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Differential smoothness of skew polynomial rings

It is shown that, under some natural assumptions, the tensor product of differentially smooth algebras and the skew-polynomial rings over differentially smooth algebras are differentially smooth.


Introduction
The study of smoothness of algebras goes back at least to Grothendieck's EGA. The concept of a formally smooth commutative (topological) algebra introduced in there [6, Définition 19. 3.1] was later extended to the non-commutative case by Schelter in [12]. An algebra is formally smooth if and only if the kernel of the multiplication map is projective as an bimodule. As argued by Schelter himself, this notion arose as a replacement of a far too general definition based on the finiteness of the global dimension. Although it plays an important role in non-commutative geometry (see e.g. [5], where such algebras are termed quasi-free), the notion of formal smoothness seems to be too restrictive. The too crude notion of smoothness based on the finiteness of the global dimensions was refined in [13], where a Noetherian algebra was said to be smooth provided that it had a finite global dimension equal to the homological dimension of all its simple modules. From the homological perspective probably most satisfying is the notion of homological smoothness introduced in [14]: an algebra is homologically smooth provided it admits a finite resolution by finitely generated projective bimodules. Algebras of this kind display a Poincaré type duality between Hochschild homology and cohomology, and retain many properties characteristic of co-ordinate algebras of smooth varieties (see e.g. [8], where this last point is strongly argued for).
A different and more constructive approach to smoothness of algebras was taken in [4]. In this approach the smoothness of an algebra A is related to the existence of a specific differential graded algebra (with A as the degree-zero part) whose size is aligned with the rate of growth of A measured by the Gelfand-Kirillov dimension, and which satisfies a strict version of the Poincaré duality in terms of an isomorphism with the corresponding complex of integral forms [3] (see Section 2 for precise definition). In view of this direct use of differential graded algebras this kind of smoothness is referred to as differential smoothness. The main advantage of this approach is its concreteness: a differentially smooth algebra comes equipped with a well-behaved differential structure and with the precisely defined concept of integration. Examples of differentially smooth algebras include the coordinate algebras of the quantum group SU q (2), the quantum 2-sphere (see [3]), the non-commutative pillow algebra, the quantum cone algebras (see [4]), the quantum polynomial algebras (see [7]), and Hopf algebra domains of Glefand-Kirillov dimension 2 that are not PI (see [2]). Although many of these examples are known to be also homologically smooth, the relationship between the differential and other types of smoothness is not clear yet.
At the root of difficulties with comparing differential and other types of smoothness is the constructive nature of the former, which prevents one from using functorial or just existential arguments. In this paper we make a few steps toward resolving such difficulties and present two general constructions which lead from differentially smooth to differentially smooth algebras. First, we show that -under some natural assumptions on differential structures and algebras -the tensor product of differentially smooth algebras is differentially smooth. This allows one to deduce quickly smoothness of polynomial and Laurent polynomial rings without necessity of constructing specific differential structure (it suffices to have such a structure for polynomials in one variable). Second, again under some natural assumptions, we prove that the skew-polynomial rings over a smooth algebra are smooth.

Preliminaries
Let F be a field. By a differential calculus over an F-algebra R we mean a differential graded algebra (ΩR, d) (i.e. a graded algebra with the degree-one square-zero linear map d : ΩR → ΩR satisfying the graded Leibniz rule) such that: (a) ΩR = n∈N Ω n R, i.e. it is non-negatively graded, and Ω 0 R = R, (b) For all n ∈ N, The requirement (b) is called the density condition. A differential calculus (ΩR, d) over R is said to be connected, provided ker (d | R ) = F. It is said to have dimension N or to be N-dimensional provided Ω N R = 0 and Ω n R = 0, for all n > N.
An N-dimensional differential calculus (ΩR, d) over R is said to admit a volume form, provided Ω N R is isomorphic to R as a both left and right R-module (but not necessarily as an R-bimodule). Any free generator v of Ω N R as a right and left R-module is referred to as a volume form. Associated to a volume form v are two maps: (a) the right R-module co-ordinate isomorphism: the twisting algebra automorphism: Note that the admittance of a volume form does not automatically imply the existence of a volume form. To any right R-linear homomorphism ϕ : Ω n R → R we associate a family of right R-module maps 3) The right R-multiplication on the space of right R-linear maps Hom R (Ω k R, R) is defined by (ψr)(γ) = ψ(rγ), for all ψ ∈ Hom R (Ω k R, R), r ∈ R and γ ∈ Ω k R. The maps ℓ k πv associated to a volume form co-ordinate isomorphism (2.1) are R-bimodule homomorphisms, provided the left R-multiplication on Hom R (Ω k R, R) is defined via the twisting automorphism θ v , Following [4] a differential calculus with a volume N-form v is said to be integrable provided all bimodule homomorphisms ℓ k πv , k ∈ {0, 1, . . . , N} are invertible. This is equivalent to the the existence of a complex of integrable forms [3] isomorphic to (ΩR, d) (see [4,Theorem 2.2]). To relieve the notation we will write ℓ k v or simply ℓ k for ℓ k πv . For the future use we thus record that if v ∈ Ω N R is a volume form, then Given an affine F-algebra R with generating subspace V, let us write V(n) for the subspace of R spanned by 1 and all words in generators of R of length at most n. The Gelfand-Kirillov dimension of R is a real number defined as if it exists, and is defined as infinity otherwise. The Gelfand-Kirllov dimension of an arbitrary F-algebra R is by definition the supremum of the Gelfand-Kirillov dimensions of affine F-subalgebras of R (see [11, 8.1.16]). Although it is not generally true that the Gelfand-Kirillov dimension of the tensor product of two algebras R, S is equal to the sum of their (finite) Gelfand-Kirillov dimensions, it is, however, the case that if GKdim(R) ≤ 2 or GKdim(S) ≤ 2, then GKdim(R ⊗ S) = GKdim(R) + GKdim(S); see [9,Proposition 3.12]. We refer the reader to [9] or [11,Chapter 8] for a detailed discussion of the Gelfand-Kirillov dimension, which, in the case of a commutative Noetherian algebra is a very good measure of geometric dimension of the underlying affine space. The version of smoothness studied in the present text is recalled in the following ). An affine algebra R of integer Gelfand-Kirillov dimension N is said to be differentially smooth, if there exists a connected, N-dimensional, integrable differential calculus on R.
Let R be an algebra and σ an algebra automorphism of R. By a skew-polynomial ring over R we mean the algebra R[z; σ] generated additionally by z and the relations zr = σ(r)z, for all r ∈ R. Similarly the Laurent skew-polynomial ring R[z ±1 ; σ] is defined. As was the case for tensor product algebras, it is not generally true that GKdim(R[z; σ]) = GKdim(R) + 1 (see [11,Example 8.2.16]). The equality holds, whenever σ is locally algebraic, i.e. if for all r ∈ R, the set {σ n (r) | n ∈ N} is contained in a finite dimensional subspace of R (see [10, Proposition 1]).
As we will often make statements that apply equally well to the skew-polynomial and the Laurent skew-polynomial rings, we reserve the symbol R[z • ; σ] to denote either

Differential smoothness of the tensor product of algebras
The aim of this section is to prove that, under some mild and geometrically natural assumptions, tensor product of integrable differential calculi on two algebras gives an integrable calculus on the tensor product algebra.
Suppose that (ΩR, d R ), where ΩR = N k=0 Ω k R, is an N-dimensional differential calculus on an F-algebra R, and that (ΩS, d S ), where ΩS = M k=0 Ω k S, is an Mdimensional differential calculus on an F-algebra S. Consider T := R ⊗ S and Components Ω i R, resp. Ω j S, are considered to be zero if i or j are not within their limits. ΩT becomes a differential graded algebra, with graded multiplication defined as for all ω ∈ Ω i R and ν ∈ Ω j S. By the density condition hence, in view of (3.1) and (3.2), Therefore, the differential graded algebra (ΩT, d T ) is a differential calculus on T .
Proposition 3.1. Let R and S be algebras with integrable differential calculi (ΩR, d R ) and (ΩS, d S ). Suppose that ΩR is a finitely generated projective right R-module and that ΩS is a finitely generated projective right S-module. Then (ΩR ⊗ ΩS, d) is an integrable differential calculus for R ⊗ S.
Proof. We write T := R ⊗ S and assume that ( Since Ω i+j R = 0 for i + j > N and Ω N +M −i−j S = 0 for i + j < N, we have for all k: This means in particular for all i ≤ k.
Since Ω N +M T = Ω N R⊗Ω M S, for all ϕ 1 ∈ Hom R (Ω N R, R) and ϕ 2 ∈ Hom S (Ω M S, S), ϕ := ϕ 1 ⊗ϕ 2 ∈ Hom T (Ω N +M T, T ), and so we can consider the maps ( If Ω N −i R and Ω M −(k−i) S are finitely generated projective as right R-modules, respectively as right S-modules, then by [15, 15.9], where components of ΩR or ΩS are zero if their degrees are not within the limits, we eventually conclude that ℓ k π is a bijection between Ω k T and Hom T (Ω N +M −k T, T ). ⊔ ⊓ then the tensor product algebra R ⊗ S is differentially smooth.
Proof. We only need to check whether the connectedness of ΩR and ΩS implies the connectedness of ΩT . Let x = i,j α ij r i ⊗ s j ∈ ker d T ⊆ R ⊗ S, where the sets {r i } and {s j } are linearly independent and α ij ∈ F. If ΩR is connected, then in view of the definition (3.2), i α ij r i is a scalar multiple of 1, for all j, i.e. x = j 1 ⊗ β j s j , for some scalars β j . If, furthermore ΩS is connected, then the definition (3.2) implies that j β j s j is a scalar multiple of 1, hence x is a scalar multiple of 1 ⊗ 1. Therefore, ΩT is connected. The assertion then follows by Proposition 3.1. ⊔ ⊓ Corollary 3.3. Let R be a differentially smooth algebra with respect to a differential calculus ΩR that is finitely generated and projective over R. Then extensions of the form R[x 1 , . . . , x n , y ±1 1 , . . . , y ±1 m ] are also differentially smooth. Proof. Both the polynomial algebra F[x] and the Laurent polynomial algebra F[y ±1 ] have Gelfand-Kirillov dimension one and are smooth. In the case of F[x] a connected one-dimensional integrable differential calculus is (freely as a module) generated by the volume one-form v = d(x) and the associated twisting automorphism θ v (f (x)) = f (qx), where q is any non-zero element of F (this determines fully the structure of ΩF ). In the case of F[y ±1 ], the volume form can be chosen as v = y −1 d(y) and then θ v is the identity map. Since j ] = 1 ≤ 2, the assertion follows by a repeated use of Corollary 3.2 and [9, Proposition 3.12]. ⊔ ⊓

Differential smoothness of skew-polynomial rings
The aim of this section is to prove the following Theorem 4.1. Let R be a an algebra with an integrable differential calculus (ΩR, d) such that ΩR is a finitely generated right R-module. For any automorphism σ of R that extends to a degree-preserving automorphism of ΩR, which commutes with d, there exists an integrable differential calculus (ΩA, d) on the skew-polynomial ring A = R[z; σ] and the Laurent skew-polynomial ring A = R[z ±1 ; σ]. If R is differentially smooth with respect to (ΩR, d) and GKdim(A) = GKdim(R)+1, then A is also differentially smooth.
Recall that the trivial extension of an algebra A by an A-bimodule M is the algebra B isomorphic to A ⊕ M as a vector space and with the multiplication (a, m)(a ′ , m ′ ) = (aa ′ , am ′ + ma ′ ), for all a, a ′ ∈ A, m, m ′ ∈ M.
If ν is an automorphism of an algebra A, then we will denote by A ν , the A-bimodule with the multiplication a · b · a ′ := abν(a ′ ), for all a, a ′ , b ∈ A.  for all homogeneous ω ∈ ΩR and integers n. Then the trivial extension ΩA = S ⊕ S σ is an N + 1-dimensional differential calculus on R[z • ; σ] with differential for all homogeneous ω, ν ∈ ΩR, where ∂ z denotes the (formal) derivative of polynomials. The grading on ΩA is given by for all homogeneous ω, ν ∈ ΩR with ν = 0.
We need to check that the map d defined by (4.4) satisfies the graded Leibniz rule. Let us take any homogeneous ω, ν ∈ ΩR and compute d(ωz n νz m ) = d(ωσ n (ν)z n+m ) = d(ωσ n (ν))z n+m + (−1) |ω|+|ν| ωσ n (ν)∂ z (z n+m )dz = d(ω)σ n (ν)z n+m + (−1) |ω| ωσ n (d(ν))z n+m where we used the definition of multiplication of the skew-polynomial algebra, (4.4) and the fact that both d on ΩR and ∂ z satisfy the (graded) Leibniz rule. Next This proves that (ΩA, d) is a differential graded algebra. It is clear that Ω N +1 A = Ω N R[z • ]dz = 0 and there are no components Ω n A if n > N + 1, hence (ΩA, d) has dimension N +1. Since every element of (ΩA, d) can be written as a linear combination of ωz n + νz m dz, with ω, ν ∈ ΩR and ΩR satisfies the density condition (over R), also ΩA satisfies this condition (over A). Therefore, (ΩA, d) is an N +1-dimensional calculus as claimed. ⊔ ⊓ Lemma 4.3. In the set-up of Lemma 4.2 assume that (ΩR, d) has a volume N-form v with the twisting automorphism θ v and the co-ordinate isomorphism π v . Let u = π v (σ(v)) and define the map Then: (1) The mapθ is an algebra automorphism of R[z • ; σ].
Proof. First note that there exists v ∈ R such that σ −1 (v) = vv. Hence uσ(v) = 1 = vσ −1 (u), i.e. u is invertible and the mapθ is well defined also in the Laurent case. Furthermoreθ is invertible, with the inversē Since, for all a ∈ R, av = vθ v (a) and Hence holds for all a in R, and thereforē This implies thatθ is an algebra map and completes the proof of the first assertion.
To prove the second assertion, first let us write A for R[z • ; σ]. As Ω N R = vR, we have Ω N R [z • ] = vA, and thus Ω N +1 A = Ω N R [z • ]dz = vAdz, i.e. any element of Ω N +1 A is of the form vf dz, for some f ∈ A. By (4.5), vf dz = vdzσ −1 (f ), hence Ω N +1 A = vdzA. Let f = a i z i ∈ A be such that vdzf = 0, then vσ(f ) = 0, which implies f = 0. Thus Ω N +1 A = vdzA is a free rank one right A-module. Moreover, for any element f ∈ A, f vdz = vθ(f )dz = vdzσ −1 (θ(f )), which also shows that Avdz = vdzA, and hence vdz is a free generator on both sides, and that the twisting automorphism has the stated form. ⊔ ⊓ Lemma 4.4. Let R ⊆ S be a ring extension and σ ∈ Aut(S) such that the restriction of σ to R is an automorphism of R. Consider the skew-polynomial rings S[z • ; σ] and its subring R[z • ; σ]. Let M be a σ-stable, right R-submodule of S and consider the right (1) The additive map ψ : is well-defined and injective. (2) If M is a finitely generated right R-module and σ(M) = M, then ψ is bijective.
Proof. (1) We will first show that ψ f z k is a right R[z • ; σ]-linear map. For all m ∈ M, r ∈ R and i, j ∈ Z, i.e. σ k (f k (m)) = 0, for all k. Hence f k = 0 for all k, showing that ψ is injective.
(2) For any k, define an additive map The maps p k are right R-linear, because ). There exist a finite indexing set I ⊂ Z and elements β ik ∈ R, for 1 ≤ i ≤ n and Hence f = ψ k∈I f k z k . ⊔ ⊓ Corollary 4.5. Let σ be an automorphism of degree 0 of a graded algebra Ω = ∞ k=0 Ω k . Set R = Ω 0 . If Ω k is finitely generated as right R-module, then With these assertions at hand we can now prove Theorem 4.1.
Proof of Theorem 4.1. Let us denote by v a volume form for ΩR, with the corresponding co-ordinate isomorphism π v : Ω N R → R. By Lemma 4.3, vdz is a volume form for the differential calculus ΩA on A = R[z • , σ] constructed in Lemma 4.2, and let π vdz : Ω N +1 A → A be the corresponding co-ordinate isomorphism. For all ω ∈ Ω N R, The following equation shows the commutation of σ and π v . Let ω ∈ Ω N R. Then (4.9) Furthermore, the invertible elements v i give rise to a linear isomorphism Using equations (4.8) and (4.9) we compute, for all A is the homomorphism from Lemma 4.4. Hence we have shown that the following diagram commutes: (4.10) In a way similar to the definition of Φ we define the linear isomorphism Φ ′ : , given by ϕ(ωz i dz) = σ −1 (ω)z i , is an isomorphism of right A-modules. The adjoint map of ϕ is the isomorphism Using again equation (4.9), Φ ′ and ϕ * we compute, for all v (Φ ′ (ωz i )))(ϕ(ω ′ z j dz)) = ϕ * (ψ(ℓ k v (Φ ′ (ωz i ))))(ω ′ z j dz).
Therefore, the following diagram commutes: Assume for all k, that the maps ℓ k v are bijective and that the right R-modules Ω k R are finitely generated. Then ψ is bijective by Corollary 4.5 and hence ψ • ℓ k−1 v • Φ is bijective as well as ϕ * • ψ • ℓ k v • Φ ′ are bijective maps. Since the diagrams (4.10) and (4.11) commute, also ℓ k vdz : Ω k A → Hom A (Ω N +1−k A, A) is bijective. Finally, if ΩR is a connected calculus then vanishing of the first component in (4.2), implies that if d(f ) = 0, for f = i a i z i ∈ R[z • ; σ], then f ∈ F[z • ] (i.e. f has scalar coefficients only). The second component in (4.2) is simply ∂ z (f ), hence it vanishes if and only if f is a scalar multiple of the identity in A. Therefore, the calculus ΩA is also connected. This completes the proof of the theorem.
Since The statement of Theorem 4.1 can be iterated in the following way.