Adjoint functors raised from Ore extensions

Let A be a k-algebra and A[t;α, δ] its Ore extension. We give a pair of adjoint functors between the module category over ker δ and the module category over A[t;α, δ]. For a kind of special Ore extensions, this pair describes an equivalence between the module category over ker δ and an appropriate subcategory of the module category over A[t;α, δ]. Applied to the case of Weyl algebras, this is exactly a Kashiwara’s theorem about D-modules. Mathematics Subject Classification 13N10 · 16G10 · 16U20

the Ore extension satisfies some special conditions. The paper is organized as follows. In Sect. 2, we first set the notations and present some basic results on Ore extensions. Then we show that the functor A⊗ A δ -is left adjoint to the functor ker t -. This leads to a result that the endomorphism algebra End B (A ⊗ A δ M) is isomorphic to End A δ (M) if A δ -module M satisfies an additional property. In this case, M is an indecomposable A δ -module if and only if A ⊗ A δ M is an indecomposable B-module.
In Sect. 3, we first show that the set T = {t i | i ≥ 0} is a left Ore set of B with the assumption that the α-derivation δ is locally nilpotent. Then any simple B-module is either T -torsion or T -torsion free. We use the functor ker t -to characterize when a simple B-module is T -torsion or T -torsion free.
Obviously, the adjoint pair given above do not in general describe an equivalence of categories. In Sect. 4, we consider a class of Ore extensions and, if we restrict to an appropriate subcategory of B-module category, then the adjoint pair do restrict to an equivalence. In case where the Ore extension algebras are Weyl algebras, this gives rise to a Kashiwara's theorem, one of the major results of the theory of D-modules.
In Sect. 5, we deal with particular Ore extension algebras, i.e., the nth Weyl algebras. We mainly describe the relationship of multiplicity as well as Gelfand-Kirillov dimension between finitely generated A δ -module M and the induced B-module A ⊗ A δ M. This leads to a result that M is a holonomic A δ -module if and only if A ⊗ A δ M is a holonomic B-module.

The adjunction of functors
In this section, we give a description of the functor A⊗ A δ -as well as ker t -. We show that the functor A⊗ A δis left adjoint to the functor ker t -. This leads to an algebra isomorphism from the endomorphism algebra End B (A ⊗ A δ M) to End A δ (M) if the A δ -module M satisfies a special condition. In this case, the property of indecomposability of the A δ -module M can be lifted to the induced B-module A ⊗ A δ M.
In the following, A is a unital associative algebra over a field k and α is an injective endomorphism of A. The k-linear map δ from A to itself is called an α-derivation of A if δ(ab) = α(a)δ(b) + δ(a)b, for any a, b ∈ A. The α-derivation δ is called locally nilpotent if for any a ∈ A, there exists some natural number n such that δ n (a) = 0. The Ore extension of A with respect to the α-derivation δ is a unital associative algebra B := A[t; α, δ], where B is generated as an algebra by the indeterminate t over A subject to ta = α(a)t + δ(a), for a ∈ A.
(2.1) It can be seen by induction on n ≥ 1 that t n a = α n (a)t n + a n−1 t n−1 + · · · + a 1 t + δ n (a), for a, a 1 , . . . , a n−1 ∈ A. (2. 2) The algebra B is a free left A-module with an A-basis {t i | i ≥ 0}. If α is an automorphism of A, then B is also a free right A-module. Note that is an algebra homomorphism [5]. This induces a left B-module structure on A defined by ⎛ Denote by A δ := ker δ. Then A δ is a subalgebra of A. Consider A as a right A δ -module, where the action is given by multiplication.
Let A δ -Mod and B-Mod denote the left module categories over A δ and B, respectively. If X is a left B-module, it follows from (2.1) that the k-subspace ker t X := {μ ∈ X | t · μ = 0} of X is a left A δ -module. In particular, for any B-module morphism g : X → Y, we have g(ker t X ) ⊆ ker t Y . Thus, the restriction of g to the subspace ker t X of X gives rise to the morphism ker t g : ker t X → ker t Y . It is obvious that ker t -is an additive k-linear covariant left exact functors from B-Mod to A δ -Mod.

Proposition 2.1 We have the following descriptions of the functors ker t -and A⊗ A δ -:
(1) For any B-module X , Hom B (A, X ) is isomorphic to ker t X as A δ -modules.
(2) There is a natural equivalence from the functor Hom B (A, −) to ker t -.
(3) The functor A⊗ A δ -is left adjoint to ker t -, or equivalently, ker t -is right adjoint to A⊗ A δ -. Namely, there exists an isomorphism for any A δ -module M and B-module X , which is functorial at M and X .
We define the following two k-linear maps: and It is straightforward to show that ψ X • ϕ X = id Hom B (A,X ) and ϕ X • ψ X = id ker t X . The rest is to check that ϕ X and ψ X are both A δ -linear. In fact, for any a ∈ A δ and f ∈ Hom B (A, X ), the A δ -module structure on On one hand we have, for any b ∈ A, ψ X (a · μ)(b) = b · (a · μ) = (ba) · μ, and on the other hand since ψ X (μ) ∈ Hom B (A, X ) and Hom B (A, X ) is an A δ -module, we have (a · ψ X (μ))(b) = ψ X (μ)(ba) = (ba) · μ. We obtain the desired result. (2) For any B-module morphism f : X → Y , we have A δ -module isomorphisms ϕ X : Hom B (A, X ) → ker t X and ϕ Y : Hom B (A, Y ) → ker t Y by Part (1). It is direct to check that the diagram is commutative in A δ -Mod. Thus, ϕ = {ϕ X } X ∈B-Mod gives a natural equivalence from the functor Hom B (A, −) to ker t -. (3) Note that the functor A⊗ A δ -is left adjoint to Hom B (A, −); that is, there is an isomorphism which is functorial at A δ -module M and B-module X . The morphism τ f is given by τ f (m)(a) = f (a ⊗m), for any m ∈ M and a ∈ A. The isomorphism ϕ X : Then the following composition gives rise to a desired isomorphism which is functorial at M and X .
According to Proposition 2.1, we are able to characterize the indecomposability and simplicity of B-module A as shown in the following.

Corollary 2.2 For the B-module A, we have the following: (1) A is indecomposable if and only if the subalgebra A δ of A contains only the trivial idempotents. (2) If the α-derivation δ is locally nilpotent, then A is simple if and only if the subalgebra A δ of A is a division algebra.
Proof By replacing the B-module X with A in Proposition 2.1 (1), we obtain an algebra isomorphism from the endomorphism algebra End B (A) to the opposite algebra (A δ ) op of A δ . Thus, Part (1) and the necessity of Part (2) follow. To prove the sufficiency of Part (2), we suppose that I is a non-zero submodule of A. Then I is a left ideal of A satisfying δ(I ) ⊆ I . For any non-zero element a ∈ I, there exists n ≥ 0 such that δ n+1 (a) = 0 while δ n (a) = 0 since the derivation δ is locally nilpotent. This implies that δ n (a) ∈ A δ ∩ I. The condition A δ is a division algebra which implies that δ n (a) is invertible, so we obtain an invertible element δ n (a) in the left ideal I of A, this yields that I = A.
Then the property of indecomposability of the A δ -module M can be lifted to the induced B-module A ⊗ A δ M.

Theorem 2.3 Let M be an A δ -module satisfying the condition (2.6). Then the endomorphism algebra End B (A ⊗ A δ M) is isomorphic to End A δ (M). In this case, M is an indecomposable A δ -module if and only if
The composition of the two isomorphisms stated above is denoted by In the following, we shall verify that this isomorphism respects the algebra structure. For any By the requirement of the map θ , we have θ(1 ⊗ θ(g(1 ⊗ m))) = θ(g(1 ⊗ m)), and hence With this preparation, we have We conclude that M is an algebra isomorphism from End The condition (2.6) looks rather strong. However, as we shall see that it is quite often satisfied.

Proposition 2.4 Let A be a free right A δ -module with a basis
Then any left ideal I of A δ satisfies the condition (2.6).
Proof We first verify that {δ( be a polynomial algebra over a field k, q ∈ k\{0, 1}, and α an automorphism of A satisfying α(x) = qx. Then the formula This derivation is locally nilpotent and is known as the q-difference operator or Eulerian derivation. Let B = A[t; α, δ]. Then B is known as a quantized Weyl algebra over k. It is obvious that

Ore quotient rings and modules
In this section, we always assume that the α-derivation δ in B = A[t; α, δ] is locally nilpotent. In this case, the set T = {t i | i ≥ 0} is a left Ore set of B. The aim of this section is to characterize simple T -torsion as well as simple T -torsion-free modules over B through the functor ker t -. (1) D is a multiplicative set of R, that is, the unity 1 ∈ D and for any two a, b ∈ D, ab ∈ D; (2) For any a ∈ D and r ∈ R, there exist b ∈ D and s ∈ R such that br = sa, that is, Dr ∩ Ra = ∅.
A right Ore set of ring R is defined as a left Ore set of opposite ring R op , and an Ore set of R is simultaneously left and right Ore set, see [5] for details.

Proposition 3.2 The set T = {t
Proof We assert that for any a ∈ B and t i ∈ T, there exist b ∈ B and t j ∈ T such that t j a = bt i . To this end, we denote by a = b 0 t + a 0 , for some b 0 ∈ B and a 0 ∈ A. Since the derivation δ is locally nilpotent, there exists n 1 ≥ 0 such that δ n 1 (a 0 ) = 0. It follows from (2.2) that t n 1 a 0 = b 1 t, for some b 1 ∈ B. Hence, Similarly, for a 1 ∈ B, there exist n 2 ≥ 0 and a 2 ∈ B such that t n 2 a 1 = a 2 t. Thus, t n 1 +n 2 a = a 2 t 2 . Repeating the argument, we obtain that t n 1 +···+n i a = a i t i , for some a i ∈ B, as required.
According to Proposition 3.2, we have the following corollary.  (1) The regular module B is T -torsion free. Moreover, any free module over B is T -torsion free.
(2) Any flat module over B is T -torsion free.
Proof (1) It is straightforward to check using the definition of a T -torsion-free module.
(2) Let X be a flat B-module. Then there exists an epimorphism φ : We have t i ν = j t i b j ν j , for some b j ∈ B and ν j ∈ ker φ. Then t i (ν − j b j ν j ) = 0, and hence ν − j b j ν j ∈ tor T (F) = 0. It follows that ν = j b j ν j ∈ ker φ, and hence μ = φ(ν) = 0.
We are ready to consider the structure of T −1 B, the left Ore localization of the ring B with respect to the left Ore set T, see [5] for a complete introduction. Let the symbol ∼ be an equivalence relation on B × T defined by (a, t i ) ∼ (b, t j ), i ≥ j if and only if there exist k, l ≥ 0 such that t k a = t l b and t k t i = t l t j , this is precisely a = t i− j b. The equivalence class of any pair (a, t i ) in B × T is denoted by a t i , and the set of these equivalence classes is denoted by T −1 B.
The addition and multiplication of elements a t i and b t j , i ≥ j in T −1 B are defined as follows: It can be seen from these definitions that T −1 B is equipped with a unital ring structure. The natural ring morphism B → T −1 B, a → a 1 is injective since (a, t i ) ∼ (0, 1) if and only if a = 0. Having constructed the left ring of fractions T −1 B with respect to T, we return to the construction of an analogous module of fraction T −1 X for any left B-module X. Define an equivalence relation ∼ on X × T as (μ, t i ) ∼ (ν, t j ), i ≥ j if and only if there exist k, l ≥ 0 such that t k μ = t l ν and t k t i = t l t j , this is precisely μ = t i− j ν provided that ker t X = 0. The equivalence class of any pair (μ, t i ) in X × T is denoted by μ t i and the set of these equivalence classes is denoted by T −1 X. T −1 X is a left T −1 B-module, where the T −1 B-module structure on T −1 X is given as follows: for any a where c ∈ B and t k ∈ T such that t k a = ct j . 1) if and only if t k μ = 0, which does not mean that μ = 0. However, if ker t X = 0, then the natural morphism is injective. We have the following descriptions of any simple T -torsion B-module. Theorem 3.6 Let X be a simple B-module. The following are equivalent: Note that ker t X ⊆ tor T (X ) ⊆ X . If ker t X = 0, then X = tor T (X ) by the simplicity of X . [5,Proposition 10.12]. We assert that T −1 B ⊗ B X = 0 with the assumption that X = tor T (X ). Indeed, for any i For any simple B-module X , X is either T -torsion or T -torsion free. By the observation of Theorem 3.6, we have the following corollary. Corollary 3.7 Let X be a simple B-module. The following are equivalent:

Kashiwara-like theorem
In this section, the field k is of characteristic zero, k × := k\{0}. We assume that there exists x ∈ A such that {x i | i ≥ 0} forms a basis of A as a free right A δ -module. We also assume that δ i (x i ) ∈ k × for i ≥ 0, and the restriction of α to the subalgebra A δ of A is an automorphism, whose inverse map is denoted by α −1 if there is no confusion. Typical examples of such Ore extensions include the nth Weyl algebras for n > 1 [4], quantized Weyl algebras as described in Example 2.5 for q not a root of unity.
The aim of this section is to give the so-called Kashiwara-like theorem for this kind of Ore extension algebras. If the Ore extension algebras are specialized to Weyl algebras, we recover the well-known Kashiwara's theorem.

Lemma 4.1 We have the following description of A:
(2) α(x) = εx + a 0 , for some a 0 ∈ A δ and ε ∈ k × which is not a root of unity.

Proposition 4.2 Let M be an A δ -module. Then M satisfies the condition (2.6). Thus, M is an indecomposable A δ -module if and only if A ⊗ A δ M is an indecomposable B-module.
Proof Note that A is a free right A δ -module with a basis {x i | i ≥ 0} and δ(x n ) = λ n−1 x n−1 + · · · , where λ n−1 = (1 + ε + · · · + ε n−1 )δ(x) ∈ k × by Lemma 4.1. For any left A δ -module M, it is straightforward to check that Recall that tor T (X ) is a submodule of X , by Corollary 3.3 based on the fact that T is a left Ore set.

Theorem 4.3 For any B-module X , there is a B-module isomorphism
Proof Given the k-linear map It follows that ϕ is injective. The next step is to prove that ϕ is surjective. Since A is freely with a basis where ker t s X is a subspace of X whose elements are annihilated by t s . We shall prove by induction on s ≥ 1 that ker t s X ⊆ i≥0 x i ker t X . If s = 1, the inclusion ker t X ⊆ i≥0 x i ker t X is trivial. Suppose that the inclusion ker t s X ⊆ i≥0 x i ker t X holds for 1 ≤ s ≤ m. Let μ ∈ ker t m+1 X . Then t m+1 μ = 0. By Lemma 4.1 (2), we have t m+1 x = α m+1 (x)t m+1 + λ m t m , where λ m = δ(x)(1 + ε + · · · + ε m ). It deduces that Thus, by the induction hypothesis, But t m (tμ) = 0 implies that tμ ∈ i x i ker t X, and hence α(x)tμ ∈ i x i ker t X. It follows that (δ(x) − λ m )μ ∈ i x i ker t X . We obtain that μ ∈ i x i ker t X since δ(x) − λ m = −ελ m−1 ∈ k × , as required.
Denote by B(T -torsion) the full subcategory of B-Mod whose objects are T -torsion modules. We are now ready to state the main result, which leads to a categorical version of Kashiwara's theorem for Weyl algebras [4, Ch 18, Theorem 3.1].

Theorem 4.4 The functor ker t -:B(T -torsion) → A δ -Mod gives an equivalence of categories with inverse functor A ⊗
are commutative in B(T -torsion) and A δ -Mod, respectively, as desired.

Applications to Weyl algebras
In this section, we present a concrete application. For any integer n ≥ 1, the nth Weyl algebra A n over a field k of characteristic zero is the algebra generated by 2n generators x 1 , . . . , x n , y 1 , . . . , y n with relations: It is known that A n is a simple noetherian domain with a k-basis {x i 1 1 · · · x i n n y j 1 1 · · · y j n n | i 1 , . . . , i n , j 1 , . . . , j n ∈ N}. The Weyl algebra A n can be viewed as the iterated Ore extensions for n ≥ 1, where ∂ n is the differential operator with respect to the independent x n and A 0 = k. It is easy to see that the subalgebra ker(∂ n ) of A n−1 [x n ] is exactly the (n − 1)th Weyl algebra A n−1 . In view of this, any left A n−1 -module M induces a left A n -module A n−1 [x n ] ⊗ A n−1 M. The following two results follow directly from Proposition 2.4 and Theorem 4.4, respectively. We shall now investigate the relationship of Gelfand-Kirillov dimension and multiplicity between the finitely generated A n−1 -module M and the induced A n -module A n−1 [x n ] ⊗ A n−1 M. We begin with the introduction of the notations of Gelfand-Kirillov dimension and multiplicity for any finitely generated A n -module. We refer to [4] for more details.

Proposition 5.1 For any A n−1 -module M, M is indecomposable if and only if the induced A n -module
Let B = {B k } k∈N be the Bernstein filtration of the Weyl algebra A n and M a finitely generated A nmodule with generators u 1 , u 2 , . . . , u s . Denote by 0 = span{u 1 , u 2 , . . . , u s } and k = B k 0 , for k ≥ 1. Then { k } k∈N is a good filtration of M with respect to the Bernstein filtration B [4, Ch 8, § 3]. Furthermore, the graded module gr M : is generated over S n by the symbols u 1 , u 2 , . . . , u s , where S n = gr B A n , which is isomorphic to the polynomial ring with 2n variables over the field k. Note that S n is a noetherian graded ring with k the homogeneous component of degree 0. This implies that k / k−1 is a finite dimensional k-vector space. Therefore, dim k ( k / k−1 ) < ∞, for any k ∈ N. The Hilbert polynomial of S n -module gr M is given by  χ(t, , M). A finitely generated A n -module is called holonomic if it is zero, or if it is of Gelfand-Kirillov dimension n. The number n is the minimal possible Gelfand-Kirillov dimension for any non-zero finitely generated A n -module by the Bernstein inequality.  Proof Note that the polynomial ring k[x 1 ] is a left A 1 -module, where the actions of x 1 and y 1 on k[x 1 ] are given, respectively, by left multiplication and the differential operator ∂ 1 . Then the tensor product k[x 1 ] ⊗ k M is naturally a left A 1 ⊗ k A n−1 -module, and so is an A n -module by the observation of the algebra isomorphism A 1 ⊗ k A n−1 ∼ = A n . By [12,Theorem 4 Then the results follow from the following A n -module isomorphism: As an immediate consequence of Theorem 5.3, we have the following corollary.

Corollary 5.4 Let M be a finitely generated A n−1 -module. Then M is a holonomic A n−1 -module if and only if A n−1 [x n ] ⊗ A n−1 M is a holonomic A n -module.
Remark 5.5 Theorem 5.3 allows us to construct holonomic A n -modules possessing any multiplicity. More explicitly, let M 0 be a finite dimensional k-vector space with dim k M 0 = m. Then M 0 is a finitely generated left module over A 0 = k. Define A n -module M n recursively as M n = A n−1 [x n ] ⊗ A n−1 M n−1 , for n ≥ 1. Then M n is holonomic with multiplicity m by Theorem 5.3.