Differential Calculi on Quantum Principal Bundles over Projective Bases

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -- the differential -- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf--Galois extensions of algebras lift to Hopf--Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles $\mathcal{O}_q(\mathrm{SL}_2(\mathbb{C}))$ and $\mathcal{O}_q(\mathrm{GL}_2(\mathbb{C}))$ over the projective space $\mathrm{P}^1(\mathbb{C})$ are discussed in detail.


Introduction
While in classical differential geometry there is a canonical (functorial) construction of the differential calculus on a differentiable manifold, a main feature of noncommutative differential geometry is the non uniqueness of the calculus.Even on quantum groups and requiring (co)invariance conditions there are interesting different calculi [47].In this perspective, the definition of a noncommutative geometry requires also a characterization of its differential calculus structure.Going beyond quantum groups we have quantum homogeneous spaces and quantum principal bundles.Their study has been very active in the last three decades when the base space is a noncommutative deformation of an affine variety.In this case, as usual in noncomutative geometry, the affine geometric objects are replaced by a deformation of the algebra of functions on them.The case of bundles on projective varieties is intrinsically more difficult since it requires patching affine opens.In noncommutative geometry, when it is necessary to go beyond the affine setting, we must take the ringed space approach as expressed by Grothendieck: to do geometry one does not need the space itself, but only the category of sheaves on that would-be space (see [18] and also [44] with refs.therein for more insight on this point of view).
In this paper we propose an approach to noncommutative differential geometry which employs the sheaftheoretic language developed in [4].This does not only cover the established affine theory but also allows to consider differential structures for quantum principal bundles over non-affine bases.
Important steps towards a sound foundation of noncommutative algebraic geometry were taken in the works [1,27,38,45].In [1], quantum projective schemes are understood as the category of modules associated with a given graded noncommutative algebra, while in [38], an affine quantum space is viewed as the spectrum of a noncommutative ring and quantum projective schemes are defined accordingly with a gluing procedure in analogy with the classical Proj construction.Indeed this is our perspective on non-affine noncommutative spaces: we shall define, following [4], a quantum space as a locally ringed space and then build quantum principal bundles over quantum spaces as sheaves with suitable (co)invariant properties with respect to a fixed quantum structure group.
While we have extensive studies on quantum principal bundles in the affine setting (see e.g .[7,12,6,19]), the more general non-affine case (considered e.g. in [43]) lacks a comprehensive and exhaustive treatment.In [4] we study this more general setting combining the sheaf approach to principal bundles with affine bases of [35] (see also [11] and [3]) with the quantum projective homogeneous spaces construction of [10,16], where a graded quantum ring is built out of the key notion of quantum section, associated to a quantum line bundle, with a gluing procedure similar to the classical Proj one in [38].In particular, we study the important example of quantum principal bundles on quantum flags of algebraic groups, the total space algebra being a quantum group.
More specifically, the main construction is based on quantizations of the classical principal bundle G → G/P , where G is a semisimple complex algebraic group and P a parabolic subgroup, so that G/P is a projective variety.In this case the P -invariant ring O(G/P ), i.e., the O(P )-coaction invariant (for short coinvariant) ring O(G/P ) is trivially C and is then replaced by the homogeneous coordinate ring Õ(G/P ) of G/P with respect to a chosen projective embedding, associated with a very ample line bundle L. This line bundle L can be recovered more algebraically via a character χ of P ; the corresponding sections are the O(P )-semi-coinvariant elements of O(G) with respect to χ and generate the homogeneous coordinate ring Õ(G/P ) of G/P .The relation between this latter and the structure sheaf O G/P of G/P is then as usual by considering projective localizations (zero degree subalgebras of the localizations) of Õ(G/P ).
Let O q (G) be a quantum deformation of G and O q (P ) a quantum deformation of a parabolic subgroup; both O q (G) and O q (P ) are Hopf algebras (see [10] Sect. 3 and [4]).Via the datum of a quantum section s ∈ O q (G), quantum version of the lift to O(G) of the character χ of P defining the line bundle L, we obtain a quantization of L. Indeed, through the quantum section we define a graded algebra Õq (G/P ) (quantum homogeneous coordinate ring): each graded component consists of the elements of O q (G), which are not coinvariant, but transform according to powers of the projection of s on O q (P ).Furthermore, the quantum section s allows to define two sheaves F G and O M on M = G/P and in [4] we prove that F G is a sheaf of O q (P )-comodule algebras on the quantum ringed space (M, O M ).Given the locally principal comodule algebra property (or faithfully flatness property) of F G , then this is a quantum principal bundle.In particular the (coproduct of the) quantum section determines an open cover {V i } of G.The projected open cover {U i := p(V i )} of M (where p : G → M = G/P ) then lead to the intersections U I := U i1 ∩ • • • ∩ U ir , I = (i 1 , . . ., i r ) which form the basis B for the topology used to define both the sheaves F G and O M .
In the present paper we develop a theory of differential calculi on sheaves which is suited but not limited to affine and non-affine examples such as quantum flags for quantum algebraic groups.In particular we construct first order differential calculi on the quantum principal bundles F G that are canonically given once a calculus on O q (G) is chosen.For a different approach to quantum differential calculi on quantum flags we refer to [21,22,34].
We start with the discussion of first order differential calculus in the affine setting, that is, when the base and total spaces are affine.Our focus is on the construction and induction of differential structures: via tensor products, algebra homomorphisms and quotients.This also on the level of covariant and bicovariant first order differential calculi.In particular, we recall the construction of covariant calculi on smash product algebras from [36], where a calculus on a module algebra and a bicovariant calculus on the structure Hopf algebra shape the so-called smash product calculus.This affine framework has been fruitfully investigated in a series of works, including [7,13,25,29,33,36].
We conclude our discussion of the affine setting with a treatment of principal covariant calculi.Associated with a commutative principal bundle we canonically have the exact sequence of horizontal forms into forms on the total space onto vertical forms.Even for principal comodule algebras (faithfully flat Hopf-Galois extensions) exactness of this sequence in the noncommutative case is an extra requirement.It is known that the exterior algebra Ω • H of a bicovariant calculus on a quantum group H is a graded Hopf algebra.We here for simplicity study the quotient algebra Ω 1  H = H ⊕ Γ H where forms of degree 2 and higher are set to zero.Similarly we can study when the (truncated) exterior algebra Ω 1 A = A ⊕ Γ A is a Ω 1 H -comodule algebra, and we can study the induced module of one forms Γ B on the subalgebra B = A coH ⊆ A of coinvariants.It was shown in [39,40] that exactness of the sequence of modules (where the cotensor product A H Γ H equals A ⊗ coH Γ H , with coH Γ H the module of left coinvariant one-forms on the quantum group H) is equivalent to principality of the graded Hopf-Galois extension (Ω 1 A ) coΩ 1 H ⊆ Ω 1 A .We revisit these results considering differential graded algebras, where the above sequence becomes that of noncommutative horizontal forms, forms on A, vertical forms.The sequence then defines a principal covariant calculus on A. We show that these calculi correspond to graded Hopf-Galois extensions where the Ω 1 H -coaction is differentiable (compatible with the differentials on A and on A ⊗ H).The main examples of differential calculi on sheaves we shall encounter satisfy such an exact sequence locally.In this section dedicated to the affine setting we further clarify the relation of principal covariant calculi to quantum principal bundles as pioneered in [7].
Building on the affine case results we proceed to introduce the sheaf approach to first order differential calculi.Namely, we define a first order differential calculus (Υ, d) on a sheaf F of comodule algebras as a sheaf Υ of covariant F -bimodules together with a morphism d : F → Υ of sheaves of comodules.We demand the Leibniz rule and surjectivity of the F -linear span of the differential only locally on stalks.These are the two characterizing properties of a first order differential calculus, and their holding true on stalks goes along with the property that stalks of a quantum principal bundle are principal comodule algebras.
The following result, proven in Theorem 4.16, shows that the construction of the sheaf F G from the Hopf algebra O q (G) developed in [4] (with G a semisimple complex algebraic group, P a parabolic subgroup) extends to the level of first order differential calculi.
Theorem.Let (Γ, d) be a right covariant first order differential calculus on the Hopf algebra O q (G).

i.) The assignment
induced by Ore extensions, defines a right O q (P )-covariant first order differential calculus on the sheaf F G .
ii.)The first order differential calculus (Υ G , d) on the sheaf F G induces a first order differential calculus iii.)If F G is a quantum principal bundle, the sheaf of base forms is isomorphic, as a sheaf of O M -bimodules, to the intersection of that of horizontal and O q (P )-coinvariant forms: For the sheaf F G , which is obtained from the Hopf algebra O q (G), the definition of the calculus on O q (G) uniquely determines the quotient calculus on O q (P ), and from Theorem 4.16, the Ore extended calculus on F G and that on the coinvariant subsheaf O M = F coOq(P ) G .Thus, despite non uniqueness of the noncommutative differential calculus, once defined the calculus on O q (G) the other calculi are canonically obtained.
We next proceed to extend our sheaf-theoretic treatment to the notion of principal covariant calculus on a quantum principal bundle, which accounts for compatibility of the calculus (Υ, d) on the total sheaf F , the calculus (Υ M , d M ) on the base sheaf O M and a bicovariant calculus (Γ H , d H ) on the structure Hopf algebra H.For a principal covariant calculus we demand the sequence of stalks to be exact for every p ∈ M , and the calculus on H to be bicovariant, cf.equation (4.14).
The rest of the article is devoted to providing three explicit examples of this canonical construction.We study calculi on the quantum principal bundles O q (SL 2 (C)) and O q (GL 2 (C)), both over P 1 (C).The first example is based on a 4D bicovariant differential calculus on O q (SL 2 (C)).The associated sequence (1.1) is well-defined but not exact.Despite the base sheaf O P 1 (C) being commutative the resulting calculus is noncommutative (the bimodule of one forms is noncommutative).
The second example is based on a 3D calculus on O q (SL 2 (C)) and the associated sequence is exact but not a principal covariant sequence since the induced calculus on the quantum parabolic subgroup O q (P) (P ⊆ SL 2 being the Borel subgroup of upper triangular matrices) is not bicovariant.
The third example has total space algebra O q (GL 2 (C)) with a 4D bicovariant calculus.This induces a principal covariant calculus on the quantum principal bundle F GLq over the base sheaf O P 1 (C) .Even though the sheaf F GL q is locally trivial, i.e., on an open cover {U i } of P 1 (C) the algebras F GL q (U i ) are isomorphic to smash product algebras, the associated smash product calculi do not give a first order differential calculus on the sheaf F GL q .This is an instance where, while the sheaf defining the quantum principal bundle is locally trivial, the sheaf of one forms is not.

Preliminary concepts
In this section we establish our notation and provide the key results we will need in the sequel.Though most of the material of this section is known, we include it by completeness, so that the reader has readily available the results in the form we need them later on.

Notation on modules and comodules
Let k be a commutative unital ring (in later sections specialized to C[q, q −1 ] and to a field) and let ⊗ be the tensor product of k-modules.Algebras over k will be associative and unital.Let H be a Hopf algebra over k.Throughout this paper we denote the coproduct of H by ∆ and its counit by ǫ.Recall that those are algebra homomorphisms ∆ : We employ Sweedler's notation ∆(h) = h 1 ⊗ h 2 for the coproduct of an element h ∈ H, i.e. we omit summation symbols and summation indices.By the coassociativity we write for all h ∈ H and similarly for higher coproducts.The antipode S : H → H of H is an anti-bialgebra homomorphism, namely S(hh ′ ) = S(h ′ )S(h), S(1) = 1, ∆(S(h)) = S(h 2 ) ⊗ S(h 1 ), ǫ • S = ǫ, and furthermore we have S(h 1 )h 2 = ǫ(h)1 = h 1 S(h 2 ) for all h, h ′ ∈ H.We always assume that H has invertible antipode and denote its inverse by S : H → H.The latter is an anti-bialgebra homomorphism such that S(h and The maps ∆ R and ∆ L are called right and left H-coaction, respectively.We use the Sweedler' Accordingly to (2.1) and (2.2), we further use the notations ) and similarly for higher coactions.A right H-comodule M which is also a left H-comodule is said to be an H-bicomodule if ∆ R and ∆ L are commuting coactions, i.e., ( In Sweedler's like notation this commutativity reads for all m ∈ M .A k-linear map satisfying (2.3) is also called right H-colinear.Similarly, left H-colinear and H-bicolinear maps are defined.

Covariant bimodules
Algebras in the category of comodules have a multiplication and unit that are compatible with the coaction.Since we are encountering several different coactions in the course of this section we utilize the convention to denote the right H-coaction on a right H-comodule algebra A by δ R : for all a, a ′ ∈ A and δ R (1) = 1 0 ⊗1 1 = 1⊗1 H .A right H-comodule algebra homomorphism is a right H-comodule homomorphism which is also an algebra homomorphism.Similarly, one defines left H-comodule algebras and H-bicomodule algebras and their morphisms.
Fix a right H-comodule algebra (A, δ R ) in the following.We now introduce A-bimodules in the category of right H-comodules.

Definition 2.2. An
1 for all a, a ′ ∈ A and m ∈ M .A right H-covariant right (resp.left) A-module is a right (resp.left) A-module, where we only ask compatibility of a right H-coaction with the right (resp.left) A-module structure.Similarly, left H-covariant A-modules and H-bicovariant A-bimodules are defined, if A is a left H-comodule algebra or an H-bicomodule algebra, respectively.
We are particularly interested in the case A = H, since H is naturally a right and a left H-comodule algebra with respect to the coactions given by the coproduct.
For H-covariant H-modules there is the following fundamental theorem (c.f.[31] Thm.1.9.4,[41]).Proposition 2.4.For any right H-covariant right H-module M , there is an isomorphism Using the inverse S of the antipode S there are analogous isomorphism of right H-covariant left H-modules and left H-covariant right H-modules, respectively.The fundamental theorem allows us to write any element m of a bicovariant H-bimodule as m = a i m i , where {m i } i ∈ I ⊆ coH M is a basis of (left) coinvariant elements and a i are coefficients in H.This is particularly useful when one is dealing with H-linear maps of bicovariant bimodules.Then, it is sufficient to specify properties on coinvariant elements.

Hopf-Galois extensions
In this section H denotes a Hopf algebra and A a right H-comodule algebra with right H-comodule action δ R : A → A ⊗ H.In particular H is a right H-comodule algebra with coaction ∆.
Definition 2.5.The algebra extension B := A coH ⊆ A is said to be i.) a Hopf-Galois extension, if the k-linear map is a bijection, where µ : A ⊗ B A → A is the product of A induced on the balanced tensor product A ⊗ B A.
ii.) a cleft extension, if there is a k-linear map j : H → A, the cleaving map, such that 1.) j is right H-colinear, i.e., j(h ) j is convolution invertible, i.e., there is a k-linear map j −1 : ) j respects the units, i.e., j(1 H ) = 1.
iii.) a trivial extension, if there is a right H-comodule algebra homomorphism j : H → A.
A trivial extension A coH ⊆ A is automatically a cleft extension since an H-comodule algebra map j : H → A is right H-colinear and unital by assumption and its convolution inverse is j −1 = j •S.This trivially implies that the composition j −1 • S : H → A is right H-colinear.In the following lemma we prove that this H-colinearity property holds for any cleft extension.
Lemma 2.6.The inverse of the cleaving map satisfies where τ H,H is the flip.In particular, j −1 • S is right H-colinear.
To complete the hierarchy of Definition 2.5, we note that cleft extensions are in particular Hopf-Galois extensions.We shall frequently call them cleft Hopf-Galois extensions.
Proposition 2.7.B := A coH ⊆ A is a cleft extension if and only if B ⊆ A is a Hopf-Galois extension and there is a left B-module and right H-comodule isomorphism B ⊗ H ∼ = A.
We conclude this section by recalling the notion of principal comodule algebra from [12] and Schneider theorem for faithfully flat Hopf-Galois extensions.Definition 2.8.A right H-comodule algebra A is said to be a principal comodule algebra if B := A coH ⊆ A is a Hopf-Galois extension and if A is right H-equivariantly projective as a left B-module.The latter means that there exists a left B-linear and right H-colinear map s : A → B ⊗ A such that m • s = id A , where m : B ⊗ A → A denotes the restricted product of A.
Any cleft extension is a principal comodule algebra with section s : A → B ⊗A, a → s(a) = a 0 j −1 (a 1 )⊗j(a 2 ).Let B := A coH ⊆ A be a Hopf-Galois extension.If k is a field, and recalling that we always consider Hopf algebras H with invertible antipode, the right H-equivariant projectivity of A is equivalent to the existence of a strong connection, it is also equivalent to faithfully flatness of A as a left B-module, see e.g.[8] Part VII, Thm.6.16, 6.19, 6.20.For faithfully flat Hopf-Galois extensions we will later use the following main result.Theorem 2.9 (Schneider,[42] Thm. 1).Let B := A coH ⊆ A be a faithfully flat Hopf-Galois extension.We have the equivalence of categories: where A M H denotes the category of right H-covariant left A-modules and B M the category of left B-modules.

Smash products
Definition 2.10.For a left H-module algebra (B, ⊲) the smash product B#H is defined as the k-module B ⊗ H endowed with the multiplication for all b, b ′ ∈ B and h, h ′ ∈ H.
In the following we write b#h instead of b ⊗ h if we want to view b ⊗ h as an element of B#H.Moreover, we set The compatibility of this coaction with the product • # is easily checked and thus B#H is a right H-comodule algebra (see also [31] Sect.4.1).The smash product algebra B#H is the trivial Hopf-Galois extension B = (B#H) coH ⊆ B#H with cleaving map defined by the algebra inclusion H → B#H.Vice versa trivial Hopf-Galois extensions are isomorphic (as H-comodule algebras) to smashed products.For a trivial Hopf Galois extension B = A coH ⊆ A, conjugation via the cleaving map j : This is easily verified using that j is an algebra map and that its convolution inverse j −1 = j • S is an antialgebra map.Furthermore, j(h 1 )bj −1 (h 2 ) ∈ B follows from the right H-colinearity of j and Lemma 2.6.The compatibility of ⊲ with the algebra structure holds because j −1 is the convolution inverse of j and j is unital.
Proof.In Proposition 2.7 it has been proven that θ is an isomorphism of right H-comodules.It remains to prove that θ is an algebra homomorphism.Trivially we have θ(1 B #1 H ) = 1 A and also which concludes the proof.
3 Covariant differential calculi on Hopf-Galois extensions We begin this section with the definition of the category of noncommutative differential calculi and discuss pullback and quotient calculi.We then specialize to covariant differential calculi and bicovariant calculi, building on previous works of [23] and [47].Examples of (bi)covariant calculi on the quantum groups O q (SL 2 (C)) and O q (GL 2 (C)) are recalled and the induced calculi on their parabolic quantum subgroups are presented.We next revisit a result of Pflaum and Schauenburg [36], where an H-module calculus and a bicovariant calculus merge to the smash product calculus: a covariant calculus on the smash product algebra.This latter is a trivial principal bundle and the smash product calculus is a FODC on it.In Section 3.5 we show that for right H-covariant FODCi on principal comodule algebras base forms are the intersection of horizontal and coinvariant forms.Furthermore, when the injection of horizontal forms into total space forms is completed in an exact sequence with vertical forms, we obtain a graded Hopf-Galois extension with compatible differentials on the total space algebra and on the quantum structure group.

Noncommutative differential calculi
In this section we give the definition and some results on noncommutative differential calculi.Though this material is well established in the context of bicovariant calculi on Hopf algebras, since we take a slightly more general point of view, we prefer to review the main points.We refer the reader to [47], [26,Chpt. 12], [5, Chpt.2] for more details.
We give the definition of a differential calculus.
We say that Γ is generated (as a left A-module) by exact forms and refer to iii.) as the surjectivity property of the FODC.A morphism between a FODC (Γ, d) on A and a FODC (Γ ′ , d ′ ) on another algebra A ′ is a couple (φ, Φ), where φ : A → A ′ is an algebra homomorphism and Φ : Γ → Γ ′ is a k-linear map such that Two FODCi are called equivalent if there is an isomorphism of FODCi between them.
The classical example of a FODC comes from differential geometry.Given a smooth manifold M , the C ∞ (M )-bimodule Ω 1 (M ) of differential 1-forms on M together with the de Rham differential is a FODC on C ∞ (M ).On any algebra A there is the so-called universal FODC (Γ u , d u ), where Γ u := ker µ A ⊆ A ⊗ A is the kernel of the multiplication µ A : A ⊗ A → A and d u a := 1 ⊗ a − a ⊗ 1 for all a ∈ A. The left and right A-module action on Γ u is given by the multiplication on the first and the second tensor factor, respectively.This FODC is universal in the sense that every FODC on A is isomorphic to a quotient of (Γ u , d u ).(c.f.e.g.[47]).
It is well-known that the tensor product A ⊗ A ′ of two algebras A and A ′ is an algebra with associative product (a This construction extends to the level of FODCi (see also [36] (3.1) where a, b, c Proof.Clearly, (3.2) defines an A ⊗ A ′ -bimodule structure and (3.1) a k-linear map.This latter satisfies the Leibniz rule as an equality of sets.Finally, associativity follows from the equality of Remark 3.3.In other words, Proposition 3.2 proves that FODCi form a monoidal category.The monoidal product is and the monoidal unit is the trivial calculus (Γ = {0}, d = 0) on the algebra A = k.
Given a FODC on A we can induce a FODC on any subalgebra and quotient algebra of A. For a surjective algebra map π : A → A ′ , we identify A ′ with A/I where I := ker π and denote by [a] := a + I the class of a ∈ A. Proposition 3.4.Let (Γ, d) be a FODC on A.

i.) An algebra map ι : A
ii.) A surjective algebra map π : is the quotient with Γ I := IdA + AdI, where I := ker π ⊆ A and d π : To see this, we first verify that the maps defined in where we used that ι is an algebra map.The maps in (3.5) are left and right A ′ -actions because ι is an algebra map.They are trivially commuting.Next, the k-linear map d ι satisfies the Leibniz rule for all a, b ∈ A and ω ∈ Γ.It follows that the map d induced on the quotient, d π : , is k-linear and satisfies the Leibniz rule: Considering the quotient with respect to Γ I gives the isomorphism Γ ι → Γ/Γ I , ω → π(ω), which equals the Φ morphism defined in (3.4).We are left to prove the compatibility of (φ = id A ′ , Φ) with the exterior derivatives, for all We call (Γ ι , d ι ) the pullback calculus, while we call (Γ π , d π ) the quotient calculus.The above proposition provides a useful tool to produce examples of noncommutative differential calculi.In the context of Drinfel'd twist deformation quantization it has been employed to describe noncommutative differential geometry on submanifolds of R n given by generators and relations [15].More abstractly, the braided Cartan calculus of a braided commutative algebra with triangular Hopf algebra symmetry is related to the braided Cartan calculus on a submanifold algebra in the above sense [46].

Covariant calculi on comodule algebras
In this section we discuss the theory for covariant calculi on comodule algebras, following the perspectives of [23,47] and [36].Let H be a Hopf algebra.
for all a, b ∈ A, extends to a well-defined k-linear map Γ → Γ ⊗ H. ii.
for all a, b ∈ A, extends to a well-defined k-linear map Γ → H ⊗ Γ.
iii.)A FODC (Γ, d) on an H-bicomodule algebra (A, δ R , δ L ) is said to be H-bicovariant if it is right and left H-covariant.
If A = H is an H-comodule algebra with respect to the coproduct we call an H-covariant FODC simply covariant.We recall the following proposition.We define a morphism of right/left/bi H-covariant FODCi (Γ, d) and (Γ ′ , d ′ ) on right/left/bi H-comodule algebras A, A ′ to be a morphism (φ, Φ) between FODCi such that φ and Φ are right/left/bi H-colinear maps.Since Φ, if it exists, is determined by φ via Φ(a The natural question arises if Proposition 3.4 generalizes to the H-covariant setting. Analogous statements hold for left H-covariant and H-bicovariant FODCi on left H-comodule algebras and H-bicomodule algebras, respectively.Proof.From Proposition 3.4 it follows that (Γ ι , d ι ) and (Γ π , d π ) are FODCi on A ′ .We prove i.) by observing that for all a i , b i ∈ A (and understanding finite sums on the indices i) The third statement trivially holds recalling that for the morphism (φ, Φ) of FODCi Φ is H-colinear if φ is H-colinear, which is the case since φ = id A ′ .
We have the following corollary of Proposition (3.7) ii.).
and those on Γ defined by as an H-bicovariant FODC on A. The projection π : A → H is left and right H-colinear so that from Proposition (3.7) the induced FODC (Γ π , d π ) on H is H-bicovariant with H-coaction given by the H-coproduct.

Examples of covariant calculi
In this section we describe covariant calculi on the quantum groups O q (GL 2 (C)), O q (SL 2 (C)) and on their parabolic subalgebras.
Example 3.9 (4D calculus on O q (GL 2 (C))).Let q ∈ C be a non-zero complex number which is not a root of unity.The free C-algebra C α, β, γ, δ generated by indeterminates α, β, γ, δ modulo the relations , where I M is the ideal in C α, β, γ, δ generated by the (3.8).The quotient algebra O q (M 2 (C)) is a bialgebra with coalgebra structure where ⊗ denotes the tensor product and matrix multiplication.Furthermore, there is a central element det q := αδ − q −1 βγ ∈ O q (M 2 (C)), satisfying ∆(det q ) = det q ⊗ det q and ǫ(det q ) = 1.On the quotient algebra O q (GL 2 (C)) := O q (M 2 (C))[r]/ r det q − 1 we induce a Hopf algebra structure by (3.9) and ∆(r) = r ⊗ r, ǫ(r) = 1, together with the antipode On the quantum group A := O q (GL 2 (C)) there is a 4-dimensional bicovariant FODC (Γ GL , d GL ) which we are going to describe following the explicit exposition of [2] 1 .Γ is a free left A-module generated by a basis {ω 1 , ω 2 , ω 3 , ω 4 } of left coinvariant 1-forms obeying the commutation relations with λ := q −1 − q.The differential is given in terms of this basis by where The basis 1-forms are then expressed explicitly as where 1 Note that in our convention we used q −1 instead of q and we rescaled ω 2 , ω 3 by q −1 for later use in Example 3.10 We now construct a bicovariant FODC (Γ PGL , d PGL ) on O q (P GL ) := A/ γ .This is done via the Hopf algebra quotient π PGL : A → O q (P GL ), a → π PGL (a) =: [a], using Corollary 3.8.We recall that Γ PGL := Γ GL /Γ Iγ , where Γ Denoting the projected generators by the right H-action and differential are explicitly given by and Example 3.10 (Bicovariant FODC on O q (SL 2 (C)) and its parabolic quotient).Consider the quotient Hopf algebra O q (SL 2 (C)) := O q (GL 2 (C))/ det q − 1 with coalgebra structure and antipode induced from (3.9) and (3.10).We denote the associated Hopf algebra map projection by pr SL : O q (GL 2 (C)) → O q (SL 2 (C)).The quantum group O q (SL 2 (C)) is the Manin deformation (see [30]) of the ring of algebraic functions on the complex special linear group SL 2 (C).It is the deformed algebra of regular functions on the complex special linear group SL 2 (C).Let O q (P ) be the deformed algebra of functions on the Borel subgroup P ⊂ SL 2 (C).We identify it with the quotient O q (SL 2 (C))/I P where I P ⊆ O q (SL 2 (C)) is the Hopf ideal generated by the element γ.On generators the Hopf algebra projection is given by or, in other words, O q (P ) = O q (SL 2 (C))/I P = C t, t −1 , p / tp − q −1 pt .Note that O q (P ) = O q (P GL )/ det q − 1 ∼ = O q (GL 2 (C))/ γ, det q and the corresponding projection pr P : O q (P GL ) → O q (P ) makes the diagram O q (P GL ) O q (P ) pr SL πP GL πP pr P (3.17) commute.
Example 3.11 (3-dim calculus on O q (SL 2 (C)) and its parabolic quotient O q (P )).Let A := O q (SL 2 (C)) and H := O q (P ) be the Hopf algebras from Example 3.10.There is a 3-dimensional left covariant FODC (Γ SL , d SL ) on A described in [26] Sect.14.1.4.Γ SL is the free left A-module generated by the basis {ω 0 , ω 1 , ω 2 } of left coinvariant 1-forms with commutation relations for j = 0, 2 and The differential d SL : A → Γ SL is determined by and thus As in Example 3.10 elements in the equivalence class H are denoted by [ω] with a representative ω ∈ Γ SL .The induced quotient calculus (Γ P , d P ) on H is the 2-dimensional left covariant FODC with Γ P being the free left H-module generated by the basis {[ω 0 ], [ω 1 ]} of left coinvariant elements (notice that [ω 2 ] = 0, while [ω 0 ], [ω 1 ] are linearly independent).The resulting commutation relations are and the differential reads

The smash product calculus
In this section we recall the construction of a covariant differential calculus on the smash product algebra B#H from an H-module calculus on an H-module algebra B and a bicovariant calculus on the Hopf algebra H, given in [36].If the H-action on B is trivial we recover the tensor product differential calculus on B ⊗ H described in Proposition 3.2.
Let B be a left H-module algebra, with action ⊲ : H ⊗ B → B. Let M be a B-bimodule, with actions and a left H-module with action that with slight abuse we denote ⊲ : H ⊗ M → M .We say that M is a relative H-module B-bimodule if the H and B actions have the compatibility, for all Similar definitions are given if M is just a left or a right B-module.
Definition 3.12.Let B be a left H-module algebra with action ⊲ : The rationale of this definition is in the following proposition.Proof.Since (3.20) holds we have a well-defined H-action given by, where we used that Compatibility of this action with the left B-action is immediate.Compatibility with the right Baction: h⊲ Given an H-module algebra B and a FODC (Γ B , d B ) and a left covariant FODC (Γ H , d H ), we consider the k-module extended linearly to all Γ # .Defining the left B-action on Γ # as the B-action on the first factors in the tensor products Γ B ⊗ H and B ⊗ Γ H we obtain the left The proof that this indeed defines an action of the algebra B#H on Γ # , actually a B#H-action on the submodules Γ B ⊗ H and B ⊗ Γ H , uses the same steps of the proof of associativity of the multiplication in B#H.

We define (ω
) 0 and extend it linearly to all Γ # .It is easy to prove that this is a right B-action on Γ # .Defining the right H-action on Γ # as the right H-action on the second factors in the tensor products Γ B ⊗ H and B ⊗ Γ H we obtain the right B#H-action on Γ # We prove commutativity of the left and right B#H-actions on Γ B ⊗ H:

Commutativity of the left and right
for all b ∈ B and h ∈ H.
Proof.We show that d # : B#H → Γ # satisfies the Leibniz rule: The smash product construction of differential calculi is compatible with right H-coactions.

Proof. Define a right H-coaction on Γ
We prove that the calculus is right H-covariant by showing right H-colinearity of the differential, cf.Proposition 3.6 i.).For all b#h ∈ B#H,

Base forms, horizontal forms and principal covariant calculi
In this section we study differential calculi on noncommutative principal bundles over affine bases.We assume the ground ring k to be a field and recall that the Hopf algebra (quantum structure group) H is assumed to have invertible antipode.We have seen that in this setting a principal comodule algebra B = A coH ⊆ A is equivalently a faithfully flat Hopf Galois extension B = A coH ⊆ A.
The B-bimodule of base forms is the intersection of those of horizontal and coinvariant forms Proof.The inclusion AΓ B ⊆ Γ A implies (AΓ B ) coH ⊆ (Γ A ) coH and, since A is a flat B-module, we have the inclusion where for the isomorphism A ⊗ B (Γ A ) coH ∼ = Γ A we used Theorem 2.9.The inclusion Γ B ⊆ (AΓ B ) coH and flatness of A over B imply the inclusion that composed with (3.24) gives injectivity of the natural map A ⊗ B Γ B → Γ A and hence the isomorphism From Theorem 2.9 we then have We where

is the cotensor product of A and Γ H , and the vertical map ver : Γ
If, in addition, the FODC (Γ A , d A ) is right H-covariant and the FODC (Γ H , d H ) is bicovariant, we say we have a principal covariant calculus on the principal comodule algebra A.
We can easily check that ver, if it is well-defined, it satisfies ver(a for all a, a ′ ∈ A and ω ∈ Γ A .The sequence is therefore a sequence of left A-modules right B-modules.
The following lemma provides a sufficient condition for the existence of the vertical map.

and the quotient map of forms by π
] is a well-defined map.We prove that this map coincides with ver.Recall that the induced right H-coaction on A is given by δ i.e. (id A ⊗ π Γ ) • ∆ ΓA L = ver, as claimed.For a right H-covariant calculus on a principal comodule algebra the second arrow in (3.26) is injective by Theorem 3.17.In this context of principal comodule algebras the notion of principal covariant calculus of Definition 3.18 is then equivalent to that of strong quantum principal bundle in [7] Def.4.9 and [5] Sect.5.4.This follows from the canonical identifications A ⊗ B Γ B ∼ = AΓ B , cf.Theorem 3.17, and A H Γ H ∼ = A ⊗ coH Γ H , where coH Γ H denotes the module of left coinvariant one forms characterizing the covariant FODC (Γ H , d H ) (see [47]).In this light Theorem 3.17, which does not assume an exact sequence, can be read as generalizing the conditions of [5] From [36] Thm.4.1 and Lem.4.2 we have the isomorphisms of right H-covariant B#H-bimodules Their inverses are given by ω ⊗ ω H 0 , respectively.Under this first isomorphism the second arrow in (3.26) becomes the inclusion Γ B #H → Γ B #H ⊕ B#Γ H . Finally, the vertical map equals the projection pr : Γ B #H ⊕ B#Γ H → B#Γ H up to the second isomorphism in (3.29) (cf.e.g.[36] Thm.2.9).
Example 3.21 (3D calculus on the q-monopole fibration).Consider A = O q (SL 2 (C)) (q real, q = ±1) with the Hopf algebra quotient The projection π induces a right H-coaction: The subalgebra B := A coH of coinvariants is given by the (complex) Podleś sphere O q (S 2 ) with generators B − = αβ, B + = γδ, B 0 = βγ and commutation relations (see [37] and refs.therein).It is known that B ⊆ A is a faithfully flat Hopf-Galois extension, i.e. that A is a principal comodule algebra [32], (cf.also [8]  We next show that for a principal covariant calculus on a principal comodule algebra A the Hopf-Galois extension B = A coH ⊆ A lifts to a Hopf-Galois extension of graded algebras.Recall that for a bicovariant H-bimodule Γ H we obtain a graded Hopf algebra Ω 1 H := H ⊕ Γ H .The multiplication is given by that in H and by the H-module structure, while the product of degree one elements is trivial.The comultiplication H has components on degree zero and one ∆ 0 := ∆ : H → H ⊗ H and )ω 0 S(ω 1 ).
Lemma 3.22.Consider a principal comodule algebra A with a principal covariant calculus (Γ A , d A ).The graded algebra given by its components ∆ 0 Defining Proof.By assumption ∆ 0

coaction and an algebra map. Similarly also ∆ Ω 1
A is an Ω 1 H -coaction and an algebra map.For the coaction property we just have to prove on 1-forms.This is a straightforward computation using the right H-covariance of (Γ A , d A ), the definition of the vertical map and the bicovariance of (Γ H , d H ). The coaction ∆ Ω 1 A is an algebra map because for all a, a ′ ∈ A and ω ∈ Γ A we have where we used that Γ A is a right H-covariant A-bimodule and that the vertical map satisfies the compatibility condition (3.28).
The algebra inclusion B ⊆ A implies the graded algebra one Ω 1 is obvious in degree zero.In degree one, recalling definition (3.30) and that the codomain of ∆ and ver(ω) = 0.These equalities respectively imply ω ∈ Γ coH A and ω ∈ Γ hor A , this latter condition following from exactness of the sequence (3.26).Then, from Theorem 3.17 we obtain Γ In the following we prove that principal covariant calculi are equivalent to graded Hopf-Galois extensions with compatible differentials.Theorem 3.23.Let A be a principal comodule algebra and (Γ A , d A ) a principal covariant calculus on A with corresponding bicovariant FODC (Γ H , d H ) on H. Then is a faithfully flat Hopf-Galois extension of graded algebras.Moreover we have the following commutative diagram where we endow A ⊗ H with the tensor product calculus (Γ A⊗H , d A⊗H ).
Conversely, if A is a graded right Ω 1 H -comodule algebra.Then from [40] Cor.5.9, A is a Hopf-Galois extension which is faithfully flat as a left Ω  (3.31) is a faithfully flat Hopf-Galois extension (so that in particular B ⊆ A is a faithfully flat Hopf-Galois extension) then from [40] Cor.5.9 the sequence is exact, where Θ is the projection of ∆ 1 If we assume commutativity of the diagram in (3.32) it follows that 1 for all a ∈ A. Since Θ and ∆ ΓA R are A-bilinear (where A ⊗ H is an A-bimodule via δ A ) the first equality implies that (Γ A , d A ) is a right H-covariant calculus, the second one that Θ is the vertical map.Thus (3.33) is the exact sequence showing the principality of the differential calculus (Γ A , d A ).

A sheaf-theoretic approach to noncommutative calculi
In this section we introduce differential calculi on quantum principal bundles in a two-step process.First we define a covariant calculus on a sheaf of comodule algebras F as a sheaf Υ of F -bimodules together with a morphism d : F → Υ of sheaves, requiring locally Leibniz rule and a surjectivity condition.In case F is a quantum principal bundle we demand an additional local compatibility of the calculi on the total sheaf, the base sheaf and the structure Hopf algebra.

Quantum principal bundles as sheaves
We start by recalling the main notions and results of [4].Those concern the (function) algebra level.In the remaining sections we generalize the definitions and findings to FODCi.
A quantum ringed space (M, O M ) is a pair consisting of a classical topological space M and a sheaf O M on M of noncommutative algebras.We call a sheaf of H-comodule algebras F an H-principal bundle or quantum principal bundle (QPB) over (M, O M ) if there exists an open covering {U i } of M such that: If the base ring is a field local principality is equivalent to faithfully flatness of the local Hopf-Galois extensions.If these Hopf-Galois extensions on the open cover {U i } are cleft or trivial we say that the QPB is locally cleft or locally trivial, respectively.Remark 4.1 (Principal comodule algebras restrict locally).In [11] Lem. 4.1 it is shown that the pushforward of a strong connection ℓ : H → A ⊗ A by a comodule algebra map φ : A → A ′ is a strong connection (φ ⊗ φ) • ℓ on A ′ .Recalling that if the base ring k is a field there is a bijective correspondence of strong connections on comodule algebras with principal comodule algebras (cf. the paragraph after Definition 2.8) it follows that the existence of a comodule algebra map φ : A → A ′ implies that if A is a principal comodule algebra, so is A ′ .
Since F is a sheaf of H-comodule algebras, for any inclusion U ⊂ U i of an open U of the topology of M in an open U i of the covering {U i }, the restriction r UUi : F (U i ) → F (U ), which by definition is a comodule algebra map, then implies that F (U ) is a principal comodule algebra.
Let now F p be the stalk of F at p ∈ M .Choosing U i such that p ∈ U i and considering the canonical map F (U i ) → F p of right H-comodule algebras, defined by s → [(U i , s)] p , where the equivalence class [(U i , s)] p ∈ F p is the germ at p of the section s ∈ F (U i ), we obtain that the stalk F p is a principal comodule algebra.
Our main examples of QPBs have a quantum group as total space.Let G be a complex semisimple algebraic group and P a parabolic subgroup.The quotient G/P is a projective variety and the projection G −→ G/P is a principal bundle.Let O q (G) and O q (P ) = O q (G)/I P be Hopf algebras over C q := C[q, q −1 ], quantizations of O(G) and O(P ), the coordinate algebras of G and P , respectively (see [4] Sect. 3, [10] Sect.3).We shall later specialize q to a complex number in order to consider Hopf-Galois extensions with C the base ring.Once we fix a projective embedding for G/P , obtained through the global sections of a very ample line bundle L, we have the graded algebra O(G/P ).We say that s ∈ O q (G) is a quantum section if where π : O q (G) −→ O q (P ), ∆ is the coproduct of O q (G) and t is a classical section, that is, it lifts to O(G) the character of P defining L (see [4] Def. 3.6).Denote by {s i } i∈I a choice of linearly independent elements in ∆(s) = i∈I s i ⊗ s i , that we assume to generate the homogeneous coordinate ring O q (G/P ) quantization of the commutative homogeneous coordinate ring O(G/P ) (L is very ample).Assume furthermore that S i = {s r i , r ∈ Z ≥0 } is Ore, denote by O q (G)S −1 i the Ore localization, and assume that subsequent Ore localizations do not depend on the order.In the specialization for q = 1, the s i 's define an open cover of G, whose projection on M = G/P gives an open cover {U i } of M (see [4], Sect. 4 for more details).Let us define the open sets with r = 0, 1, 2, . . .n (here n is the cardinality of I and the empty set ∅ corresponds to r = 0).We consider on M the topology generated by the open sets U I , which hence form a basis B = {U I } I∈II , where II is the set of ordered multi-indices I = (i 1 , . . ., i r ), 1 ≤ i 1 < • • • < i r ≤ n with r = 1, 2, . . .n (and we also consider the case r = 0 corresponding to the empty set).
The main result in [4] is Theorem 4.8 that states the following.
Theorem 4.2.Let the notation be as above.Then i.) the assignment defines a sheaf F G of right O q (P )-comodule algebras on the topological space M , iii.) Remark 4.3.We clarify the sheafification implicitly understood in [4] Thm. 4.8.The restriction morphisms U I ⊆ U J (i.e.J ⊆ I), for O M and F G are given by the natural morphisms.Thus, (4.1) and (4.2) determine presheaves O M and F G on the basis B.
Define F coOq(P ) G (U I ) := F G (U I ) coOq(P ) for all I and define the restriction morphisms which are well-defined because H-colinearity of the r IJ in (4.3) implies r IJ (F coH G (U J )) ⊆ F coH G (U I ).The restriction morphisms r coH IJ inherit the B-presheaf properties from the r IJ , so that F coOq(P ) G is a B-presheaf.Using the technique of [4] Prop.4.7, one can prove the B-presheaf equality F coOq(P ) G = O M .Using the following observation these B-presheaves are extended to sheaves on the full topological space, these are the sheaves of Theorem 4.2.
Observation 4.4.If we have a presheaf G B defined on a basis B for the topology, we can always extend it to the presheaf G, where G(U ) := lim ← − G B (V ) for V ⊂ U , V ∈ B, provided the target category admit inverse limits (see [17] Chpt.0, Sect.3.2.1).If B is finite the existence of lim ← − G B (V ) is granted.Then, once we have a presheaf, we can always proceed to the sheafification and obtain a sheaf on the topology with the same target category (groups, H-comodule algebras, etc).By universality, a morphism of presheaves on a basis will extend uniquely first to a morphisms of presheaves and then to a morphisms of their sheafification (cf.[20] Chpt. 2 and also [14] for more details on this standard construction).To ease the notation, we shall use the same letter to denote a presheaf on B, its extension to a presheaf and the sheafification.
The topology of M used in Theorem 4.2 is finite since it is induced by a finite cover {U i } i∈I of M via the finite basis B = {U I } I∈II .We can then define the minimal opens Given a presheaf F on M , its stalk at p ∈ M coincides with its sections on U p , i.e.F p = F (U p ).This is immediate from the definition of stalk since any open neighbourhood of p includes U p .Now the stalks of the sheafification of F coincide with the stalks of the initial presheaf, and so F coincides with its sheafification on the opens U p .

Covariant calculi on sheaves of comodule algebras
We now give the definition of a covariant FODC on a sheaf of comodule algebras.The bimodule property is entirely captured in a sheaf-theoretic language, while we demand the Leibniz rule and surjectivity property of the differential on stalks.To account for coinvariance we have to consider the differential as a morphism of sheaves of comodules.We give some fundamental examples of covariant FODCi on sheaves and discuss the sheafs of base forms, horizontal forms and coinvariant forms.Given a morphism ϕ : F → G of sheaves on M we denote the induced morphism on the stalks at p ∈ M by ϕ p : F p → G p .We also recall that, for F a sheaf of algebras on a topological space M , a sheaf of F -modules associates to each open U in M an F (U )-module with compatible restriction maps.Definition 4.5.Let F be a sheaf of algebras on a topological space M .A FODC on F is a sheaf Υ of F -bimodules on M with a morphism d : F −→ Υ of sheaves, such that for all p ∈ M the induced maps on stalks d p : If H is a Hopf algebra and F a sheaf of right H-comodule algebras over M we call a FODC (Υ, d) on F right H-covariant if Υ is a sheaf of right H-covariant F -bimodules and d is a morphism of sheaves of right H-comodules.
Similarly one defines left H-covariant and H-bicovariant FODCi on sheaves of left H-comodule algebras and H-bicomodule algebras, respectively.Example 4.6.
i.) Let M be an algebraic variety and G an (affine) algebraic group acting on M .Then, O M , the structural sheaf of M , carries an H = O(G) coaction, where O(G) denotes the global sections of the structural sheaf of G, which carries a natural Hopf algebra structure.Define Ω as the sheaf of Kähler differentials with d : O M −→ Ω as in [20], Sec. 8, II.As one can readily check, using the results in [20], (Ω, d) is a FODC on O M .Theorem 4.10.Let the base ring k be a field.For any right H-covariant FODC (Υ, d) on a QPB F we have an isomorphism Proof.The natural inclusion Υ M → Υ coH ∩ Υ hor (by which we mean the sheafification of the corresponding morphism of presheaves) is a morphism of sheaves.Recalling from Remark 4.1 that stalks are principal comodule algebras, the induced inclusion on stalks, (Υ M ) p → Υ coH p ∩ Υ hor p , is an isomorphism thanks to Theorem 3.17.Thus Υ M → Υ coH ∩ Υ hor is an isomorphism of sheaves.

Ore extension of covariant calculi
As a tool for the following sections, we now discuss the Ore extension of covariant calculi.We recall that if a ∈ A is a (right) Ore element, we denote by AS −1 or by A[a −1 ] the Ore localization of A at the multiplicative right Ore set S = {a m } m∈Z ≥ 0 (for more details see [28] Chpt.4, Sect. 10 and [24] Sect.I.7.).
Let (Γ, d) be a FODC on an algebra A. Let a be an Ore element in A. We define the A[a −1 ]-bimodule and the k-linear map where we extend d a to A[a −1 ] by the Leibniz rule.
Proof.We need to verify the properties of Definition 3.1.The first two properties are satisfied by definition.
Moreover by the equality where a ′ , a ′′ ∈ A and n ∈ N, we see that we can express any element of Γ a as an element in With a slight abuse of terminology, we call (Γ a , d a ) the localization of (Γ, d) via the Ore element a ∈ A.
We now show that the covariant properties of the FODC Γ are preserved under localization.
Lemma 4.12.Let (Γ, d) be a right H-covariant FODC on a right H-comodule algebra (A, δ R ) and a ∈ A an Ore element.Assume that δ R (a and by construction it is compatible with the algebra structure of given by the diagonal coaction.We show that ∆ a R structures Γ a as a right and ω i ∈ Γ, sum over i understood.Since, according to i.), δ a R is an algebra homomorphism, we obtain where in the last expression we used right H-colinearity of d.This implies The Leibniz rule of d a , the algebra homomorphism property of δ a R , the A-bilinearity of ∆ a R and (4.10) then imply that d a is right H-colinear on any power of a −1 and furthermore on any element of A[a −1 ].
Let A be an algebra, a i ∈ A, i = 1, . . ., n be Ore elements and S i = {a r i , r ∈ Z ≥0 }.For a multi-index I = (i 1 , . . ., i k ) ∈ {1, . . ., n} k we use the notation A I = AS −1 i1 . . .S −1 i k , or equivalently the notation for the k-fold Ore localization.Given a FODC (Γ, d) on A we similarly define the k-fold Ore localization of the A-bimodule Γ by and of the exterior derivative by Lemma 4.13.Let A be an algebra and a i ∈ A, i = 1, . . ., n be Ore elements such that subsequent localizations do not depend on the order.Then, subsequent localizations of a FODC on A are independent of the order, as well.
Proof.Consider a FODC (Γ, d) on A. For two Ore elements a i , a j ∈ A, using that A (i,j) = A (j,i) we have as an equation of A (i.j) -bimodules.In particular, d (i,j) , d (j,i) : A (i.j) → Γ (i,j) have the same domain and codomain.They coincide on A and by definition (4.9) they also agree on a −1 i and a −1 j .Both differentials are extended by the Leibniz rule, therefore d (i,j) = d (j,i) .An easy inductive argument on the number of Ore localization then proves the lemma.
This lemma shows in particular that the k-fold Ore localization of the A-bimodule Γ is Γ I = A I ⊗ A Γ ⊗ A A I , that for short we frequently write as Γ I = A I ΓA I .Recalling also Lemma 4.12 the following proposition is straightforward.
Proposition 4.14.Let (Γ, d) be a FODC on a right H-comodule algebra (A, δ R ) and a i ∈ A, i = 1, . . ., n be Ore elements such that δ R (a i ) are invertible elements of A i ⊗ H. Then If subsequent Ore localizations of A are independent on the order also the subsequent localizations of the FODC (Γ, d) are order independent.
Remark 4.15.We have considered FODCi on Ore extended algebras.In Section 3.5, associated with a FODC (Γ A , d A ) there is the graded algebra of 0-and 1-forms It is natural to consider also Ore extensions of this graded algebra (which are necessarily by degree zero elements).In this case an Ore element a is in particular Ore in the zero degree subalgebra A and we also require to be able to order on the right (and on the left) of 1-forms the negative powers a.Thus, if a ∈ A is Ore in Ω 1 A the graded algebra A a ⊕ Γ a with Γ a defined in (4.8) is the Ore extension of A ⊕ Γ corresponding to a.This Ore condition on 1-forms is met in the examples we shall consider.

Covariant differential calculi on sheaves over quantum projective varieties
In this section we consider all modules over the complex numbers C. When considering the quantizations O q (G), O q (P ) we therefore specialize q to be in C. Fix a complex semisimple algebraic group G with parabolic subgroup P , quantizations O q (G), O q (P ) and the sheaves F G and O M as in Theorem 4.2, where the QPB property of F G is with respect to the finite open cover {U i } i∈I , I = {1, . . ., n} determined by a quantum section.Consider the (finite) topology generated by the open cover; it will have topological basis B = {U I } I∈II , where we recall that II is the set of ordered multi-indices I = (i 1 , . . ., i r ), 1 ≤ i 1 < • • • < i r ≤ n with r = 1, 2, . . .n (and we also consider the case r = 0 corresponding to the empty set) and U I := ∩ i∈I U i .
Given a right O q (P )-covariant FODC (Γ, d) on O q (G) we induce a O q (P )-covariant FODC on F according to Section 4.3 as follows.We set on the basis B of open sets and define restriction morphisms where f, g ∈ F G (U I ) and ω ∈ Γ.Since r JI are algebra homomorphisms which are the identity on O q (G) the expression (4.13) is well-defined on the algebraic tensor product.The right O q (P )-comodule algebra map r JI defines an F G (U I )-bimodule structure on Υ G (U J ) and r Υ JI : Υ G (U I ) → Υ G (U J ) is a map of right O q (P )-covariant Υ G (U J )-bimodules, i.e., it is O q (P )-colinear and r Υ JI (f • θ • g) = r JI (f )r Υ JI (θ)r JI (g) for all f, g ∈ F G (U I ) and θ ∈ Υ G (U I ).The equality r Υ KJ • r Υ JI = r Υ KI for any three opens U K ⊆ U J ⊆ U I then follows from that for the restriction morphisms of F G .We have defined the B-presheaf Υ G of right O q (P )-covariant F G -bimodules.We denote its sheafification by the same symbol Υ G .Theorem 4.16 (Induced calculus on the sheaf F G ). Let (Γ, d) be a right covariant FODC on the Hopf algebra O q (G) and F G , Υ G the sheaves of right O q (P )-comodule algebras and of right O q (P )-covariant F G -bimodules defined above.iii.)If F G is a QPB the sheaf of base forms is isomorphic, as a sheaf of O M -bimodules, to the intersection of that of horizontal and H-coinvariant forms: Proof.Let F G be the B-presheaf defined in (4.2) and Υ G the B-presheaf defined in (4.12), (4.13).We show that the assignment U I → d I defines a morphism of B-presheaves d : By the Leibniz rule, cf.Proposition 4.14, this B-presheaf property extends to any element f ∈ F G (U I ), d J (r JI (f )) = r Υ JI (d I f ).Recalling from Proposition 4.14 that (Υ G (U I ), d I ) is a right O q (P )-covariant FODC and that F G and Υ G are B-presheaves of right O q (P )-comodules, we see that the B-presheaf morphism d : F G → Υ G is compatible with the right O q (P )-comodule structure.Furthermore, the Leibniz rule and surjectivity condition are satisfied for all opens U I in B. We sheafify according to Observation 4.4 and obtain the sheaf Υ G of right O q (P )-covariant F G -bimodules with the sheaf morphism d : F G → Υ G .Recalling that stalks are preserved under sheafification and that the stalks at p are respectively (F G ) p = F G (U p ), (Υ G ) p = Υ G (U p ) we infer that the Leibniz rule and the surjectivity condition for the sheafified morphism d : F G → Υ G are satisfied on the stalks.This proves i.).Recalling that the sheaf equality F coOq(P ) G = O M was proven in Theorem 4.2 we have that ii.) follows directly from Corollary 4.9.The last point iii.) follows from Theorem 4.10.

Principal covariant calculi on quantum principal bundles
A principal covariant calculus on a QPB F is a right H-covariant FODC (Υ, d) on the sheaf F (see Definition 4.5) with an exact sequence on the stalks.Definition 4.17.Consider a QPB F on a quantum ringed space (M, O M ) and a FODC (Υ, d) on F .Given a left covariant FODC (Γ H , d H ) on H we call (Υ, d) a principal calculus on F if for all p ∈ M we have the exact sequence of stalks 0 where ver p is the vertical map defined in Section 3.5 and Υ M denotes the sheaf of base forms defined in Proposition 4.8.If, in addition, (Υ, d) is right H-covariant and (Γ H , d H ) is bicovariant, we say that (Υ, d) is a principal covariant calculus on the sheaf F .
We next apply Theorem 4.16 and consider three FODCi on QPBs over P 1 (C), the second one being a principal calculus and the third one a principal covariant calculus.The first two have total space algebra O q (SL 2 (C)) with total space calculus the 4D and 3D ones of Examples 3.10 and 3.11.The third example has total space algebra O q (GL 2 (C)) with total space calculus the 4D one of Example 3.9.Consider the principal bundle ℘ : SL 2 (C) −→ SL 2 (C)/P ≃ P 1 (C), where P is the upper Borel in SL 2 (C).Take V i to be the subset of matrices in SL 2 (C) with entry (i, 1) not equal to zero.We observe that {V . Explicitly, U 1 , U 2 are the opens obtained by removing the north or south pole.The coaction The quantum deformation of this construction has been studied in [4] as an example of QPB over projective base.In summary, starting with the Hopf algebra A := O q (SL 2 (C)) and its parabolic quotient Hopf algebra H := O q (P ) as in (3.16) we define the noncommutative localizations A 1 := A[α −1 ] and A 2 := A[γ −1 ] and structure them as right H-comodule algebras in complete analogy to the classical setting.The subalgebras of right H-coinvariants are given by and the ringed space (P 1 (C), O P 1 (C) ) can then be constructed as i ∈ {1, 2}, with the only non-trivial restriction map given by r 12,2 : B 2 → B 12 , v → u −1 .Moreover we observe that , defines a sheaf of right O q (P )-comodule algebras on P 1 (C), which we denote by F SL q .Endowed with the cleaving maps, F SLq becomes a QPB on P 1 (C) (on A 12 we can choose j 12 = j 1 : H → A 1 ⊂ A 12 as a cleaving map or alternatively j 2 ).Notice that the stalk (F SL q ) p of F SL q at p ∈ P Proof.The A I -bimodules Υ SL q (U I ) = Γ I = A I ΓA I of the induced calculus of Theorem 4.16 are constructed in Proposition 4.14.We show that Γ I = A I Γ.The proof then follows using that Γ is a free left A-module and ordering the invertible elements of A I on the left.The only nontrivial cases are Γ i , i ∈ {1, 2}, and Γ 12 .From the commutation relations (3.11) (recall that the differential and the bimodule structure of the 4-dimensional bicovariant calculi on SL q (2) and GL q (2) coincide up to the identification det q = 1) it immediately follows that in Γ 1 and Recalling that a general element of A 1 is of the form j a j α −nj ∈ A 1 for n j ∈ N, a j ∈ A and that Γ is an A-bimodule which is freely generated as a left A-module by ω 1 , ω 2 , ω 3 , ω 4 , the commutation relations (4.17) imply ΓA Similarly we prove Γ 2 = A 2 Γ, and Γ 12 = A 12 Γ using both (4.17) and (4.18).
In the proof of Proposition 4.18 it was shown that for all U I ∈ B the right H-covariant A I -bimodule Υ SL q (U I ) = span AI {ω 1 , ω 2 , ω 3 , ω 4 } is freely generated from the left.This implies that Υ SL q (U 1 ∪ U 2 ) = Γ, i.e. that the presheaf constructed by the Ore extensions is already a sheaf and there is no sheafification required.Proposition 4.19.The induced FODC (Υ P 1 (C) , d P 1 (C) ) of base forms on O q (P 1 (C)) (described in Corollary 4.9) is determined by are free left modules and we have the sheaf isomorphism Proof.Using (3.12), (4.17) and δ = α −1 + q −1 α −1 βγ we obtain Then the equalities in (4.19) follow, the last one also recalling that  are well-defined for all U I ∈ B with the vertical map given by ver I : Γ ] for all a i ∈ A I (the proof is as in Lemma 3.19).However (Υ SL q , d) is not a principal calculus on the QPB F SL q .For example the sequence (4.22) for I = 1 is not exact.Indeed first observe that, since Γ 1 and Γ B1 are free modules, ω 1 / ∈ A 1 Γ B1 = span A1 {ω 2 }.Then, since ω 1 is in the kernel of the projection Γ → Γ H it follows that ver 1 (ω 1 ) = 1 ⊗ [ω 1 ] = 0 and therefore A 1 Γ B1 ker ver 1 .

Principal calculus via
Ore localization of 3D calculus on O q (SL 2 (C)) In this section we show that the failure of (4.22) to be an exact sequence can be cured by considering a 3dimensional left covariant FODC instead of a 4-dimensional bicovariant FODC on A = O q (SL 2 (C)) as the global calculus of F SL q .We choose a left covariant calculus on A so to canonically obtain a left covariant calculus on the quotient H = O q (P ).As in the previous section we consider the QPB F SLq with the cleaving maps (4.16).Lemma 4.21.Let (Γ, d) = (Γ SL , d SL ) be the 3-dimensional left covariant FODC on A of Example 3.11.The induced FODC (Υ SL q , d) on F SL q is a free left F SL q -module of dimension 3.
As in the previous section the freeness of the presheaf Υ SL q implies that there is no sheafification required and in particular Υ SL q (P 1 (C)) = Γ.
We have seen that the differential calculus (Υ SLq , d) is principal.Note that it is not principal covariant since the induced left covariant FODC (Γ H , d H ) on H is not bicovariant.This is the case because the ideal I = span H {(t−1)(t−q 2 ), p 2 , (t−q 2 )p} characterizing (Γ H , d H ) is not closed under the adjoint right coaction, e.g.Ad R (p 2 ) / ∈ I ⊗ H (it contains a polinomial in t with roots different from 1 and q 2 .See [47] for the classification of bicovariant FODCi on Hopf algebras in terms of ideals closed under the right adjoint coaction).Consider the principal bundle GL 2 (C) → GL 2 (C)/P GL ∼ = P 1 (C) with P GL the upper Borel in GL 2 (C).As before, localizing with respect to α and γ gives the opens U 1 and U 2 with corresponding topology {∅, U 1 , U 2 , U 12 , P 1 (C)} on P 1 (C).Although GL 2 (C) is not semisimple, a quantum deformation of this bundle, which is a QPB F GL q on P 1 (C), is obtained via Ore extensions of the quantum group A = O q (GL 2 (C)).Explicitly, we define the sheaf of algebras by F GL q (∅) := {0}, F GL q (P 1 (C)) := A, The Hopf algebra quotient A → H = O q (P GL ) = A/ γ defined from Example 3.9 induces a natural comodule algebra structure on F GL q .This is a locally trivial QPB with cleaving maps j 1 : H → A 1 , j 2 : H → A 2 and j 12 = j 1 : H → A 1 ⊂ A 12 given by j 1 t p 0 s = α β 0 α −1 det q , j 1 (r ′ ) = r , j 2 t p 0 s = γ δ 0 γ −1 det q , j 2 (r ′ ) = r , ( where r ∈ A and r ′ ∈ H are the inverses of the quantum determinants.It is straightforward to verify that the sheaf O P 1 (C) of coinvariants of the QPB F GL q coincides with the one defined in (4.15).
Proposition 4.23.The Ore extension of the bicovariant 4-dimensional FODC (Γ, d) := (Γ GL , d GL ) of Example 3.9 gives a right H-covariant FODC (Υ GLq , d) on the QPB F GLq .Explicitly, the sheaf Υ GLq of right H-covariant F GL q -bimodules is defined by Υ GL q (∅) = {0}, Υ GL q (P 1 (C)) = Γ and Together with the pullback calculus on the sheaf O P 1 (C) of coinvariants and the bicovariant quotient calculus on H this gives a principal covariant calculus on F GL q .
Using the freeness of the presheaf Υ GL q it is not hard to check that Υ GL q (U 1 ∪ U 2 ) = Υ GL q (P 1 (C)), proving that Υ GLq is a sheaf of right H-covariant F GLq -bimodules.By Proposition 4.14 the extended differential d I : A I → Υ GL q (U I ) defines a right H-covariant FODC for each open U I ∈ B and it is straightforward to verify that this determines a morphism d : F GL q → Υ GL q of sheaves of right H-comodules.
We next prove that the sequences As a corollary, applying Theorem 4.10 we have that Υ P 1 (C) ∼ = Υ coH GLq ∩ Υ hor GLq are isomorphic sheaves.The sheaf F GL q is locally trivial since A I ∼ = B I #H according to (4.25).Considering the FODC on (Υ P 1 (C) , d P 1 (C) ) and the bicovariant calculus (Γ H , d H ) on H, we can construct locally the smash product calculus (Γ BI #H , d BI #H ) of Section 3.4, which can be pulled back to A I via ϕ I : A I → B I #H (the module of one forms being Γ BI #H since ϕ I are isomorphisms).However this local data does not give a FODC on F GL q .Indeed for that to be the case the restriction morphisms r Υ # IJ of the would be sheaf Υ # of covariant F GL q -bimodules would have to be left F GL q -linear and compatible with the differentials: Recalling that Γ BI #H are free left B I #H-bimodules it follows form straightforward computations that there is not such map r Υ # 12,2 : Γ B2#H → Γ B12#H .As a consequence we have that in this example the Ore extended differential calculi (Γ AI , d I ) on A I ∼ = B I #H are not isomorphic to the smash product calculi (Γ BI #H , d BI #H ).While the latter, from a bottom up approach, could seem to be the canonical choice, it turns out that it is the first one to provide a FODC on the sheaf F GL q .This is yet another instance of the non uniqueness of the differential calculus construction.

Corollary 3 . 8 .
Let π : A → H be a Hopf algebra epimorphism.Any right/left/bicovariant FODC (Γ, d) on the Hopf algebra A gives a right/left H-covariant or H-bicovariant FODC on A and projects to a right/left/bicovariant FODC (Γ π , d π ) on the Hopf algebra H. Proof.Here we prove the case of a bicovariant FODC (Γ, d) on A. The others are similarly proven.The left and right H-coactions on A defined by δ

P
By the commutativity of the square diagram in (3.17) and recalling that the modules of 1-forms are generated by the algebras and differentials we obtain the commutative cube

Proposition 3 .
13. (Γ B , d B ) is an H-module FODC if and only if Γ B is a relative H-module B-bimodule and dB : B → Γ B is an H-module map: for all h ∈ H, b ∈ B, h ⊲ d B b = d B (h ⊲ b).

Corollary 3 . 15 .
Let H be a Hopf algebra, B a left H-module algebra, (Γ B , d B ) an H-module FODC on B and (Γ H , d H ) a bicovariant FODC on H.The FODC (Γ # , d # ) of Theorem 3.14 is then right H-covariant.

Definition 3 . 16 .
Let (Γ A , d A ) be a FODC on a right H-comodule algebra A. We call the pullback calculus (Γ B , d B ) := (Bd A | B B, d A | B ) on B := A coH ⊆ A the FODC of base forms.We further call Γ hor := AΓ B , the (A, B)-bimodule of horizontal forms.If (Γ A , d A ) is a right H-covariant FODC, base and horizontal forms can be further characterized.Theorem 3.17.Let A be a principal comodule algebra, B := A coH and ( Cor. 5.53 for the equality Γ B = Γ hor A ∩ Γ coH A to hold true.We present two examples of principal covariant calculus, further examples including a principal calculus are in Section 4.5.Example 3.20 (Smash product calculus).Let (Γ H , d H ) be a bicovariant FODC on a Hopf algebra H and (Γ B , d B ) an H-module FODC on a left H-module algebra B. Then, the smash product calculus (Γ # , d # ) of Section 3.4 is a principal covariant calculus on B#H.Proof.Recall from Section 2.3.1 that B = (B#H) coH ⊆ B#H is a trivial Hopf-Galois extension.We show that the sequence in (3.26) is equivalent to the exact sequence 0 on a right H-comodule algebra A and (Γ H , d H ) a bicovariant FODC on H such that (3.31) is a faithfully flat Hopf-Galois extension of graded algebras and the diagram in (3.32) commutes, then A is a principal comodule algebra and (Γ A , d A ) a principal covariant calculus on A. Proof.If (Γ A , d A ) is a principal covariant calculus on A B = A coH ⊆ A is a faithfully flat Hopf-Galois extension, the sequence (3.26) is exact and by Lemma 3.22 Ω 1 i.) The linear mapsd I : F G (U I ) −→ Υ G (U I ) ,defined in (4.11) for A I = F G (U I ) and Γ I = Υ G (U I ), define a morphism of sheaves of right O q (P )comodules d : F G → Υ G .This gives a right O q (P )-covariant FODC (Υ G , d) on the sheaf F G .ii.)The FODC (Υ G , d) on the sheaf F G induces a FODC (Υ M , d M ) on the sheaf F coOq(P ) G = O M .

4. 5 . 1
First order differential calculus via Ore localization of 4D Calculus on O q (SL 2 (C)) ] respectively are isomorphic to C[u] and C[v] with u := α −1 γ and v := αγ −1 and they are the coordinate rings of the affine algebraic varieties corresponding to the opens U 1 , U 2 in SL 2 (C)/P ≃ P 1 (C).

0
→ A I ⊗ A Γ BI → Γ I verI −−→ A I H Γ H → 0 (4.27) are exact for all U I ∈ B. From Theorem 3.17 we have injectivity of the second arrow.Recall from Example 3.9 that the bicovariant quotient calculus (Γ H , d H ) on H is 3-dimensional (instead of 2-dimensional in the framework of Proposition 4.18) with left coinvariant basis {[ω 1 ], [ω 3 ], [ω 4 ]} and that by Lemma 3.19 the vertical map ver I : Γ I → A I H Γ H is well-defined for all U I ∈ B. Now take an arbitrary element a ℓ ω ℓ ∈ Γ I (sum understood).Because of freeness 0 = ver I (a ℓ ω ℓ ) = a ℓ 0 ⊗ a ℓ 1 [ω ℓ ] if and only if a 1 = a 3 = a 4 = 0. Thus ker ver I = span AI {ω 2 } = A I Γ BI .Surjectivity of ver I is verified as in Proposition 4.22.Hence the sequences (4.27) are exact and (Υ GL q , d) is a principal covariant calculus on F GL q .
and hence we simply write m −1 ⊗ m 0 ⊗ m 1 .If M is an H-bicomodule then M coH and coH M are left and right H-subcomodules, respectively.A right H-comodule morphism is a k-linear map φ : M → N between right H-comodules (M, ∆ M ) and (N, ∆ N Note that in(2.4)thetensor product M coH ⊗ H is a right H-covariant right H-module with respect to the diagonal coaction and the right H-action (m ⊗ h) • h ′ := m ⊗ (hh ′ ), where m ∈ M coH and h, h ′ ∈ H.If M is furthermore a bicovariant H-bimodule we endow M coH ⊗ H with the diagonal left H-coaction and the left H-action h • (m ⊗ h ′ ) := h 1 mS(h 2 ) ⊗ h 3 h ′ .Similarly in (2.5), H ⊗ coH M becomes a left H-covariant left H-module via the diagonal coaction and the left H-action given by h • (h ′ ⊗ m) := (hh ′ ) ⊗ m, where h, h ′ ∈ H and m ∈ coH M .If M is furthermore a bicovariant H-bimodule we endow H ⊗ coH M with the diagonal right H-coaction and the right H-action )of left H-covariant left H-modules.If M is a bicovariant H-bimodule, (2.5) extends to an isomorphism of bicovariant H-bimodules.
bimodule and d is right H-colinear.In this case ∆ R is determined by (3.6).ii.)A FODC (Γ, d) on a left H-comodule algebra (A, δ L ) is left H-covariant if and only if (Γ, ∆ L ) is a left H-covariant A-bimodule and d is left H-colinear.In this case ∆ L is determined by (3.7).
Theorem 3.14.Let H be a Hopf algebra and B a left H-module algebra.Given an H-module FODC (Γ B , d B ) on B and a left covariant FODC (Γ H , d H ) on H there is a FODC (Γ # , d # ) on B#H, where the k-module Γ # := Γ B ⊗ H ⊕ B ⊗ Γ H is endowed with the B#H-bimodule actions (3.22), (3.23) and the exterior derivative d now consider a FODC (Γ H , d H ) on H, with (Γ A , d A ) that is not necessarily right H-covariant.Definition 3.18.A FODC (Γ A , d A ) on a principal comodule algebra A together with a left covariant FODC (Γ H , d H ) on H, is called a principal calculus on A if we have the short exact sequence 0 [5]]6.26).The 3-dimensional left covariant FODC on A introduced in[48]is H-bicovariant.It induces a 2-dimensional FODC on B via the algebra embedding B → A and a 1-dimensional bicovariant FODC on H via the quotient map π.This data is a principal covariant calculus on A, see[5]Ex.5.51.
1 B -module.Moreover, the diagram (3.32) commutes by the right H-covariance of (Γ A , d A ) and the definition (3.27) of the vertical map.
Conversely, if A is a right H-comodule algebra with a FODC (Γ A , d A ) and a bicovariant FODC (Γ H , d H ) on H such that