Adjoint functors raised from Ore extensions

Arabian Journal of Mathematics, Aug 2016

Let A be a \({\mathbb{k}}\)-algebra and \({A[t; \alpha,\delta]}\) its Ore extension. We give a pair of adjoint functors between the module category over ker \(\delta\) and the module category over \({A[t; \alpha,\delta]}\). For a kind of special Ore extensions, this pair describes an equivalence between the module category over ker \({\delta}\) and an appropriate subcategory of the module category over \({A[t; \alpha,\delta]}\). Applied to the case of Weyl algebras, this is exactly a Kashiwara’s theorem about D-modules.

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Adjoint functors raised from Ore extensions

Arab. J. Math. Adjoint functors raised from Ore extensions Zhihua Wang Libin Li 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 Let A be a unital associative algebra over a field k, α a k-linear injective endomorphism of A and δ a k-linear αderivation of A. The Ore extension of A with respect to the α-derivation δ is denoted by B := A[t ; α, δ]. It is the algebra freely generated by the algebra A adjoined by an indeterminate t subject to the rule t a = α(a)t + δ(a) for a ∈ A. These Ore extensions are very important in constructing interesting mathematical objects and hence they have received a lot of attention. We refer to [3,9] for their global dimension and Krull dimension, to [6,10,11] for their ring-theoretical properties, to [1,2,7,8] for their representation theory. Observe that Aδ := ker δ is a subalgebra of A and hence A is a right Aδ-module under multiplication. The algebra homomorphism from B to Endk( A) given by i≥0 ai δi (see e.g., [5]) induces a left i≥0 ai t i → B-module structure on A. Moreover, it is easy to see that A is a B- Aδ-bimodule. This leads to a functor A⊗Aδ - from the category of left Aδ-modules into the category of left B-modules. In addition, if X is a left B-module, then the k-subspace of X kert X := {μ ∈ X | t · μ = 0} is a left Aδ-module. So we obtain a functor kert - from the category of left B-modules to that of left Aδ-modules. The aim of the paper is to study the relationship between the functors A⊗Aδ - and kert -. We show that these functors form an adjoint pair of functors. Moreover, the pair gives an equivalence of appropriate categories if - 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⊗Aδ - is left adjoint to the functor kert -. This leads to a result that the endomorphism algebra EndB ( A ⊗Aδ M ) is isomorphic to EndAδ (M ) if Aδ-module M satisfies an additional property. In this case, M is an indecomposable Aδ-module if and only if A ⊗Aδ 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 kert - 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 Aδ-module M and the induced B-module A ⊗Aδ M . This leads to a result that M is a holonomic Aδ-module if and only if A ⊗Aδ M is a holonomic B-module. 2 The adjunction of functors In this section, we give a description of the functor A⊗Aδ - as well as kert -. We show that the functor A⊗Aδ is left adjoint to the functor kert -. This leads to an algebra isomorphism from the endomorphism algebra EndB ( A ⊗Aδ M ) to EndAδ (M ) if the Aδ-module M satisfies a special condition. In this case, the property of indecomposability of the Aδ-module M can be lifted to the induced B-module A ⊗Aδ 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 δn(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 for a ∈ A. It can be seen by induction on n ≥ 1 that t na = αn (a)t n + an−1t n−1 + · · · + a1t + δn(a), for a, a1, . . . , an−1 ∈ A. 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 i≥0 i≥0 i≥0 for ai , a ∈ A. i≥0 is an algebra homomorphism [5]. This induce (...truncated)


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Zhihua Wang, Libin Li. Adjoint functors raised from Ore extensions, Arabian Journal of Mathematics, 2016, pp. 177-186, Volume 5, Issue 3, DOI: 10.1007/s40065-016-0150-4