi ) is also an example of a differentially rational F algebra, and it will be clear below that every example is a quotient of such a tensor product. In particular, the Picard–Vessiot ring of a Picard–Vessiot closure of F is a differentially rational F algebra. An R[D] morphism f between R[D] modules M and N is an additive homomorphism satisfying f (rm) = rf (m) and DN (f (m)) = f (DM (m)) for r ∈ R and m ∈ M . If in addition M is an R · D module then so is f (M ). An R[D] morphism between R · D modules is called an R · D morphism. The R[D] modules form a category of which the R · D modules are a full subcategory. If f is an R[D] morphism, its kernel and image as an R module homomorphism are R[D] modules, and it follows that the categories of R[D] and R · D modules are abelian. If M and N are R[D] modules, so is M ⊗R N (with DM ⊗N (m ⊗n) = DM (m) ⊗n +m ⊗DN (n)) and HomR (M, N ) (with DHom(M,N ) (φ)(m) = DN (f (m)) − f (DM (m))). If M and N are further R · D modules, so is M ⊗R N , but this need not be the case for HomR (M, N ). It will be an R · D module provided M is finitely generated as an R module (see Corollary 4 below). With the above actions, the map HomR (M, N ) ⊗R M → N by f ⊗ m → f (m) is an R[D] homomorphism. When V is an F [D] module, the F [D] modules R ⊗F V and HomF (R, V ) are R[D] modules under the R actions r · (s ⊗ v) = rs ⊗ v and (r · f )(s) = f (sr). When V is an F · D module, so is R ⊗F V , but in general HomF (R, V ) will not be an R · D module. Let R[X; D] denote the twisted polynomial ring over R: as additive R modules, R[X; D] coincides with the polynomial ring R[X], but with multiplication determined by Xr = rX + D(r) [9, Defn. 1.6.16, p. 93]. Clearly F [X; D] is a subring of R[X; D]. An R[D] module M becomes an R[X; D] module via Xm = DM (m), and an R[X; D] module becomes an R[D] module via DM (m) = Xm. Regarded as an R[X; D] module, an R · D module is an R[X; D] module every element of which is annihilated by a monic element of F [X; D]. In other words, the category of R · D modules is the category of F [X; D]-torsion R[X; D] modules. If M is an R[X; D] module and m ∈ M then AnnR[X;D] (m) = {f ∈ R[X; D]|f m = 0} is a left ideal. M is an R · D module provided that AnnF [X;D] (m) = 0 for all m ∈ M . It is known that left ideals of F [X; D] are principal [9, Prop. 1.6.24, p. 95]. It is trivial that non-zero principal left ideals of F [X; D] are of F finite codimension, from which it follows that an F [X; D] module is torsion if and only if all its cyclic submodules
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(i.e. singly generated) are finite F dimensional, and this latter is equivalent to being a sum (not necessarily finite) of F finite dimensional F [X; D] modules. Thus if M is a torsion F [X; D] module and m ∈ M there is an F finite dimensional F [X; D] submodule W ⊆ M with m ∈ W . Lemma 1. Let M be an R[D] module and let V be an F finite dimensional F [D] submodule. Then RV is an R · D module. Proof. RV is clearly an R[D] module. Let r ∈ R, and let W be an F finite dimensional F [D] submodule of R containing r. Then W V is an F finite dimensional F [D] submodule of RV containing rV . It follows that RV is the sum of its F finite dimensional F [D] submodules. 2 Corollary 1. Let M be an R[D] module. The sum Mt of all the F finite dimensional F [D] submodules of M is an R · D submodule which contains all R · D submodules of M . Moreover, Mt is the sum of all the R · D submodules of M . Proof. If m ∈ Mt there is an F finite dimensional F [D] submodule of M with m ∈ V . By Lemma 1 RV is an R · D module and hence a sum of F finite dimensional F [D] submodules of M . Thus RV ⊆ Mt and in particular Rm ⊂ Mt . It follows that Mt is an R[D] module and hence an R · D module. Since any R · D submodule of M is contained in Mt , it further follows that Mt is also the sum of the R · D submodules of M . 2 Corollary 2. Let V be an F · D module. Then R ⊗F V is an R · D module. Proof. It is clear that R ⊗ V is an R[D] module. By Corollary 1 applied to the case R = F , V is the sum of its finite dimensional F [D] submodules W , so R ⊗ V is the sum of the R ⊗ W . The latter is an R · D module by Lemma 1. 2 If M is a finitely generated R · D module and S is a finite generating set, then S is contained in a finite dimensional F · D submodule V of M , and there is an R · D module surjection R ⊗F V → M . We record this: Corollary 3. A finitely generated R · D module M is a quotient of one of the form R ⊗F V where V is a finite dimensional F · D module. If V is cyclic [as an F [D] module] then M is cyclic [as an R[D] module]. It will always be the case that V is F [D] cyclic when DF is non-trivial. The penultimate sentence in Corollary 3 is obvious and added for use below. The final sentence is the Cyclic Vector Theorem [10, Prop. 2.9, p. 40]. As noted above, for R · D modules M and N , the R[D] module HomR (M, N ) need not be an R · D module.
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For example, let R = F [x] with Dx = 1. Take N = R, and let M be a countable direct sum of copies of R (both the latter being R · D modules). Then HomR (M, N ) is a countable direct product of copies of R. Consider the element y = (x, x2 , x3 , . . .) of this product. If HomR (M, N ) is an R · D module, then there is an F finite dimensional F [D] submodule V containing y. Suppose V has F dimension m. Then for any n, the projection of V on the nth factor is an F [D] submodule of R of dimension at most m containing xn . This is impossible for n ≥ m. However, if M is finitely generated, we have the following: Corollary 4. Let M and N be R · D modules with M finitely generated over R. Then HomR (M, N ) is an R · D module. Proof. By Corollary 3, there is a finite dimensional F · D module V and a surjection R ⊗ V → M . This induces an inclusion HomR (M, N ) → HomR (R ⊗ V, N ), and it suffices to show the latter is an R · D module. There is an R module isomorphism HomR (R ⊗ V, N ) → V ∗ ⊗F N , which is seen to be an R[D] morphism as well. Since V ∗ ⊗F N is R[D] isomorphic to (R ⊗F V ∗ ) ⊗R N and by Corollary 2 the first factor is an R · D module, so is the tensor product and the result follows. 2 The notation introduced in Corollary 1 will be useful below, so we single it out in a definition: Definition 1. Let M be an R[D] module. Then Mt denotes the largest R · D submodule of M . 2. F · D modules The category of F · D modules is investigated in [6] via the module which we here denote I which consists of all the elements of the Picard–Vessiot closure F1 of F which satisfy a linear differential equation over F . Note that I is an F · D module, although typically not finite dimensional over F . If Π(F ) denotes the proalgebraic group (over C) of the differential automorphisms of F1 over F , then V → HomF [D] (V, I) is an antiequivalence between the category of F finite dimensional F · D modules and the category of C finite dimensional rational Π(F ) modules. It follows that I is an injective F · D module, in fact an injective cogenerator for the category of F · D modules. It is also a subring of F1 , and contains a unique copy of the one dimensional trivial F module. We write this later inclusion as a morphism F → I and call it the canonical inclusion. If V is an F · D module, the morphism V → V ⊗F I by V → v ⊗ 1 is called the canonical inclusion for V . Proposition 1. Let V and E be F · D modules with E injective. Then V ⊗F E is injective.
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Proof. We need to show that Hom(·, V ⊗E) is exact, and in particular converts injections to surjections. Since F · D modules are directed unions of finite dimensional submodules, the usual Zorn’s Lemma arguments reduce the problem to showing that injections X → Y with Y finite dimensional are converted to surjections. So suppose we have a morphism X → V ⊗ E. Since X is finite dimensional, we can assume the image lies in W ⊗ E for some finite dimensional submodule W of V , so we have a morphism f : X → W ⊗ E. As usual, we identify W ⊗ E and Hom(W ∗ , E). By adjoint associativity (see Theorem 2 below, using R = F ), f corresponds to a morphism X ⊗W ∗ → E, which by the injectivity of E extends to a morphism Y ⊗ W ∗ → E. By adjoint associativity, this corresponds to a morphism Y → W ⊗ E which extends f . 2 It follows from Proposition 1 that the canonical inclusion V → V ⊗ I is an embedding of the F · D module in an injective F · D module, which we note is functorial in V . It follows that in the category of F · D modules we have functorial injective resolutions. For later use, we also provide another construction of an injective cogenerator for F ·D modules which does not rely on the use of the Picard–Vessiot closure. We begin with some observations on notation. If L is a left ideal of F [X; D], then F [X; D]/L is an F [D] module with D action D(f + L) = Xf + L. Then (F [X; D]/L)∗ = HomF (F [X; D]/L, F ) is an F [D] module with D action D(φ)(f + L) = D(φ(f + L)) − φ(D(f ) + L). We have similar considerations for the F duals of non-cyclic F [D] modules. Theorem 1. Assume that DF = 0. Let F be the set of left ideals L of F [X; D] such that F [X; D]/L is finitely generated as a F module. Regard F as directed by reverse inclusion. Then E = lim(F [X; D]/L)∗ −−→ F
is an injective cogenerator for F · D modules. Proof. Since DF is non-trivial, by the Cyclic Vector Theorem [10, Prop. 2.9, p. 40] every F finitely generated F [D] module V is cyclic, that is of the form F [X; D]/L for some L ∈ F. Applying this to V ∗ shows that V is also of the form (F [X; D]/L1 )∗ . We begin by showing that E is injective. As in the proof of Proposition 1, this reduces to the consideration of an F · D injection X → Y with Y finitely generated over F and an F · D morphism X → E. The image of this latter lies in some (F [X; D]/L)∗ . Taking duals and applying the Cyclic Vector Theorem then shows that we can assume that X ∗ = F [X; D]/LX and Y ∗ = F [X; D]/LY are cyclic and the surjection comes from the inclusion LY ⊆ LX . Let L1 = L ∩ LY . The morphism F [X; D]/L1 → F [X; D]/LY → F [X; D]/LX coincides with F [X; D]/L1 → F [X; D]/L → F [X; D]/LX , and taking duals this means that X → (F [X; D]/L)∗ → (F [X; D]/L1 )∗ → E lifts to Y → (F [X; D]/L1 0)∗ → E. This shows that E is an injective F · D module.
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Any non-zero F · D module M contains a non-zero F finitely generated submodule V . As previously remarked, V is of the form (F [X; D]/L)∗ and hence has a non-zero map to E. Since E is injective, this map extends to a non-zero map from M to E, showing that this latter is a cogenerator. 2 3. Injective We are going to show the construction of some injective R · D modules. The first step is an adjoint associativity property: Theorem 2. Let M and P be R · D modules and let N be an F · D module. Then there is a natural isomorphism HomF ·D (M ⊗R P, N ) → HomR·D (M, HomF (P, N )t ) Proof. Recall that HomF (P, N ) is an R module via r · f (p) = f (rp). The usual adjoint associativity isomorphism HomF (M ⊗R P, N ) → HomR (M, HomF (P, N )) carries R[D] homomorphisms to R[D] homomorphisms. Since the image of an R · D module under an R[D] homomorphism is an R · D module, R[D] morphisms M → HomF (P, N ) have their image in HomR (P, N )t . The result follows. 2 Corollary 5. Let E be an injective F · D module. Then HomF (R, E)t is an injective R · D module. If E is a cogenerator, so is HomF (R, E)t Proof. By Theorem 2, HomR·D (·, HomF (R, E)t ) equals HomF ·D (· ⊗R R, E) = HomF ·D (·, E). Since this last is exact so is the first. Suppose that M is a non-zero R · D module and E an injective cogenerator for F · D modules. As above, HomR·D (M, HomF (R, E)t ) equals HomF ·D (M, E). Since there is a non-zero map of M as an F · D module to E, there is non-zero map of M as an R · D module to HomF (R, E)t . 2 Corollary 5 applies when E is either of the injective cogenerators I or E of Section 2. In particular, the category of R · D modules has explicit injective cogenerators. We can also hope to construct an injective cogenerator from dual cyclic modules along the lines of Theorem 1. Here duality would be R duals with respect to R: HomR (·, R). Examining the proof of Theorem 1 we see we would (1) need this to be a category self-equivalence on R finitely generated R · D modules; (2) need submodules of R finitely generated R · D modules to be R finitely generated to apply (1); and (3) need R finitely generated R · D modules to be cyclic. As pointed out in Corollary 3, (3) is satisfied provided DF is non-trivial, but the other two points must be assumed. We will see later that they are satisfied if R is R[D] simple (Theorem 5). For now, we simply note that Theorem 1 is true if they are assumed.
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Theorem 3. Assume that DF = 0. Assume further that 1. HomR (·, R) is a category self-equivalence on R finitely generated R · D modules. 2. Submodules of R finitely generated R · D modules are R finitely generated. Let L be the set of left ideals L of R[X; D] such that R[X; D]/L is finitely generated as an R module. Regard L as directed by reverse inclusion. Then E = lim(R[X; D]/L)∗ −−→ L
is an injective cogenerator for R · D modules. There are, of course, injective R[D] modules: these are the same as injective R[X; D] modules. In general, these will not be R · D modules. Nor should we expect injective R · D modules to be R or R[D] injective. However, injective R · D modules share some properties with injective R modules, as we now see. The following lemma relates R and R[D] morphisms between R · D modules. Lemma 2. Let M and N be R · D modules and let f : M → N be an R homomorphism. Then there are an F [D] module V , an R module homomorphism g : M → M ⊗F V , and an R[D] homomorphism h : M ⊗F V → N such that f = h ◦ g. If M and N are R · D modules with M finitely generated then V may be taken finite dimensional over F . Proof. Let V be the F [D] submodule of HomR (M, N ) generated by f . The map g is given by m → m ⊗ f . The map h is given by m ⊗ v → v(m). It is immediate that f = h ◦ g. One checks that g and h are homomorphisms of the asserted type. If M and N are R · D modules with M finitely generated, then HomR (M, N ) is an R · D module by Corollary 4 so V is finite dimensional. 2 Using Lemma 2, we can show that injective R · D modules have an extension property with respect to R morphisms and R · D morphisms. Theorem 4. Let I be an injective R · D module and let φ : M → N be a one–one R · D morphism of R · D modules with M R finitely generated. Then the induced morphism HomR (N, I) → HomR (M, I) is surjective. Proof. Let f : M → I be an R homomorphism. Factor f as in Lemma 2 as f = h ◦ g : M → M ⊗F V → I, where V is F finite dimensional. Since ⊗F is exact, φ ⊗F 1V is a one–one R · D morphism between the R · D modules M ⊗F V and N ⊗F V . As I is R · D injective, the R · D morphism h : M ⊗F V → I extends to a morphism N ⊗F V → I. Preceding this with N → N ⊗F V by n → n ⊗f gives a map q : N → I with q ◦φ = f . 2
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Definition 2. R is said to be differentially Noetherian if R · D submodules of R finitely generated R · D modules are R finitely generated. If R is differentially Noetherian and M is an R finitely generated R·D module, we can construct an R · D module resolution {Pi } of M where each Pi is R free of finite rank: let P0 = R ⊗F V0 where V0 is an F finite dimensional F · D module and R ⊗F V0 → M is an R · D surjection. Let K1 be its kernel and let P1 = R ⊗F V1 where V1 is an F finite dimensional F · D module and R ⊗F V1 → K1 is an R · D surjection. The resolution arises by continuing this process. Corollary 6. Assume that R is differentially Noetherian. Let I be an injective R·D module and let M be an R finitely generated R · D module. Then ExtpR (M, I) = 0 for p > 0. Proof. Let {Pi } be the resolution of M constructed above. Then ExtpR (M, I) = H p (HomR (P∗ , I)). By Theorem 4, HomR (·, I) is exact on exact sequences of R finitely generated R · D modules. Since the kernels and cokernels of the transition maps in {Pi} are R finitely generated by the differentially Noetherian assumption, HomR (·, I) has no homology in degree above zero. So the result follows. 2 In the proof of Corollary 6, we actually showed that for any R · D module N , ExtpR (M, N ) = H p (HomR (P∗ , N )) where {Pi } is a resolution of M by R finitely generated R · D modules, with R · D morphisms as the transition maps. This makes HomR (P∗ , N ) a sequence of R · D modules, and hence its homology groups are R · D modules. Consequently, each ExtpR (M, N ) is an R · D module. Applying Corollary 6 and standard dimension shifting arguments, we also obtain similar results for injective resolutions: Corollary 7. Assume that R is differentially Noetherian. Let M and N be R · D modules with M R finitely generated. Let {E i } be a resolution of N by injective R · D modules. Then ExtpR (M, N ) = H p (HomR (M, E ∗ )). 4. Simple In this section we consider some of the strong algebraic consequences for R·D modules which follow when R has no proper differential ideals. We fix terminology: Definition 3. An R · D module M is said to be differentially simple if the only R · D submodules of M are {0} and M . The ring R is said to be differentially simple if R is a differentially simple R · D module, i.e. has no proper differential ideals. Let M be an R[D] module. The annihilator of M , AnnR (M ) = {r ∈ R|rm = 0∀m ∈ M } is a differential ideal, since if rM = 0, then 0 = D(r)m +rD(m) = D(r)m. Although in most of this section R will be an integral domain, we will define the torsion submodule
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of M directly: Mtor = {m ∈ M |rm = 0 for some non-zero divisor r ∈ R}. Note that Mtor = Mt . Lemma 3. Let M be an R[D] module. Then Mtor is an R[D] submodule. Proof. If mi ∈ M and ri mi = 0 for some non-zero divisor ri ∈ R, i = 1, 2, then r1 r2 (m1 + m2 ) = 0, and r1 r2 is not a zero divisor and m1 + m2 ∈ Mtor . If r ∈ R then r1 (rm1 ) = 0, so rm1 ∈ Mtor . Finally, r1 m1 = 0 implies that D(r1 )m1 + r1 D(m1 ) = 0. Multiply by r1 : then r12 D(m1 ) = 0, and since r12 is not a zero divisor, this means that D(m1 ) ∈ Mtor . 2 If S is the set of non-zero divisors of R then S −1 R is a differential ring and S −1 M = S R ⊗R M is an S −1 R[D] module, and Mtor is the kernel of the map M → S −1 M by m → 1 ⊗ m. It is possible that AnnR (Mtor ) = 0 even for Mtor = 0. However, we do have the following: 1
Lemma 4. Let M be an R · D module and let X be a finite subset of Mtor . Let N be the R[D] submodule of M generated by X. Then AnnR (N ) is a non-zero differential ideal of R. Proof. N is an R finitely generated submodule of Mtor , say generated by x1 , . . . xn . Let ri be a non-zero divisor with ri xi = 0. Then r = r1 . . . rn is a non-zero element of AnnR (N ). 2 We look at the implications of Lemma 4 when R is differentially simple; that is, R is the only non-zero ideal. We recall that in this case R is an integral domain. We let qf(R) denote its quotient field. Note that qf(R) = S −1 R where S is the set of non-zero divisors of F . The following result, which reveals the key consequences of the differential simplicity property, is due to Yves André [1, Theorem 2.2.1]; in fact that more general result deals with R[D] modules. For completeness of exposition, we have included both the statement of the result and its proof here. Theorem 5. Assume that R is differentially simple, and let M be an R · D module. 1. 2. 3. 4.
Mtor = 0. M → qf(R) ⊗ M is injective. M = 0 if and only if qf(R) ⊗ M = 0. Let f : A → B be a morphism of R · D modules. The f is one–one (onto) if and only if qf(R) ⊗ f is one–one (onto). 5. If M is R finitely generated, it is R projective. 6. M is R flat.
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Proof. Let x ∈ Mtor and let N be the R[D] submodule of M generated by x. By Lemma 4, AnnR (N ) = 0. Since R is simple, this means that AnnR (N ) = R, so that x = 1x = 0, proving 1. Since Mtor is the kernel of M → qf(R) ⊗ M , this map is injective, proving 2. Assertion 3 is immediate from 2. Let K be the kernel of f . Then qf(R) ⊗ K is the kernel of qf(R) ⊗ f . Since by 3, K = 0 if and only if qf(R) ⊗ K = 0, f is one–one if and only if qf(R) ⊗ f is. The argument for the onto case is similar. This establishes 4. For 5, we begin by noting that by Corollary 3 there are a finitely generated F · D module V and a surjection R ⊗F V → M , which determines a morphism f : HomR (M, R ⊗F V ) → HomR (M, M ). Since both homomorphism spaces are R · D modules by Corollary 4, by 4 f is surjective if and only if qf(R) ⊗ f is. Since the modules in question are R finitely generated, tensoring with qf(R) (or localizing at S) distributes over Hom and reduces to the case where S −1 M is a vector space over the field S −1 R, so that qf(R) ⊗ f is surjective. Then f is surjective, and this means that M is an R direct summand of the free R module R ⊗F V , and hence M is projective, proving 5. In general, since every finitely generated R · D submodule of M is projective, M is flat, proving 6. 2 Part 5 of Theorem 5 has many consequences. Recall that a simple R is an integral domain, so that projective modules have constant rank. Corollary 8. Assume that R is differentially simple, and let M be a finitely generated R · D module. 1. R · D submodules of M are R direct summands. 2. M has the ascending chain condition and the descending chain condition on submodules. 3. HomR (M, ·) is an exact functor of finitely generated R · D modules. 4. M → M ∗∗ (double duality with respect to R) is an isomorphism. 5. Even if M is not finitely generated, HomR (·, M ) is an exact functor of finitely generated R · D modules. Proof. If N is an R · D submodule of M then M/N is R projective and hence N is an R direct summand of M . So if N = M , N has smaller rank; 1 and 2 follow. For 3, HomR (M, ·) is exact on all R modules, because M is projective. Double duality is an isomorphism for finitely generated free R modules, hence for projective ones, giving 4. For N finitely generated projective, HomR (N, M ) = (N )∗ ⊗ M , and this is an exact functor of N since N is projective and M is flat, proving 5. 2 A Zorn’s Lemma argument applied to part 1 of Corollary 8 shows that the same result is true even if M is not finitely generated: Proposition 2. Assume that R is differentially simple, and let M be a R · D module.
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1. R · D submodules of M are R direct summands. 2. Exact sequences of R · D modules are R split. Proof. Part 2 follows from 1. So assume N is an R·D submodule of M , and consider pairs (Mα , fα ) where Mα is an R · D submodule of M which contains N and fα : Mα → N is an R homomorphism such that fα (n) = n for all n ∈ N . For example, (N, idN ) is such a pair. Order the pairs so that (Mα , fα ) ≤ (Mβ , fβ ) if Mα ⊆ Mβ and fβ (x) = fα (x) for x ∈ Mα . Any chain {(Mα , fα )|α ∈ A} has an upper limit P = (∪Mα , g) where g(x) = fα (x) for x ∈ Mα . So Zorn’s Lemma implies the existence of a maximal pair (M0 , f0 ). If M0 = M , then M = N ⊕ Ker(f0 ). So suppose not, and let x ∈ M − M0 . Let Q be the (finitely generated) R · D module generated by x, and let M1 = M0 + Q. Note that M1 is an R · D module containing N and that M0 M1 . Now M0 ∩ Q is an R · D submodule of Q, and hence, by Corollary 8, is an R direct summand. Let M2 be an R module complement, so that Q = (M0 ∩ Q) ⊕ M2 . Then we have M1 = M0 ⊕ M2 as R modules: Q ⊆ M0 + M2 so both Q and M0 belong to M0 + M2 , which implies M1 = M0 + M2 ; and, since M2 ⊆ Q, any element in M0 ∩ M2 also belongs to Q and hence to M2 ∩ Q ∩ M2 = 0, so the sum is direct. We define f1 : M1 → N to be f0 ⊕ 0. Then (M1 , f1 ) is strictly larger than (M0 , f0 ). This contradiction implies that M0 = M so that, as R modules, M = N ⊕ Ker(f0 ), proving 1. 2 Definition 4. A finitely generated R · D module M is called constantly generated if it is generated as an R module by a finite set of elements whose derivatives are zero. For example, R is constantly generated, as is R(n) using component-wise differentiation. (Here and below we use the notation (·)(n) for the direct sum of n copies of (·).) When R is differentially simple the set of elements RD of R of derivative 0 is a field. When R is a Picard–Vessiot ring, this field is C. Proposition 3. Assume that R is differentially simple and let V = R(n) be a free R module with component-wise differentiation. Then every differential submodule W of V is (isomorphic to) a free R module with component-wise differentiation and is an R · D direct summand of V . Proof. Let e1 , . . . , en be the standard basis of R(n) . We show by induction that there is a basis v1 , . . . , vn of V consisting of vectors with constant entries (called a constant basis) such that W = Rv1 + . . . + Rvm . We assume W = 0. Choose w ∈ W such that w = 0 and the number k of initial zero coordinates of w is maximal for W . The image of the projection V → R on the k + 1 coordinate restricted to W is then a non-zero differential R submodule of R, hence equal to R. In particular, we may assume the k + 1 coordinate of w is 1. But then the derivative of w has k + 1 initial zero coordinates, and hence is zero. This means all the coordinates of w are in RD . Let v1 = w. The usual replacement for RD vector spaces then gives a constant basis for V beginning with v1 . Project on the
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span of the last n − 1 elements of this basis, apply induction, and then pull back to V . Because Rv1 ⊆ W , the result follows. 2 Proposition 3 implies that all constantly generated R · D modules are tuples: Corollary 9. Assume that R is differentially simple. Then any finitely generated constantly generated R · D module M is (isomorphic to) a free R module with component-wise differentiation. Proof. Let m1 , . . . , mn be elements of derivative zero generating M . Then M is the epimorphic image of R(n) with component-wise differentiation via the differential R ri mi . The kernel V is a differential submodule to which morphism (r1 , . . . , rn ) → Proposition 3 applies. 2 Corollary 9 then implies that exact sequences of constantly generated R · D modules are R · D split. Corollary 10. Assume that R is differentially simple. Let f : M → N be a morphism of finitely generated constantly generated R · D modules. Then the kernel (respectively image) of f is a direct summand of M (respectively N ). In particular, if f is injective (respectively surjective) then it has a left (respectively right) inverse. Proof. Apply Corollary 9 to see that M and N are free R modules with component-wise differentiation and then apply Proposition 3. 2 We extend the result of Corollary 10 to non-finitely generated modules via a Zorn’s Lemma argument as in Proposition 2. An examination of the proof of that proposition shows, in the notation of that proof, that the key step is that R · D submodules of a certain finitely generated R · D submodule Q of the ambient R · D module are direct summands. By Corollary 10, if Q is constantly generated then such submodules are R · D module summands. So if we assume that all finitely generated R · D submodules are constantly generated, we get that any R · D submodule is an R · D direct summand: Corollary 11. Let f : M → N be an injective R · D morphism, and assume every finitely generated R · D submodule of N is constantly generated. Then f has an R · D left inverse. We saw above (Corollary 9) that finitely generated constantly generated R·D modules are isomorphic to direct sums of copies of R. We now use Corollary 11 to show that the same is true for arbitrary R · D modules all of whose finitely generated R · D submodules are constantly generated:
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Corollary 12. Let M be an R · D module and assume every finitely generated R · D submodule of M is constantly generated. Then M is isomorphic as an R · D module to a direct sum of copies of R. Proof. M is the sum of its finitely generated R · D submodules, and by Corollary 9 each of these is a (finite) direct sum of copies of R. Thus M is a sum of copies of R. Since R is a simple R · D module, standard arguments show that M is actually a direct sum of copies of R: use a Zorn’s Lemma argument to find a maximal submodule M0 of M which is a direct sum of copies of R. If M0 M , there would be a copy of R contained in M but not in M0 . Then R ∩ M0 = {0} by simplicity of R, so that M0 + R = M0 ⊕ R is a larger direct sum of copies of R contained in M . Hence M0 = M . 2 Corollary 11 applies in case every finitely generated R · D module is constantly generated. An example of this is the case that R is the Picard–Vessiot ring of a Picard–Vessiot closure: Proposition 4. Let R be the Picard–Vessiot ring of a Picard–Vessiot closure of F . Then every finitely generated R · D module is constantly generated. Proof. Obviously a homomorphic image of a constantly generated R · D module is constantly generated. Since every R · D module is a quotient of one of the form R ⊗F V where V is a finite dimensional differential F module (i.e., an F · D module) it suffices to prove that R ⊗F V is constantly generated. So let R0 ⊂ R be the Picard–Vessiot ring of a Picard–Vessiot extension E ⊇ F for V . By definition, E ⊗F V is differentially isomorphic to a direct sum E (n) , from which it follows that R0 ⊗F V is isomorphic to (n) R0 so that R ⊗F V is isomorphic to R(n) , and in particular that R ⊗F V is constantly generated. 2 We conclude this section with some remarks on examples. On the one hand, we have that any simple rationally differential F algebra can be embedded properly in a larger one: Proposition 5. Let R be a simple rationally differential F algebra. Then there is a simple rationally differential F algebra S properly containing R. Proof. Let {xα |α ∈ A} be a set of indeterminates over R, and consider the polynomial ring S0 = R[xα , α ∈ A] as a differential ring extension of R where D(xα ) = 0 for all α. As an R module, S0 is an (infinite) direct sum of copies of R, and hence an F · D module. A similar remark applies to any ring generated over R by constants, and in particular 1 to the ring S1 = R[y][{ y−c |c ∈ R, c = 0}] where D(y) = 0. Let S = S1 /P where P is a maximal differential ideal. Then S is a simple rationally differential F algebra S containing R. If the containment is not proper, there is an element r ∈ R with y − r ∈ P .
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This implies that y − r ∈ P ∩ R = 0. So r = 0 which means that y − r is a unit of S1 and hence not in P after all. So R ⊂ S is indeed proper. 2 On the other hand, if we do not allow new constants, all simple rationally differential F algebras are subalgebras of the Picard–Vessiot ring of a Picard–Vessiot closure of F : Proposition 6. Let R be a simple rationally differential F algebra whose constants are those of F . Then R is a differential subring of the Picard–Vessiot ring of a Picard–Vessiot closure of F . Proof. Let E denote the quotient field of R, let F1 be a Picard–Vessiot closure of F and let E1 be a Picard–Vessiot closure of E. Since F ⊆ E, we may take E1 to contain F1 . Let S be the Picard–Vessiot ring of F1 . Let a ∈ R and let L(Y ) = 0 be a linear homogeneous differential equation over F that a satisfies. Let V be the set of solutions of L in F1 and let W be the set of solutions of L in E1 . Both V and W are vector spaces over the constants of the same dimension (the order of L). Hence they will be equal provided E1 and F1 have the same constants, which means that E and F have the same constants. The ideal in R of the denominators of a constant of E is a non-zero differential ideal, and hence coincides with R, so that every constant of E is in R, and by assumption the constants of R are those of F . So V = W , which means that a ∈ F1 , and therefore a ∈ S. 2 5. Square zero In this section we assume that R is a rationally differential F algebra which in most cases is assumed to be differentially simple. Definition 5. A differential square zero extension of R is a commutative differential F algebra A and a differential F algebra surjection f : A → R whose kernel I is finitely generated and satisfies I 2 = 0. The extension is split if there is an F algebra homomorphism g : R → A with f g = 1; it is differentially split if g can be taken to be a differential homomorphism. In the definition, I is an R module and hence an R[D] module. When I is an R · D module, A is a rationally differential F algebra. We recall briefly some notations from commutative algebra: For any commutative rings B ⊆ C, the kernel of the multiplication map C ⊗B C → C is denoted J(C/B) and the quotient J(C/B)/J(C/B)2 is denoted ΩC/B . Note that, in the definition, when I is an R · D module so that A is an F · D algebra then J(A/F ) and ΩA/F are F · D modules. Square zero extensions as in the definition are known to split if and only if the map δ : I → ΩA/F ⊗A R
(∗)
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by δ(x) = (x ⊗ 1 − 1 ⊗ x) ⊗A 1 has a left inverse [7, Thm. 57, p. 186]. Note that when I is an R · D module then so is ΩA/F ⊗A R and so δ is an R · D morphism. Since δ is an R · D morphism, then an examination of the construction of the splitting in [7, Thm. 57, p. 186] shows that differential square zero extensions differentially split if and only if δ has a differential left inverse. When R is simple, we will show that differential square zero extensions are split as commutative rings. By Proposition 2 and the above commutative algebra result, this is equivalent to showing that δ is injective. Theorem 6. Assume that R is a differentially simple domain and that A is a differential square zero extension of R with kernel P . Assume that P is an R · D module. Then the extension is F algebra split. Proof. We note that P is a prime ideal, as R is a domain, and also a maximal differential ideal, as R is simple. The annihilator of P is also a differential ideal, and since P is square zero it contains P and therefore coincides with it. We localize A at P . Then AP is a square zero ring extension of K = (AP )−1 R with kernel PP ; PP is a maximal ideal so the quotient K is a field, of characteristic zero since it contains F . Square zero ring extensions of characteristic zero fields split. Thus δP : PP → ΩAP /F ⊗AP K has a left inverse and is in particular injective. We have a commutative diagram δ
P −−−−→ ⏐ ⏐q
ΩA/F ⊗A R ⏐ ⏐
δP
PP −−−−→ ΩAP /F ⊗AP K The vertical map q : P → PP is injective since the annihilator of P is contained in P . So δ is a right factor of the injective map δP q and hence is injective. Now by Proposition 2, the injective R · D module map δ has a left inverse, which then implies that the differential square zero extension A → R is split as F algebras. 2 The proof of Theorem 6 amounts to showing that, for a square zero differential extension with kernel an R · D module, the R · D morphism δ is always injective. So if every
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R · D submodule of ΩA/F ⊗A R is an R · D direct summand, the extension is differentially split. This will happen, by Corollary 11, if every finitely generated R · D submodule of ΩA/F ⊗A R is constantly generated. We record this fact for later reference: Corollary 13. Let A → R be a square zero differential extension with kernel an R·D module, and assume that every finitely generated R · D submodule of ΩA/F ⊗A R is constantly generated. Then A → R is differentially split. In particular, if R is the Picard–Vessiot ring of a Picard–Vessiot closure of F , every square zero differential extension of R with kernel an R · D module is differentially split. Proof. For the final assertion, we apply Proposition 4.
2
Theorem 6 says that a differential square zero extension A → R whose kernel I is an R · D module is F algebra isomorphic to I R with component-wise addition and multiplication (i, r)(j, s) = (rj + si, rs); we regard this isomorphism as equality. Let DA be the derivation of A. Because A → R is differential, and because I is a differential ideal of A, we must have that DA (i, r) = (DI (i) + h(r), DR (r)), where h : R → I. For DA to be additive, h must be additive. For DA to satisfy DA ((i, r)(j, s)) = (i, r)DA (j, s) + (j, s)DA (i, r) requires (DI (rj + si) + h(rs), DR (rs)) to equal (DR (s)i + r(DI (j) + h(s)), rDR (s)) + (DR (r)j + s(DI (i) + h(r), sDR (r)) which means that h(rs) = rh(s) + sh(r); in other words, h is a (C) derivation. For f ∈ F , f → (0, f ) makes A an F algebra. For A to be a differential F algebra requires that DA ((0, f )) = (0, DF (f )) and hence that h(f ) = 0. Thus h is an F derivation, or an element of DerF (R, I). Conversely, any F derivation h : R → I defines a derivation on I R making it a square zero differential extension with kernel I. Now suppose that Λ : R → A is a differential splitting, which can be written as Λ(r) = (λ(r), r) where λ : R → I is a map. We first consider what it means for Λ to be multiplicative: (λ(rs), rs) = (λ(r), r)(λ(s), s) entails that λ(rs) = rλ(s) +sλ(r). Then for Λ to be F linear requires that λ be F linear and for Λ to be an F algebra map requires that Λ(a) = (0, a) for a ∈ F so λ(F ) = 0. Thus λ is an F derivation, or an element of DerF (R, I). Now for Λ to be differential requires that Λ(DR (r)) = DA (λ(r), r) which requires that λ(DR (r)) = DI (λ(r)) + h(r), or that h = λDR − DI λ. Finally, we note that M = DerF (R, I) is an R[D] module with DM (g) = DI g − gDR . Thus if the derivation DA is determined by the derivation h, there is a differential splitting of A → R if and only if h = −DM (λ) for some λ. We summarize this discussion in the following proposition: Proposition 7. Let R be a differentially simple domain and let I be an R · D module. Then all square zero extensions of R with kernel I are differentially split if and only if every element of DerF (R, I) is a derivative. Equivalently, all square zero extensions of R with kernel I are differentially split if and only if every element of HomR (ΩR/F , I) is a derivative.
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Proof. Only the final assertion needs proof. It is standard commutative algebra that DerF (R, I) = HomR (ΩR/F , I). The identification is as follows: given an R homomorphism φ : ΩR/F → I, define a derivation d by d(r) = φ(r ⊗ 1 − 1 ⊗ r + J(R/F )2 ). This identification can be checked to be as R[D] modules. 2 Now, in the notation of Proposition 7, assume that both ΩR/F and I have the property that all their finitely generated R · D submodules are constantly generated. Then, by Corollary 12, both are direct sums of copies of R as differential modules. It follows that HomR (ΩR/F , I) is a product of direct sums of copies of R (both the product and the direct sums may be infinite). Thus every element of HomR (ΩR/F , I) will be a derivative if (and only if) every element of R is a derivative. So we conclude: Corollary 14. Let R be a differentially simple domain and let I be an R · D module. Assume: 1. Both ΩR/F and I have the property that all their finitely generated R · D submodules are constantly generated. 2. Every element of R is a derivative. Then all square zero extensions of R with kernel I are differentially split. The Picard–Vessiot ring R of a Picard–Vessiot closure F1 of F has the property that all its finitely generated R · D modules are constantly generated by Proposition 4. It is also true that every element of R is a derivative (note that this need not be true for F1 ): suppose a ∈ R is a solution of the differential equation L(Y ) = 0, where L(Y ) is a monic linear homogeneous differential operator of order n over F , and let V be its full set of solutions in F1 . Then V is a vector space of dimension n over C, and V ⊂ R. Consider the equation L(Y ) = 0, which is of order n +1 and also monic, linear, and homogeneous, and let W be its full set of solutions in F1 . Then W has dimension n + 1 over C, and W ⊂ R. Consider the map W → V by derivation. This is C linear and its kernel is a subspace of C, which implies that derivation is surjective. Thus both hypotheses of Corollary 14 obtain when R is the Picard–Vessiot ring of a Picard–Vessiot closure of F . Consequently, by the conclusion of Corollary 14, all square zero extensions of R with R · D module kernel are differentially split, as we saw by other means in Corollary 13. We can also use Proposition 7 to provide examples of non-splitting: Example 1. Let F = C and let R = C[x, x−1 ] with D(x) = x. It is straightforward to verify that ΩR/F is a one dimensional free R module so that for any R · D module I HomR (ΩR/F , I) is isomorphic to I as R · D module. Take I = R. Then 1 is not a derivative in I (in R, derivatives have no constant term), so the corresponding element of HomR (ΩR/F , I) is not a derivative. This implies that the corresponding square zero extension with R · D module kernel R R is not differentially split.
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References [1] Y. André, Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence, Ann. Sci. Éc. Norm. Supér. 47 (2014) 449–467. [2] I. Doriaswamy, Projectivity of modules with suitable group actions, Comm. Algebra 10 (1982) 787–795. [3] A. Magid, Cohomology of rings with algebraic group action, Adv. Math. 59 (1986) 124–151. [4] A. Magid, Equivariant completions of rings with reductive group action, J. Pure Appl. Algebra 49 (1987) 173–185. [5] A. Magid, Lectures on Differential Galois Theory, Univ. Lecture Ser., vol. 7, American Mathematical Society, Providence, RI, 1997 (second printing with corrections). [6] A. Magid, Universal covers and category theory in polynomial and differential Galois theory, in: Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute, 2004, pp. 373–385. [7] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970. [8] B. Parshall, L. Scott, An imprimitivity theorem for algebraic groups, Indag. Math. 42 (1980) 39–47. [9] L. Rowen, Ring Theory, vol. I, Academic Press, San Diego, 1988. [10] M. van der Put, M. Singer, Galois Theory of Linear Differential Equations, Springer-Verlag, New York, 2003.
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Advances in Applied Mathematics www.elsevier.com/locate/yaama
Differential étale extensions and differential modules over differential rings Andy R. Magid Department of Mathematics, University of Oklahoma, Norman, OK 73019, United States
a r t i c l e
i n f o
Article history: Received 12 December 2014 Received in revised form 7 September 2015 Accepted 7 September 2015 Available online 19 September 2015 MSC: 12H05 Keywords: Picard–Vessiot Differential Galois
a b s t r a c t This paper studies differential square zero extensions and differential modules of a commutative differential algebra R over a differential field F where the field of constants of F is algebraically closed and of characteristic 0. All elements of R and the differential R modules and algebras considered are assumed to satisfy linear homogeneous differential equations over F . For R differentially simple, we describe the invectives, and, using that all considered differential modules are R flat, provide a criterion for all square zero extensions to be differentially split. © 2015 Elsevier Inc. All rights reserved.
0. Introduction Let F be a differential field of characteristic zero whose field of constants C is algebraically closed. Suppose an order n linear monic homogeneous differential equation with coefficients in F is given. A Picard–Vessiot, or differential Galois, extension E of F for this equation is a minimal differential field extension E containing n solutions of the equation which are linearly independent over the field of constants of E, where the latter is required to be the constant field C of F . This is the exact analogue of the usual, E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aam.2015.09.006 0196-8858/© 2015 Elsevier Inc. All rights reserved.
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or polynomial, Galois extension: given a monic polynomial of degree n over a field K, a Galois extension of K for this polynomial is a minimal extension field L containing n distinct roots of the equation. In both cases there is a Galois group consisting of all automorphisms of L (respectively differential automorphisms of E) fixing K (respectively F ). This Galois group is finite for L/K and an algebraic group over C for E/F . Moreover, in algebraic geometry language, L is essentially the function field/coordinate ring of a principal homogeneous space (respectively E is essentially the function field of a principal homogeneous space) for the Galois group. In the differential case, the coordinate ring of the space is essentially the Picard–Vessiot ring R of E/F , which can be defined as the set of elements of E which satisfy a linear monic homogeneous differential equation with coefficients in F . The Picard–Vessiot ring R is a differential F algebra, and it has no proper differential ideals. For Picard–Vessiot theory, see [5] and [10]. The ordinary Galois extension L/K corresponds to a finite étale cover of schemes Spec(L) → Spec(K). In terms of the coordinate rings L and K the étale condition is the following: suppose T → S is a surjective homomorphism of commutative K algebras whose kernel I satisfies I 2 = 0. Then any K algebra homomorphism L → S lifts to a homomorphism L → T . (This is a way of saying that the K polynomial of which L is the splitting field is separable.) It is natural to inquire whether E/F , or more precisely R/F , shares a similar property. In previous work, we considered the related question for commutative rings on which a reductive algebraic group acted. It turned out that rings with reductive algebraic group H action with no H stable ideals have a sort of “equivariant étale” property: H ring homomorphisms from them to H equivariant square zero extensions lift [4, Thm. 1.2, p. 176]. Although this statement only involves rings with group action, the proof used required considering modules which were simultaneously modules for the group and the ring, in fact it occurs naturally in the context of a general investigation of such modules. To describe this context, and its differential analogue, we retain the meaning of the symbols F and C, but redefine R and G as needed. Let G be a C algebraic group. Consider those rational G modules which are simultaneously modules for a commutative C algebra R on which G acts rationally and such that the G action and R actions on the module are compatible. Such modules, which may be infinite dimensional, arise in the consideration of induced modules for quotients of G and were studied in [8]; in that paper they were called R · G modules, a convention since followed for the consideration of such modules in a general context. In the previous work [3], as well as the equivariant étale property, we considered in general injective modules and homology for R · G modules. In this paper, we consider similar questions for differential modules over a differential commutative F algebra R. The category of differential vector spaces over a differential field bears a close analogy to the category of rational modules for an algebraic group. Indeed, for the differential field F with field of constants C, and if the differential vector spaces are finite dimensional, their category is antiequivalent to the category of finite dimensional rational modules for a certain (pro)algebraic group over C [6].
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For the differential case, in addition to the obvious compatibility of differential and R actions, we require, analogously to what is built into the definition of rational module for an algebraic group, a local finiteness condition on R and the allowed differential modules, which we also call rational. Fortunately, for the main example of rational differential rings, namely Picard–Vessiot rings [10, Defn. 1.15, p. 12], this rationality condition obtains. Following the notation from the group case, we call this the category of R · D modules. Picard–Vessiot rings are differentially simple. For rings R with algebraic group G action, if the ring has no group stable ideals, finitely generated R · G modules are R projective [2, Prop. 3.5, p. 794] (and hence all are R flat), a fact which has a number of useful consequences. Yves André [1] has proved that a similar result obtains in the differential case as we recall below. This then allows us to establish the differential analogue of the lifting property for homomorphisms to square zero extensions, although the requirement is more stringent than differentially simple, and we show by example that not all Picard–Vessiot rings have this “differential étale” property. The requirement is met, however, for Picard–Vessiot rings of Picard–Vessiot closures. This paper is organized as follows: the initial section, Basics, gives the relevant definitions and standard properties of R · D modules. The next section F · D modules considers the special case that R is the base field F ; we look at the construction of injectives in this case. The following section, Injective, looks at the construction of injective R · D modules for general R. The next section, Simple, considers R · D modules when R has no proper differential ideals. We recall here André’s theorem that all R · D modules are R flat, and R projective if R finitely generated. The final section, Square zero, considers what we have called above the “differential étale” property and obtains a condition for when it holds. Most of the results in all but the final section of this paper were presented in a Special Session in Boston in January 2012 dedicated to the memory of Jerald Kovacic; the rest of the results were obtained as a consequence of questions raised at a BIRS workshop in June 2012. I am grateful to the organizers of both the session and the workshop for the opportunities provided, and I dedicate this work in its entirety to Jerry Kovacic, whose leadership in differential algebra was and is an inspiration. 1. Basics Let F be a differential field of characteristic zero with derivation DF whose field of constants C is algebraically closed. Let R be a commutative differential F algebra with derivation DR satisfying DR |F = DF . We assume that every element of R satisfies a monic linear homogeneous differential equation over F . We call such an R a differentially rational algebra over F . An R[D] module is an R module M furnished with a C endomorphism DM such that DM (rv) = DR (r)v + rDM (v) for all r ∈ R and v ∈ M . Thus R is an R[D] module. An R[D] module M such that every element of M satisfies
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a monic linear homogeneous differential equation over F is called an R · D module. Thus R is an R · D module. If M is an R[D] module and N is an R[D] submodule then M is an R · D module if and only if both N and M/N are R · D modules. For an R[D] module M , the kernel M D of DM is a vector space over C, and more generally a module over the ring RD . If V is an F finite dimensional F [D] module, it is an F · D module. Similarly, the symmetric algebra SF (V ), being a direct sum of F finite dimensional F [D] modules, is an F ·D module. This provides an example R = SF (V ) of a differentially rational F algebra. If S = {Vi |i ∈ I} is a set of F finite dimensional F [D] modules, then R = ⊗i∈I SF (Vi ) is also an example of a differentially rational F algebra, and it will be clear below that every example is a quotient of such a tensor product. In particular, the Picard–Vessiot ring of a Picard–Vessiot closure of F is a differentially rational F algebra. An R[D] morphism f between R[D] modules M and N is an additive homomorphism satisfying f (rm) = rf (m) and DN (f (m)) = f (DM (m)) for r ∈ R and m ∈ M . If in addition M is an R · D module then so is f (M ). An R[D] morphism between R · D modules is called an R · D morphism. The R[D] modules form a category of which the R · D modules are a full subcategory. If f is an R[D] morphism, its kernel and image as an R module homomorphism are R[D] modules, and it follows that the categories of R[D] and R · D modules are abelian. If M and N are R[D] modules, so is M ⊗R N (with DM ⊗N (m ⊗n) = DM (m) ⊗n +m ⊗DN (n)) and HomR (M, N ) (with DHom(M,N ) (φ)(m) = DN (f (m)) − f (DM (m))). If M and N are further R · D modules, so is M ⊗R N , but this need not be the case for HomR (M, N ). It will be an R · D module provided M is finitely generated as an R module (see Corollary 4 below). With the above actions, the map HomR (M, N ) ⊗R M → N by f ⊗ m → f (m) is an R[D] homomorphism. When V is an F [D] module, the F [D] modules R ⊗F V and HomF (R, V ) are R[D] modules under the R actions r · (s ⊗ v) = rs ⊗ v and (r · f )(s) = f (sr). When V is an F · D module, so is R ⊗F V , but in general HomF (R, V ) will not be an R · D module. Let R[X; D] denote the twisted polynomial ring over R: as additive R modules, R[X; D] coincides with the polynomial ring R[X], but with multiplication determined by Xr = rX + D(r) [9, Defn. 1.6.16, p. 93]. Clearly F [X; D] is a subring of R[X; D]. An R[D] module M becomes an R[X; D] module via Xm = DM (m), and an R[X; D] module becomes an R[D] module via DM (m) = Xm. Regarded as an R[X; D] module, an R · D module is an R[X; D] module every element of which is annihilated by a monic element of F [X; D]. In other words, the category of R · D modules is the category of F [X; D]-torsion R[X; D] modules. If M is an R[X; D] module and m ∈ M then AnnR[X;D] (m) = {f ∈ R[X; D]|f m = 0} is a left ideal. M is an R · D module provided that AnnF [X;D] (m) = 0 for all m ∈ M . It is known that left ideals of F [X; D] are principal [9, Prop. 1.6.24, p. 95]. It is trivial that non-zero principal left ideals of F [X; D] are of F finite codimension, from which it follows that an F [X; D] module is torsion if and only if all its cyclic submodules
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(i.e. singly generated) are finite F dimensional, and this latter is equivalent to being a sum (not necessarily finite) of F finite dimensional F [X; D] modules. Thus if M is a torsion F [X; D] module and m ∈ M there is an F finite dimensional F [X; D] submodule W ⊆ M with m ∈ W . Lemma 1. Let M be an R[D] module and let V be an F finite dimensional F [D] submodule. Then RV is an R · D module. Proof. RV is clearly an R[D] module. Let r ∈ R, and let W be an F finite dimensional F [D] submodule of R containing r. Then W V is an F finite dimensional F [D] submodule of RV containing rV . It follows that RV is the sum of its F finite dimensional F [D] submodules. 2 Corollary 1. Let M be an R[D] module. The sum Mt of all the F finite dimensional F [D] submodules of M is an R · D submodule which contains all R · D submodules of M . Moreover, Mt is the sum of all the R · D submodules of M . Proof. If m ∈ Mt there is an F finite dimensional F [D] submodule of M with m ∈ V . By Lemma 1 RV is an R · D module and hence a sum of F finite dimensional F [D] submodules of M . Thus RV ⊆ Mt and in particular Rm ⊂ Mt . It follows that Mt is an R[D] module and hence an R · D module. Since any R · D submodule of M is contained in Mt , it further follows that Mt is also the sum of the R · D submodules of M . 2 Corollary 2. Let V be an F · D module. Then R ⊗F V is an R · D module. Proof. It is clear that R ⊗ V is an R[D] module. By Corollary 1 applied to the case R = F , V is the sum of its finite dimensional F [D] submodules W , so R ⊗ V is the sum of the R ⊗ W . The latter is an R · D module by Lemma 1. 2 If M is a finitely generated R · D module and S is a finite generating set, then S is contained in a finite dimensional F · D submodule V of M , and there is an R · D module surjection R ⊗F V → M . We record this: Corollary 3. A finitely generated R · D module M is a quotient of one of the form R ⊗F V where V is a finite dimensional F · D module. If V is cyclic [as an F [D] module] then M is cyclic [as an R[D] module]. It will always be the case that V is F [D] cyclic when DF is non-trivial. The penultimate sentence in Corollary 3 is obvious and added for use below. The final sentence is the Cyclic Vector Theorem [10, Prop. 2.9, p. 40]. As noted above, for R · D modules M and N , the R[D] module HomR (M, N ) need not be an R · D module.
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For example, let R = F [x] with Dx = 1. Take N = R, and let M be a countable direct sum of copies of R (both the latter being R · D modules). Then HomR (M, N ) is a countable direct product of copies of R. Consider the element y = (x, x2 , x3 , . . .) of this product. If HomR (M, N ) is an R · D module, then there is an F finite dimensional F [D] submodule V containing y. Suppose V has F dimension m. Then for any n, the projection of V on the nth factor is an F [D] submodule of R of dimension at most m containing xn . This is impossible for n ≥ m. However, if M is finitely generated, we have the following: Corollary 4. Let M and N be R · D modules with M finitely generated over R. Then HomR (M, N ) is an R · D module. Proof. By Corollary 3, there is a finite dimensional F · D module V and a surjection R ⊗ V → M . This induces an inclusion HomR (M, N ) → HomR (R ⊗ V, N ), and it suffices to show the latter is an R · D module. There is an R module isomorphism HomR (R ⊗ V, N ) → V ∗ ⊗F N , which is seen to be an R[D] morphism as well. Since V ∗ ⊗F N is R[D] isomorphic to (R ⊗F V ∗ ) ⊗R N and by Corollary 2 the first factor is an R · D module, so is the tensor product and the result follows. 2 The notation introduced in Corollary 1 will be useful below, so we single it out in a definition: Definition 1. Let M be an R[D] module. Then Mt denotes the largest R · D submodule of M . 2. F · D modules The category of F · D modules is investigated in [6] via the module which we here denote I which consists of all the elements of the Picard–Vessiot closure F1 of F which satisfy a linear differential equation over F . Note that I is an F · D module, although typically not finite dimensional over F . If Π(F ) denotes the proalgebraic group (over C) of the differential automorphisms of F1 over F , then V → HomF [D] (V, I) is an antiequivalence between the category of F finite dimensional F · D modules and the category of C finite dimensional rational Π(F ) modules. It follows that I is an injective F · D module, in fact an injective cogenerator for the category of F · D modules. It is also a subring of F1 , and contains a unique copy of the one dimensional trivial F module. We write this later inclusion as a morphism F → I and call it the canonical inclusion. If V is an F · D module, the morphism V → V ⊗F I by V → v ⊗ 1 is called the canonical inclusion for V . Proposition 1. Let V and E be F · D modules with E injective. Then V ⊗F E is injective.
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Proof. We need to show that Hom(·, V ⊗E) is exact, and in particular converts injections to surjections. Since F · D modules are directed unions of finite dimensional submodules, the usual Zorn’s Lemma arguments reduce the problem to showing that injections X → Y with Y finite dimensional are converted to surjections. So suppose we have a morphism X → V ⊗ E. Since X is finite dimensional, we can assume the image lies in W ⊗ E for some finite dimensional submodule W of V , so we have a morphism f : X → W ⊗ E. As usual, we identify W ⊗ E and Hom(W ∗ , E). By adjoint associativity (see Theorem 2 below, using R = F ), f corresponds to a morphism X ⊗W ∗ → E, which by the injectivity of E extends to a morphism Y ⊗ W ∗ → E. By adjoint associativity, this corresponds to a morphism Y → W ⊗ E which extends f . 2 It follows from Proposition 1 that the canonical inclusion V → V ⊗ I is an embedding of the F · D module in an injective F · D module, which we note is functorial in V . It follows that in the category of F · D modules we have functorial injective resolutions. For later use, we also provide another construction of an injective cogenerator for F ·D modules which does not rely on the use of the Picard–Vessiot closure. We begin with some observations on notation. If L is a left ideal of F [X; D], then F [X; D]/L is an F [D] module with D action D(f + L) = Xf + L. Then (F [X; D]/L)∗ = HomF (F [X; D]/L, F ) is an F [D] module with D action D(φ)(f + L) = D(φ(f + L)) − φ(D(f ) + L). We have similar considerations for the F duals of non-cyclic F [D] modules. Theorem 1. Assume that DF = 0. Let F be the set of left ideals L of F [X; D] such that F [X; D]/L is finitely generated as a F module. Regard F as directed by reverse inclusion. Then E = lim(F [X; D]/L)∗ −−→ F
is an injective cogenerator for F · D modules. Proof. Since DF is non-trivial, by the Cyclic Vector Theorem [10, Prop. 2.9, p. 40] every F finitely generated F [D] module V is cyclic, that is of the form F [X; D]/L for some L ∈ F. Applying this to V ∗ shows that V is also of the form (F [X; D]/L1 )∗ . We begin by showing that E is injective. As in the proof of Proposition 1, this reduces to the consideration of an F · D injection X → Y with Y finitely generated over F and an F · D morphism X → E. The image of this latter lies in some (F [X; D]/L)∗ . Taking duals and applying the Cyclic Vector Theorem then shows that we can assume that X ∗ = F [X; D]/LX and Y ∗ = F [X; D]/LY are cyclic and the surjection comes from the inclusion LY ⊆ LX . Let L1 = L ∩ LY . The morphism F [X; D]/L1 → F [X; D]/LY → F [X; D]/LX coincides with F [X; D]/L1 → F [X; D]/L → F [X; D]/LX , and taking duals this means that X → (F [X; D]/L)∗ → (F [X; D]/L1 )∗ → E lifts to Y → (F [X; D]/L1 0)∗ → E. This shows that E is an injective F · D module.
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Any non-zero F · D module M contains a non-zero F finitely generated submodule V . As previously remarked, V is of the form (F [X; D]/L)∗ and hence has a non-zero map to E. Since E is injective, this map extends to a non-zero map from M to E, showing that this latter is a cogenerator. 2 3. Injective We are going to show the construction of some injective R · D modules. The first step is an adjoint associativity property: Theorem 2. Let M and P be R · D modules and let N be an F · D module. Then there is a natural isomorphism HomF ·D (M ⊗R P, N ) → HomR·D (M, HomF (P, N )t ) Proof. Recall that HomF (P, N ) is an R module via r · f (p) = f (rp). The usual adjoint associativity isomorphism HomF (M ⊗R P, N ) → HomR (M, HomF (P, N )) carries R[D] homomorphisms to R[D] homomorphisms. Since the image of an R · D module under an R[D] homomorphism is an R · D module, R[D] morphisms M → HomF (P, N ) have their image in HomR (P, N )t . The result follows. 2 Corollary 5. Let E be an injective F · D module. Then HomF (R, E)t is an injective R · D module. If E is a cogenerator, so is HomF (R, E)t Proof. By Theorem 2, HomR·D (·, HomF (R, E)t ) equals HomF ·D (· ⊗R R, E) = HomF ·D (·, E). Since this last is exact so is the first. Suppose that M is a non-zero R · D module and E an injective cogenerator for F · D modules. As above, HomR·D (M, HomF (R, E)t ) equals HomF ·D (M, E). Since there is a non-zero map of M as an F · D module to E, there is non-zero map of M as an R · D module to HomF (R, E)t . 2 Corollary 5 applies when E is either of the injective cogenerators I or E of Section 2. In particular, the category of R · D modules has explicit injective cogenerators. We can also hope to construct an injective cogenerator from dual cyclic modules along the lines of Theorem 1. Here duality would be R duals with respect to R: HomR (·, R). Examining the proof of Theorem 1 we see we would (1) need this to be a category self-equivalence on R finitely generated R · D modules; (2) need submodules of R finitely generated R · D modules to be R finitely generated to apply (1); and (3) need R finitely generated R · D modules to be cyclic. As pointed out in Corollary 3, (3) is satisfied provided DF is non-trivial, but the other two points must be assumed. We will see later that they are satisfied if R is R[D] simple (Theorem 5). For now, we simply note that Theorem 1 is true if they are assumed.
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Theorem 3. Assume that DF = 0. Assume further that 1. HomR (·, R) is a category self-equivalence on R finitely generated R · D modules. 2. Submodules of R finitely generated R · D modules are R finitely generated. Let L be the set of left ideals L of R[X; D] such that R[X; D]/L is finitely generated as an R module. Regard L as directed by reverse inclusion. Then E = lim(R[X; D]/L)∗ −−→ L
is an injective cogenerator for R · D modules. There are, of course, injective R[D] modules: these are the same as injective R[X; D] modules. In general, these will not be R · D modules. Nor should we expect injective R · D modules to be R or R[D] injective. However, injective R · D modules share some properties with injective R modules, as we now see. The following lemma relates R and R[D] morphisms between R · D modules. Lemma 2. Let M and N be R · D modules and let f : M → N be an R homomorphism. Then there are an F [D] module V , an R module homomorphism g : M → M ⊗F V , and an R[D] homomorphism h : M ⊗F V → N such that f = h ◦ g. If M and N are R · D modules with M finitely generated then V may be taken finite dimensional over F . Proof. Let V be the F [D] submodule of HomR (M, N ) generated by f . The map g is given by m → m ⊗ f . The map h is given by m ⊗ v → v(m). It is immediate that f = h ◦ g. One checks that g and h are homomorphisms of the asserted type. If M and N are R · D modules with M finitely generated, then HomR (M, N ) is an R · D module by Corollary 4 so V is finite dimensional. 2 Using Lemma 2, we can show that injective R · D modules have an extension property with respect to R morphisms and R · D morphisms. Theorem 4. Let I be an injective R · D module and let φ : M → N be a one–one R · D morphism of R · D modules with M R finitely generated. Then the induced morphism HomR (N, I) → HomR (M, I) is surjective. Proof. Let f : M → I be an R homomorphism. Factor f as in Lemma 2 as f = h ◦ g : M → M ⊗F V → I, where V is F finite dimensional. Since ⊗F is exact, φ ⊗F 1V is a one–one R · D morphism between the R · D modules M ⊗F V and N ⊗F V . As I is R · D injective, the R · D morphism h : M ⊗F V → I extends to a morphism N ⊗F V → I. Preceding this with N → N ⊗F V by n → n ⊗f gives a map q : N → I with q ◦φ = f . 2
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Definition 2. R is said to be differentially Noetherian if R · D submodules of R finitely generated R · D modules are R finitely generated. If R is differentially Noetherian and M is an R finitely generated R·D module, we can construct an R · D module resolution {Pi } of M where each Pi is R free of finite rank: let P0 = R ⊗F V0 where V0 is an F finite dimensional F · D module and R ⊗F V0 → M is an R · D surjection. Let K1 be its kernel and let P1 = R ⊗F V1 where V1 is an F finite dimensional F · D module and R ⊗F V1 → K1 is an R · D surjection. The resolution arises by continuing this process. Corollary 6. Assume that R is differentially Noetherian. Let I be an injective R·D module and let M be an R finitely generated R · D module. Then ExtpR (M, I) = 0 for p > 0. Proof. Let {Pi } be the resolution of M constructed above. Then ExtpR (M, I) = H p (HomR (P∗ , I)). By Theorem 4, HomR (·, I) is exact on exact sequences of R finitely generated R · D modules. Since the kernels and cokernels of the transition maps in {Pi} are R finitely generated by the differentially Noetherian assumption, HomR (·, I) has no homology in degree above zero. So the result follows. 2 In the proof of Corollary 6, we actually showed that for any R · D module N , ExtpR (M, N ) = H p (HomR (P∗ , N )) where {Pi } is a resolution of M by R finitely generated R · D modules, with R · D morphisms as the transition maps. This makes HomR (P∗ , N ) a sequence of R · D modules, and hence its homology groups are R · D modules. Consequently, each ExtpR (M, N ) is an R · D module. Applying Corollary 6 and standard dimension shifting arguments, we also obtain similar results for injective resolutions: Corollary 7. Assume that R is differentially Noetherian. Let M and N be R · D modules with M R finitely generated. Let {E i } be a resolution of N by injective R · D modules. Then ExtpR (M, N ) = H p (HomR (M, E ∗ )). 4. Simple In this section we consider some of the strong algebraic consequences for R·D modules which follow when R has no proper differential ideals. We fix terminology: Definition 3. An R · D module M is said to be differentially simple if the only R · D submodules of M are {0} and M . The ring R is said to be differentially simple if R is a differentially simple R · D module, i.e. has no proper differential ideals. Let M be an R[D] module. The annihilator of M , AnnR (M ) = {r ∈ R|rm = 0∀m ∈ M } is a differential ideal, since if rM = 0, then 0 = D(r)m +rD(m) = D(r)m. Although in most of this section R will be an integral domain, we will define the torsion submodule
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of M directly: Mtor = {m ∈ M |rm = 0 for some non-zero divisor r ∈ R}. Note that Mtor = Mt . Lemma 3. Let M be an R[D] module. Then Mtor is an R[D] submodule. Proof. If mi ∈ M and ri mi = 0 for some non-zero divisor ri ∈ R, i = 1, 2, then r1 r2 (m1 + m2 ) = 0, and r1 r2 is not a zero divisor and m1 + m2 ∈ Mtor . If r ∈ R then r1 (rm1 ) = 0, so rm1 ∈ Mtor . Finally, r1 m1 = 0 implies that D(r1 )m1 + r1 D(m1 ) = 0. Multiply by r1 : then r12 D(m1 ) = 0, and since r12 is not a zero divisor, this means that D(m1 ) ∈ Mtor . 2 If S is the set of non-zero divisors of R then S −1 R is a differential ring and S −1 M = S R ⊗R M is an S −1 R[D] module, and Mtor is the kernel of the map M → S −1 M by m → 1 ⊗ m. It is possible that AnnR (Mtor ) = 0 even for Mtor = 0. However, we do have the following: 1
Lemma 4. Let M be an R · D module and let X be a finite subset of Mtor . Let N be the R[D] submodule of M generated by X. Then AnnR (N ) is a non-zero differential ideal of R. Proof. N is an R finitely generated submodule of Mtor , say generated by x1 , . . . xn . Let ri be a non-zero divisor with ri xi = 0. Then r = r1 . . . rn is a non-zero element of AnnR (N ). 2 We look at the implications of Lemma 4 when R is differentially simple; that is, R is the only non-zero ideal. We recall that in this case R is an integral domain. We let qf(R) denote its quotient field. Note that qf(R) = S −1 R where S is the set of non-zero divisors of F . The following result, which reveals the key consequences of the differential simplicity property, is due to Yves André [1, Theorem 2.2.1]; in fact that more general result deals with R[D] modules. For completeness of exposition, we have included both the statement of the result and its proof here. Theorem 5. Assume that R is differentially simple, and let M be an R · D module. 1. 2. 3. 4.
Mtor = 0. M → qf(R) ⊗ M is injective. M = 0 if and only if qf(R) ⊗ M = 0. Let f : A → B be a morphism of R · D modules. The f is one–one (onto) if and only if qf(R) ⊗ f is one–one (onto). 5. If M is R finitely generated, it is R projective. 6. M is R flat.
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Proof. Let x ∈ Mtor and let N be the R[D] submodule of M generated by x. By Lemma 4, AnnR (N ) = 0. Since R is simple, this means that AnnR (N ) = R, so that x = 1x = 0, proving 1. Since Mtor is the kernel of M → qf(R) ⊗ M , this map is injective, proving 2. Assertion 3 is immediate from 2. Let K be the kernel of f . Then qf(R) ⊗ K is the kernel of qf(R) ⊗ f . Since by 3, K = 0 if and only if qf(R) ⊗ K = 0, f is one–one if and only if qf(R) ⊗ f is. The argument for the onto case is similar. This establishes 4. For 5, we begin by noting that by Corollary 3 there are a finitely generated F · D module V and a surjection R ⊗F V → M , which determines a morphism f : HomR (M, R ⊗F V ) → HomR (M, M ). Since both homomorphism spaces are R · D modules by Corollary 4, by 4 f is surjective if and only if qf(R) ⊗ f is. Since the modules in question are R finitely generated, tensoring with qf(R) (or localizing at S) distributes over Hom and reduces to the case where S −1 M is a vector space over the field S −1 R, so that qf(R) ⊗ f is surjective. Then f is surjective, and this means that M is an R direct summand of the free R module R ⊗F V , and hence M is projective, proving 5. In general, since every finitely generated R · D submodule of M is projective, M is flat, proving 6. 2 Part 5 of Theorem 5 has many consequences. Recall that a simple R is an integral domain, so that projective modules have constant rank. Corollary 8. Assume that R is differentially simple, and let M be a finitely generated R · D module. 1. R · D submodules of M are R direct summands. 2. M has the ascending chain condition and the descending chain condition on submodules. 3. HomR (M, ·) is an exact functor of finitely generated R · D modules. 4. M → M ∗∗ (double duality with respect to R) is an isomorphism. 5. Even if M is not finitely generated, HomR (·, M ) is an exact functor of finitely generated R · D modules. Proof. If N is an R · D submodule of M then M/N is R projective and hence N is an R direct summand of M . So if N = M , N has smaller rank; 1 and 2 follow. For 3, HomR (M, ·) is exact on all R modules, because M is projective. Double duality is an isomorphism for finitely generated free R modules, hence for projective ones, giving 4. For N finitely generated projective, HomR (N, M ) = (N )∗ ⊗ M , and this is an exact functor of N since N is projective and M is flat, proving 5. 2 A Zorn’s Lemma argument applied to part 1 of Corollary 8 shows that the same result is true even if M is not finitely generated: Proposition 2. Assume that R is differentially simple, and let M be a R · D module.
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1. R · D submodules of M are R direct summands. 2. Exact sequences of R · D modules are R split. Proof. Part 2 follows from 1. So assume N is an R·D submodule of M , and consider pairs (Mα , fα ) where Mα is an R · D submodule of M which contains N and fα : Mα → N is an R homomorphism such that fα (n) = n for all n ∈ N . For example, (N, idN ) is such a pair. Order the pairs so that (Mα , fα ) ≤ (Mβ , fβ ) if Mα ⊆ Mβ and fβ (x) = fα (x) for x ∈ Mα . Any chain {(Mα , fα )|α ∈ A} has an upper limit P = (∪Mα , g) where g(x) = fα (x) for x ∈ Mα . So Zorn’s Lemma implies the existence of a maximal pair (M0 , f0 ). If M0 = M , then M = N ⊕ Ker(f0 ). So suppose not, and let x ∈ M − M0 . Let Q be the (finitely generated) R · D module generated by x, and let M1 = M0 + Q. Note that M1 is an R · D module containing N and that M0 M1 . Now M0 ∩ Q is an R · D submodule of Q, and hence, by Corollary 8, is an R direct summand. Let M2 be an R module complement, so that Q = (M0 ∩ Q) ⊕ M2 . Then we have M1 = M0 ⊕ M2 as R modules: Q ⊆ M0 + M2 so both Q and M0 belong to M0 + M2 , which implies M1 = M0 + M2 ; and, since M2 ⊆ Q, any element in M0 ∩ M2 also belongs to Q and hence to M2 ∩ Q ∩ M2 = 0, so the sum is direct. We define f1 : M1 → N to be f0 ⊕ 0. Then (M1 , f1 ) is strictly larger than (M0 , f0 ). This contradiction implies that M0 = M so that, as R modules, M = N ⊕ Ker(f0 ), proving 1. 2 Definition 4. A finitely generated R · D module M is called constantly generated if it is generated as an R module by a finite set of elements whose derivatives are zero. For example, R is constantly generated, as is R(n) using component-wise differentiation. (Here and below we use the notation (·)(n) for the direct sum of n copies of (·).) When R is differentially simple the set of elements RD of R of derivative 0 is a field. When R is a Picard–Vessiot ring, this field is C. Proposition 3. Assume that R is differentially simple and let V = R(n) be a free R module with component-wise differentiation. Then every differential submodule W of V is (isomorphic to) a free R module with component-wise differentiation and is an R · D direct summand of V . Proof. Let e1 , . . . , en be the standard basis of R(n) . We show by induction that there is a basis v1 , . . . , vn of V consisting of vectors with constant entries (called a constant basis) such that W = Rv1 + . . . + Rvm . We assume W = 0. Choose w ∈ W such that w = 0 and the number k of initial zero coordinates of w is maximal for W . The image of the projection V → R on the k + 1 coordinate restricted to W is then a non-zero differential R submodule of R, hence equal to R. In particular, we may assume the k + 1 coordinate of w is 1. But then the derivative of w has k + 1 initial zero coordinates, and hence is zero. This means all the coordinates of w are in RD . Let v1 = w. The usual replacement for RD vector spaces then gives a constant basis for V beginning with v1 . Project on the
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span of the last n − 1 elements of this basis, apply induction, and then pull back to V . Because Rv1 ⊆ W , the result follows. 2 Proposition 3 implies that all constantly generated R · D modules are tuples: Corollary 9. Assume that R is differentially simple. Then any finitely generated constantly generated R · D module M is (isomorphic to) a free R module with component-wise differentiation. Proof. Let m1 , . . . , mn be elements of derivative zero generating M . Then M is the epimorphic image of R(n) with component-wise differentiation via the differential R ri mi . The kernel V is a differential submodule to which morphism (r1 , . . . , rn ) → Proposition 3 applies. 2 Corollary 9 then implies that exact sequences of constantly generated R · D modules are R · D split. Corollary 10. Assume that R is differentially simple. Let f : M → N be a morphism of finitely generated constantly generated R · D modules. Then the kernel (respectively image) of f is a direct summand of M (respectively N ). In particular, if f is injective (respectively surjective) then it has a left (respectively right) inverse. Proof. Apply Corollary 9 to see that M and N are free R modules with component-wise differentiation and then apply Proposition 3. 2 We extend the result of Corollary 10 to non-finitely generated modules via a Zorn’s Lemma argument as in Proposition 2. An examination of the proof of that proposition shows, in the notation of that proof, that the key step is that R · D submodules of a certain finitely generated R · D submodule Q of the ambient R · D module are direct summands. By Corollary 10, if Q is constantly generated then such submodules are R · D module summands. So if we assume that all finitely generated R · D submodules are constantly generated, we get that any R · D submodule is an R · D direct summand: Corollary 11. Let f : M → N be an injective R · D morphism, and assume every finitely generated R · D submodule of N is constantly generated. Then f has an R · D left inverse. We saw above (Corollary 9) that finitely generated constantly generated R·D modules are isomorphic to direct sums of copies of R. We now use Corollary 11 to show that the same is true for arbitrary R · D modules all of whose finitely generated R · D submodules are constantly generated:
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Corollary 12. Let M be an R · D module and assume every finitely generated R · D submodule of M is constantly generated. Then M is isomorphic as an R · D module to a direct sum of copies of R. Proof. M is the sum of its finitely generated R · D submodules, and by Corollary 9 each of these is a (finite) direct sum of copies of R. Thus M is a sum of copies of R. Since R is a simple R · D module, standard arguments show that M is actually a direct sum of copies of R: use a Zorn’s Lemma argument to find a maximal submodule M0 of M which is a direct sum of copies of R. If M0 M , there would be a copy of R contained in M but not in M0 . Then R ∩ M0 = {0} by simplicity of R, so that M0 + R = M0 ⊕ R is a larger direct sum of copies of R contained in M . Hence M0 = M . 2 Corollary 11 applies in case every finitely generated R · D module is constantly generated. An example of this is the case that R is the Picard–Vessiot ring of a Picard–Vessiot closure: Proposition 4. Let R be the Picard–Vessiot ring of a Picard–Vessiot closure of F . Then every finitely generated R · D module is constantly generated. Proof. Obviously a homomorphic image of a constantly generated R · D module is constantly generated. Since every R · D module is a quotient of one of the form R ⊗F V where V is a finite dimensional differential F module (i.e., an F · D module) it suffices to prove that R ⊗F V is constantly generated. So let R0 ⊂ R be the Picard–Vessiot ring of a Picard–Vessiot extension E ⊇ F for V . By definition, E ⊗F V is differentially isomorphic to a direct sum E (n) , from which it follows that R0 ⊗F V is isomorphic to (n) R0 so that R ⊗F V is isomorphic to R(n) , and in particular that R ⊗F V is constantly generated. 2 We conclude this section with some remarks on examples. On the one hand, we have that any simple rationally differential F algebra can be embedded properly in a larger one: Proposition 5. Let R be a simple rationally differential F algebra. Then there is a simple rationally differential F algebra S properly containing R. Proof. Let {xα |α ∈ A} be a set of indeterminates over R, and consider the polynomial ring S0 = R[xα , α ∈ A] as a differential ring extension of R where D(xα ) = 0 for all α. As an R module, S0 is an (infinite) direct sum of copies of R, and hence an F · D module. A similar remark applies to any ring generated over R by constants, and in particular 1 to the ring S1 = R[y][{ y−c |c ∈ R, c = 0}] where D(y) = 0. Let S = S1 /P where P is a maximal differential ideal. Then S is a simple rationally differential F algebra S containing R. If the containment is not proper, there is an element r ∈ R with y − r ∈ P .
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This implies that y − r ∈ P ∩ R = 0. So r = 0 which means that y − r is a unit of S1 and hence not in P after all. So R ⊂ S is indeed proper. 2 On the other hand, if we do not allow new constants, all simple rationally differential F algebras are subalgebras of the Picard–Vessiot ring of a Picard–Vessiot closure of F : Proposition 6. Let R be a simple rationally differential F algebra whose constants are those of F . Then R is a differential subring of the Picard–Vessiot ring of a Picard–Vessiot closure of F . Proof. Let E denote the quotient field of R, let F1 be a Picard–Vessiot closure of F and let E1 be a Picard–Vessiot closure of E. Since F ⊆ E, we may take E1 to contain F1 . Let S be the Picard–Vessiot ring of F1 . Let a ∈ R and let L(Y ) = 0 be a linear homogeneous differential equation over F that a satisfies. Let V be the set of solutions of L in F1 and let W be the set of solutions of L in E1 . Both V and W are vector spaces over the constants of the same dimension (the order of L). Hence they will be equal provided E1 and F1 have the same constants, which means that E and F have the same constants. The ideal in R of the denominators of a constant of E is a non-zero differential ideal, and hence coincides with R, so that every constant of E is in R, and by assumption the constants of R are those of F . So V = W , which means that a ∈ F1 , and therefore a ∈ S. 2 5. Square zero In this section we assume that R is a rationally differential F algebra which in most cases is assumed to be differentially simple. Definition 5. A differential square zero extension of R is a commutative differential F algebra A and a differential F algebra surjection f : A → R whose kernel I is finitely generated and satisfies I 2 = 0. The extension is split if there is an F algebra homomorphism g : R → A with f g = 1; it is differentially split if g can be taken to be a differential homomorphism. In the definition, I is an R module and hence an R[D] module. When I is an R · D module, A is a rationally differential F algebra. We recall briefly some notations from commutative algebra: For any commutative rings B ⊆ C, the kernel of the multiplication map C ⊗B C → C is denoted J(C/B) and the quotient J(C/B)/J(C/B)2 is denoted ΩC/B . Note that, in the definition, when I is an R · D module so that A is an F · D algebra then J(A/F ) and ΩA/F are F · D modules. Square zero extensions as in the definition are known to split if and only if the map δ : I → ΩA/F ⊗A R
(∗)
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by δ(x) = (x ⊗ 1 − 1 ⊗ x) ⊗A 1 has a left inverse [7, Thm. 57, p. 186]. Note that when I is an R · D module then so is ΩA/F ⊗A R and so δ is an R · D morphism. Since δ is an R · D morphism, then an examination of the construction of the splitting in [7, Thm. 57, p. 186] shows that differential square zero extensions differentially split if and only if δ has a differential left inverse. When R is simple, we will show that differential square zero extensions are split as commutative rings. By Proposition 2 and the above commutative algebra result, this is equivalent to showing that δ is injective. Theorem 6. Assume that R is a differentially simple domain and that A is a differential square zero extension of R with kernel P . Assume that P is an R · D module. Then the extension is F algebra split. Proof. We note that P is a prime ideal, as R is a domain, and also a maximal differential ideal, as R is simple. The annihilator of P is also a differential ideal, and since P is square zero it contains P and therefore coincides with it. We localize A at P . Then AP is a square zero ring extension of K = (AP )−1 R with kernel PP ; PP is a maximal ideal so the quotient K is a field, of characteristic zero since it contains F . Square zero ring extensions of characteristic zero fields split. Thus δP : PP → ΩAP /F ⊗AP K has a left inverse and is in particular injective. We have a commutative diagram δ
P −−−−→ ⏐ ⏐q
ΩA/F ⊗A R ⏐ ⏐
δP
PP −−−−→ ΩAP /F ⊗AP K The vertical map q : P → PP is injective since the annihilator of P is contained in P . So δ is a right factor of the injective map δP q and hence is injective. Now by Proposition 2, the injective R · D module map δ has a left inverse, which then implies that the differential square zero extension A → R is split as F algebras. 2 The proof of Theorem 6 amounts to showing that, for a square zero differential extension with kernel an R · D module, the R · D morphism δ is always injective. So if every
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R · D submodule of ΩA/F ⊗A R is an R · D direct summand, the extension is differentially split. This will happen, by Corollary 11, if every finitely generated R · D submodule of ΩA/F ⊗A R is constantly generated. We record this fact for later reference: Corollary 13. Let A → R be a square zero differential extension with kernel an R·D module, and assume that every finitely generated R · D submodule of ΩA/F ⊗A R is constantly generated. Then A → R is differentially split. In particular, if R is the Picard–Vessiot ring of a Picard–Vessiot closure of F , every square zero differential extension of R with kernel an R · D module is differentially split. Proof. For the final assertion, we apply Proposition 4.
2
Theorem 6 says that a differential square zero extension A → R whose kernel I is an R · D module is F algebra isomorphic to I R with component-wise addition and multiplication (i, r)(j, s) = (rj + si, rs); we regard this isomorphism as equality. Let DA be the derivation of A. Because A → R is differential, and because I is a differential ideal of A, we must have that DA (i, r) = (DI (i) + h(r), DR (r)), where h : R → I. For DA to be additive, h must be additive. For DA to satisfy DA ((i, r)(j, s)) = (i, r)DA (j, s) + (j, s)DA (i, r) requires (DI (rj + si) + h(rs), DR (rs)) to equal (DR (s)i + r(DI (j) + h(s)), rDR (s)) + (DR (r)j + s(DI (i) + h(r), sDR (r)) which means that h(rs) = rh(s) + sh(r); in other words, h is a (C) derivation. For f ∈ F , f → (0, f ) makes A an F algebra. For A to be a differential F algebra requires that DA ((0, f )) = (0, DF (f )) and hence that h(f ) = 0. Thus h is an F derivation, or an element of DerF (R, I). Conversely, any F derivation h : R → I defines a derivation on I R making it a square zero differential extension with kernel I. Now suppose that Λ : R → A is a differential splitting, which can be written as Λ(r) = (λ(r), r) where λ : R → I is a map. We first consider what it means for Λ to be multiplicative: (λ(rs), rs) = (λ(r), r)(λ(s), s) entails that λ(rs) = rλ(s) +sλ(r). Then for Λ to be F linear requires that λ be F linear and for Λ to be an F algebra map requires that Λ(a) = (0, a) for a ∈ F so λ(F ) = 0. Thus λ is an F derivation, or an element of DerF (R, I). Now for Λ to be differential requires that Λ(DR (r)) = DA (λ(r), r) which requires that λ(DR (r)) = DI (λ(r)) + h(r), or that h = λDR − DI λ. Finally, we note that M = DerF (R, I) is an R[D] module with DM (g) = DI g − gDR . Thus if the derivation DA is determined by the derivation h, there is a differential splitting of A → R if and only if h = −DM (λ) for some λ. We summarize this discussion in the following proposition: Proposition 7. Let R be a differentially simple domain and let I be an R · D module. Then all square zero extensions of R with kernel I are differentially split if and only if every element of DerF (R, I) is a derivative. Equivalently, all square zero extensions of R with kernel I are differentially split if and only if every element of HomR (ΩR/F , I) is a derivative.
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Proof. Only the final assertion needs proof. It is standard commutative algebra that DerF (R, I) = HomR (ΩR/F , I). The identification is as follows: given an R homomorphism φ : ΩR/F → I, define a derivation d by d(r) = φ(r ⊗ 1 − 1 ⊗ r + J(R/F )2 ). This identification can be checked to be as R[D] modules. 2 Now, in the notation of Proposition 7, assume that both ΩR/F and I have the property that all their finitely generated R · D submodules are constantly generated. Then, by Corollary 12, both are direct sums of copies of R as differential modules. It follows that HomR (ΩR/F , I) is a product of direct sums of copies of R (both the product and the direct sums may be infinite). Thus every element of HomR (ΩR/F , I) will be a derivative if (and only if) every element of R is a derivative. So we conclude: Corollary 14. Let R be a differentially simple domain and let I be an R · D module. Assume: 1. Both ΩR/F and I have the property that all their finitely generated R · D submodules are constantly generated. 2. Every element of R is a derivative. Then all square zero extensions of R with kernel I are differentially split. The Picard–Vessiot ring R of a Picard–Vessiot closure F1 of F has the property that all its finitely generated R · D modules are constantly generated by Proposition 4. It is also true that every element of R is a derivative (note that this need not be true for F1 ): suppose a ∈ R is a solution of the differential equation L(Y ) = 0, where L(Y ) is a monic linear homogeneous differential operator of order n over F , and let V be its full set of solutions in F1 . Then V is a vector space of dimension n over C, and V ⊂ R. Consider the equation L(Y ) = 0, which is of order n +1 and also monic, linear, and homogeneous, and let W be its full set of solutions in F1 . Then W has dimension n + 1 over C, and W ⊂ R. Consider the map W → V by derivation. This is C linear and its kernel is a subspace of C, which implies that derivation is surjective. Thus both hypotheses of Corollary 14 obtain when R is the Picard–Vessiot ring of a Picard–Vessiot closure of F . Consequently, by the conclusion of Corollary 14, all square zero extensions of R with R · D module kernel are differentially split, as we saw by other means in Corollary 13. We can also use Proposition 7 to provide examples of non-splitting: Example 1. Let F = C and let R = C[x, x−1 ] with D(x) = x. It is straightforward to verify that ΩR/F is a one dimensional free R module so that for any R · D module I HomR (ΩR/F , I) is isomorphic to I as R · D module. Take I = R. Then 1 is not a derivative in I (in R, derivatives have no constant term), so the corresponding element of HomR (ΩR/F , I) is not a derivative. This implies that the corresponding square zero extension with R · D module kernel R R is not differentially split.
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References [1] Y. André, Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence, Ann. Sci. Éc. Norm. Supér. 47 (2014) 449–467. [2] I. Doriaswamy, Projectivity of modules with suitable group actions, Comm. Algebra 10 (1982) 787–795. [3] A. Magid, Cohomology of rings with algebraic group action, Adv. Math. 59 (1986) 124–151. [4] A. Magid, Equivariant completions of rings with reductive group action, J. Pure Appl. Algebra 49 (1987) 173–185. [5] A. Magid, Lectures on Differential Galois Theory, Univ. Lecture Ser., vol. 7, American Mathematical Society, Providence, RI, 1997 (second printing with corrections). [6] A. Magid, Universal covers and category theory in polynomial and differential Galois theory, in: Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute, 2004, pp. 373–385. [7] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970. [8] B. Parshall, L. Scott, An imprimitivity theorem for algebraic groups, Indag. Math. 42 (1980) 39–47. [9] L. Rowen, Ring Theory, vol. I, Academic Press, San Diego, 1988. [10] M. van der Put, M. Singer, Galois Theory of Linear Differential Equations, Springer-Verlag, New York, 2003.