arxiv:q-alg/9707022v1 16 jul 1997 · 1. introduction let h be a ﬁnite dimensional hopf algebra...

arX

iv:q

-alg

/970

7022

v1 1

6 Ju

l 199

7

ALGEBRAIC GEOMETRY OF HOPF-GALOIS

EXTENSIONS.

DMITRIY RUMYNIN

Abstract. We continue the investigation of Hopf-Galois exten-sions with central invariants started in [29]. Our objective is notto imitate algebraic geometry using Hopf-Galois extension but tounderstand their geometric properties.

Let H be a finite-dimensional Hopf algebra over a ground field k.Our main object of study is an H-Galois extension U ⊇ O such that Ois a central subalgebra of U . Let us briefly discuss geometric propertiesof the object. By Kreimer-Takeuchi theorem the module UO is projec-tive. Thus, it defines a vector bundle of algebras on the spectrum of Oby Serre theorem. The fibers carry a structure of Frobenius algebra. Asimilar structure was of interest to geometers for a while because com-mutative Frobenius algebras naturally arise in the study of symmetricPoisson brackets of hydrodynamical type [1]. More recently, a conceptof Frobenius manifold was introduced [15]; it is a manifold such thattangent spaces carry a structure of a commutative Frobenius algebrawhich multiplication has a generating function.Our set-up is different: we have a vector bundle rather than the tan-

gent bundle and our algebras are not necessarily commutative. How-ever, we have more structure involved: a Hopf-Galois extension maybe regarded as a “quantum” principal bundle [30, 8]. If H is commu-tative (i.e. an algebra of functions on a finite group scheme G) thena commutative H-Galois extension U ⊇ O is a G-principal bundle onthe spectrum of O.Finally, we emphasize that centrality of invariants is a crucial prop-

erty for a geometrical treatment of Hopf-Galois extensions. We willillustrate this claim throughout the paper.We introduce Hopf-Galois extensions with central invariants and dis-

cuss examples in Section 1. The notion of inverse image of a Hopf-Galois extension is discussed in Section 2. A geometric object shouldbecome trivial if one looks at it locally. We prove a weak version of thisprinciple in Section 3. A Hopf-Galois extension starts enjoying being

Date: July 15, 1997.1991 Mathematics Subject Classification. Primary 16W30; Secondary 14F05.

1

http://arxiv.org/abs/q-alg/9707022v1

2 DMITRIY RUMYNIN

a geometrical object on an affine scheme in the next section where weshow that it may be pasted from local datum. The next three sections(5,6,7) are devoted to investigation of various features a Hopf-Galoisextension carries. In section 8 we give a definition of Hopf-Galois ex-tension (or H-torsor) over any scheme.The author is grateful to A. Mavlyutov, I. Taimanov, I. Mirkovic

and E. Markman for the fruitful discussions.

1. Introduction

Let H be a finite dimensional Hopf algebra over a ground field k

from now on. One could extend a number of results to the case ofa ground ring but it is beyond our interest in the present paper. Anassociative algebra with unity U is an H-comodule algebra if there isa map ρ : U −→ U ⊗ H , written as rho(x) = x0 ⊗ x1 such thatx0ε(x1) = x, x0 ⊗ ∆(x1) = ρ(x0) ⊗ x1, and ρ(xy) = ρ(x)ρ(y) for allx, y ∈ U . We use the oxymoronic variant of the Sweedler’s Σ-notationwith Σ eliminated. The subalgebra O = UH = {x ∈ U | ρ(x) = x⊗ 1}is called the subalgebra of invariants; one says that U ⊇ O is an H-extension. Throughout the paper we make various assumptions on Oas finitely generated, reduced (semiprime), or affine (finitely generatedsemiprime). An H-extension is an extension with central invariants ifO is contained in the center of U . An H-extension is Hopf-Galois (orspecifically H-Galois) if the canonical map can:U ⊗O U −→ U ⊗ Hdefined by can(x⊗O y) = (x⊗ 1)(y0 ⊗ y1) is a bijection. [18, 24].Let us discuss some examples of Hopf-Galois extensions. A Galois

extension of fields K ⊇ k with the Galois group G is Hopf-Galois forthe Hopf algebra kG∗. Geometrically, it means that the point over Kis a G-principal bundle on the point over k. The vector bundle whichstructure it carries has the k-vector space K as a fiber at the point.However, a Hopf-Galois extension of fields is not necessarily Galois.

A purely inseparable extension may be Hopf-Galois for the dual of arestricted universal enveloping algebra [24]. Moreover, a finite separa-ble not normal field extension may be Hopf-Galois as well [12]. Fromthe geometric prospective it means that the point over K is a principalbundle on the point over k for some finite non-reduced group scheme.A criterion for a finite separable field extension to be Hopf-Galois

was proven in [12]. One should consider the normalization K of K

over k. Let S(M) be the symmetric group of the set M = Gal(K |k)/Gal(K | K). There is a natural embedding Gal(K | k) → S(M).The extension is Hopf-Galois if and only if there exists a subgroup

ALGEBRAIC GEOMETRY OF HOPF-GALOIS EXTENSIONS. 3

N ⊆ S(M), normalized by Gal(K | k), such that the left N -set M isisomorphic to the left regular N -set N .Interesting Hopf-Galois extensions may be constructed as crossed

products. More precisely, crossed products constitute all cleft Hopf-Galois extensions. An inquisitive reader should consult [24] for a niceaccount of the subject. We do not need the general construction. Werestrict our attention to twisted products, crossed products with thetrivial action, because only they produce central invariants. Let R bea commutative k-algebra and σ : H ⊗ H −→ R be a linear map. LetRσ[H ] be a vector space R ⊗ H with an algebra structure defined bythe formula

a⊗ g · b⊗ h = abσ(h1, g1)⊗ h2g2 , a, b ∈ R, g, h ∈ H (1)

It may not be a structure of an associative algebra in general. Thefollowing lemma is straightforward [24, Lemma 7.12].

Lemma 1. The formula 1 defines a structure of an associative algebrawith an identity element 1⊗ 1 if and only if for each g, h, t ∈ H

σ(h1 ⊗ g1)σ(h2g2 ⊗ t) = σ(g1 ⊗ t1)σ(h⊗ g2t2)

σ(h⊗ 1) = σ(1⊗ h) = ε(h)1

If σ is invertible with respect to the convolution [24] and satisfies theconditions of Lemma 1 we call it a cocycle, although a correspondingcochain complex has been constructed only for cocommutative H [33].We use the term twisted product for Rσ[H ] if σ is a cocycle. Thenatural question is what is happening if the map σ is not invertible butnevertheless formula 1 defines a structure of an associative algebra.It may happen if there is a family of cocycles σt converging in anappropriate sense to a linear map σ0; the latter inherits the identitiesof Lemma 1 but can fail to be convolution invertible. In other words,it is a closed condition to define an associative algebra but it is anopen one to be invertible. For such a “non-invertible cocycle” one stillgets an H-comodule algebra which is not Hopf-Galois. However, animportant piece of the structure theory breaks down in this situation.

Lemma 2. Assume that U ⊇ O is an H-extension. The followingstatements are equivalent1. U is isomorphic to a twisted product Oσ[H ] as an H-comodule

algebra.2. There exists a convolution invertible right H-comodule map γ :

H −→ U and the subalgebra O is central.3. There is a linear map from U to O⊗H conducting an isomorphism

of left O-modules and right H-comodules and O is central.

4 DMITRIY RUMYNIN

4. U is a free O ⊗H∗-module of rank 1 and UH is central.

Proof. The equivalence of 1 through 3 follows from Theorems 7.2.2and 8.2.4 of [24] and Lemma 11 of [29]. 4 follows from 1 because 1#Λfor a non-zero left integral Λ is clearly a basis of the left O⊗H∗-moduleOσ[H ]. Finally, 4 implies 3 since O⊗H is a free left O⊗H∗-module ofrank 1. Thus, there exists an O⊗H∗-module isomorphism between Uand O⊗H which must be an isomorphism of left O-modules and rightH-comodules. ✷Now it is clear why only crossed products with the trivial action are

of interest to us: an action must be trivial to produce central invariants.Let us look at an example. We define the following cocycle on CZn.

Denoting a generator of Zn by t we set for each 0 ≤ k,m < n

σ(tk ⊗ tm) =

{

1 if k +m < nq if k +m ≥ n

(2)

It is a cocycle if and only if q ∈ O is an invertible element. The firstchoice may be q ∈ O = C[q, q−1]. The twisted product C[q, q−1]σ[CZn]is the quantum cohomology ring of the projective space CP

n−1. Themultiplication coefficients of it in an appropriate basis are 3-pointGromov-Witten invariants. One should consult [23] for a good accountof the subject. Another choice one can make is q = 0 ∈ C. The corre-sponding ring Cσ[CZn] is the de Rham cohomology ring of CPn−1. It isan example of a “non-invertible twisted product”: it may be thoughtof as a limit of twisted products depending on q ∈ C \ {0} as q → 0.One can do another choice of a ring: let q = ex1 ∈ C∞(Cn). Afteridentification tk−1 = ∂

∂xkone gets a structure of Frobenius manifold on

Cn [15].This example seems to demonstrate that one can encounter an in-

teresting geometry studying Hopf-Galois extensions with central in-variants. From now on U ⊇ O is an H-Galois extension with centralinvariants unless a contrary is specified. Let Λ be a non-zero left in-tegral of H∗. By the definition Λ · x = x0Λ(x1). The following factswere first proven in [18, 1.7, 1.9, 1.10]. One may find an interestingdiscussion concerning the last two properties in [4].

Lemma 3. The following statements hold under our assumptions.1. UO is a projective finitely generated module.2. UO is faithfully flat.3. There exists an O submodule P ⊆ U such that U = O ⊕ P .4. There exists an element x ∈ U such that Λ · x = 1.

We can now explain why centrality of invariants is essential. Onemay hope, for instance, that O being commutative suffices to build


an interesting geometry. Indeed, there is a one-to-one correspondencebetween algebraic vector bundles on the spectrum of O and finitelygenerated projective modules by Serre theorem [31, Corollary 2.4.50].If O is not central then left and right actions are different. Thus, we geta bi-bundle (i.e. a space carrying two commuting structures of a vectorbundle) rather than a vector bundle. Geometry of bi-bundles is lessexploded than that of vector bundles. Thus, if we want to remain inthe realm of vector bundles we should require invariants being central.In further sections we will see more properties which do not work forHopf-Galois extensions with non-central invariants.We finish the introduction with the lemma which is handy if one

wants to test whether an H-extension is Hopf-Galois. It is a slightgeneralization of [18, Lemma 1.3].

Lemma 4. If there exists a subalgebra A ⊆ O such that the map Γ :U ⊗A U −→ U ⊗ H defined by Γ(x ⊗O y) = (x ⊗ 1)(y0 ⊗ y1) is ontothen U ⊇ O is H-Galois.

Proof. The map Γ factors through the canonical map.

U ⊗O U U ⊗H✲can

U ⊗A U

✻

✑✑✑✑✑✑✸

Γ

Thus, the canonical map is onto. It must be also one-to-one by [24,Theorem 8.3.1.].✷

We should notice that Γ being one-to-one in the last lemma does notimply A = O. A proper embedding A → O may be an epimorphism inthe category of rings making the tensor products U ⊗O U and U ⊗A Uindistinguishable.

2. Pull-back.

This section is devoted to the crucial geometric property of Hopf-Galois extensions with central invariants, the pull-back. Intuitively,given a map of schemes X −→ Y and a geometrical object on Y , weshould be able to induce a similar object using some kind of the fiberproduct. So far, we can work only on the level of rings. The followinglemma provides the existence of pull-back on the level of rings. Thisproperty fails for Hopf-Galois extensions with non-central invariants.

6 DMITRIY RUMYNIN

Lemma 5. Let φ : O → R be a map of commutative algebras. ThenU = U ⊗O R ⊇ R is an H-Galois extension with central invariants.

Proof. The natural map R→ U is one-to-one by Lemma 3. Indeed,U = O ⊕ P as O-modules and a split extension is pure. In down-to-earth terms, U = (O ⊕ P )⊗O R = O ⊗O R⊕ P ⊗O R ⊇ O ⊗O R ∼= R.The coaction of H on U is given by (u ⊗O r)0 ⊗ (u ⊗O r)1 = (u ⊗O

r0) ⊗ r1. Let S be the subalgebra of invariants UH . It is clear thatS ⊇ R. Let us consider the diagram.

U ⊗S U U ⊗H ∼= S ⊗O (U ⊗H)✲can

U ⊗O U

✻

✑✑✑✑✑✑✸

Γ

The map Γ is onto since the original extension is Hopf-Galois. ByLemma 4 the extension U ⊇ S is H-Galois.It remains to show that S = R. Let x ∈ U be an element such that

Λ · x = 1 which existence is provided by Lemma 3. It is clear thatΛ · (x ⊗O 1) = 1 ⊗O 1 = 1 ∈ R. It implies Λ · U = UH . But Λ · U ⊆Λ · ((U ⊗O 1)R) ⊆ R(Λ · U ⊗O 1) ⊆ R((Λ · U)⊗O 1) ⊆ R(O ⊗O 1) ⊆ R✷

The module UO defines a vector bundle of algebras U on Spec O[29]. Let us denote by O(χ) the local ring of algebraic functions atχ ∈ SpecO, mχ its maximal ideal, and Kχ the quotient field O(χ)/mχ.We remind the main characters of the play.The germ algebra may be defined by

U(χ) = U ⊗O O(χ) ⊇ O(χ). (3)

Here is another way to describe it. S = O \χ is a multiplicative subsetof U . The generalized algebra of quotients US−1 is isomorphic to U(χ).The fiber algebra is

Uχ = U ⊗O Kχ ⊇ Kχ. (4)

One may introduce the n-jet algebra generalizing the germs and thefibers. It is important for the study of indecomposable modules in thespirit of [28].

U (n)χ = U ⊗O O(χ)/m

nχ ⊆ O(χ)/m

nχ. (5)


It is obvious that U (n)χ∼= U(χ)/U(χ)m

nχ. Conventionally, m

∞χ = 0. Thus,

the algebra of ∞-jets U (∞)χ is the same as the germ algebra U(χ). An-

other interesting choice is n = 1: U (1)χ is the fiber algebra Uχ at the

point χ. The last algebra we want to introduce is the restriction to theclosure of a point χ:

U[χ] = U/Uχ ⊇ O/χ.

If the point χ is closed then U[χ] is isomorphic to Uχ.Now we can prove the main theorem of the section improving The-

orems 15 and 16 of [29].

Theorem 6. UH[χ] ⊇ O/χ and U (n)

χ ⊇ Oχ/mnχ are H-Galois extensions

for each χ and n (including n = ∞). If the extension U ⊇ O is cleftthen so are the jet extensions and the restrictions.

Proof. The first statement is an immediate corollary of Lemma5. To prove the second statement we assume that γ : H −→ U is asplitting map. Then the composition H

γ−→ U −→ U (n)

χ is a splitting

map for the jet extension U (n)χ ⊇ O(χ)/m

nχ. Similarly, the composition

Hγ−→ U −→ U/Uχ is a splitting map for the restriction extension

U/Uχ ⊇ O/χ. ✷This theorem is not the best statement one can get. We will prove

in the next section that the jet algebras are always cleft for n <∞ andgenerically cleft for n = ∞ under mild restrictions on O. Let us giveone more definition. Let GALH(O) be a set of isomorphism classes ofH-Galois extensions U ⊇ O. We understand an isomorphism of H-comodule algebras by an isomorphism. Following [29], we denote thesubset of isomorphism classes of cleft H-Galois extensions by GalH .The following theorem is an immediate consequence of Lemma 5. Itimproves [29, Theorem 14].

Theorem 7. GALH is a covariant functor from the category of ringsto the category of sets and GalH is its subfunctor.

3. Local structure.

Usually if one looks at a geometrical object locally it seems trivial. Atrivial H-Galois extension is the tensor product O⊗H ⊇ O. It is falsethat anH-Galois extension is locally trivial: if U ⊇ O is a proper Galoisextensions of fields with Galois group G then O is already localized,nevertheless U is not isomorphic toO⊗kG∗. However, one could expectthat a Hopf-Galois extension is locally cleft. We don’t know whether itholds. The following question is interesting from this prospective. In

8 DMITRIY RUMYNIN

this section we prove a weaker version that a Hopf-Galois extension isgenerically locally cleft.Question. Let U ⊇ O be an H-Galois with central O and finite-

dimensional H . Assume that O is a local algebra. Is the extensioncleft?First, we want to understand what is happening if the vector bundle

defined by a Hopf-Galois extension is trivial. If we managed to showthat this implied that the Hopf-Galois extension were cleft it would bevery nice. Indeed, any vector bundle is locally trivial. The followingfact is the best we can do. It first appeared in [2, Lemma 2.9]. Weshould mention that trivial bundles correspond to free modules on thealgebraic side.

Lemma 8. If UO is a free module of rank n then Un is a free leftO ⊗H∗-module of rank n.

Proof. Using the canonical map we get a chain of isomorphisms:

Un ∼= U ⊗O U ∼= U ⊗H ∼= U ⊗O (O⊗H) ∼= (O⊗H)n ∼= (O⊗H∗)n ✷

Thus, U is a projective module of rank 1 if the rank is well-defined forprojective O ⊗H∗-modules. The next idea is to put some constraintson O and H to ensure that any projective O ⊗ H∗-module of rank 1is free. A reader may supplement the next corollary with other similarstatements providing cleftness.

Corollary 9. U is cleft if one of the following conditions holds:1. O is artiniian and UO is free;2. O is a local artiniian ring;3. O is a field;4. O ⊗H∗ is self-injective and UO is free;5. SpecO is a toric variety and O ⊗H∗ is self-injective.

Proof. Each of the conditions 1-5 implies the module UO is free. Itis explicitly stated in 1 and 4. Any finitely generated projective moduleover local ring or field is free. Algebra of functions on a toric varietyis a semigroup algebra of a normal monoid. A finitely generated pro-jective module is free over such an algebra according to the Andersonconjecture proved in [14, Theorem 2.1].Using Lemma 8, it suffices to understand some kind of uniqueness

of decomposition theorem in each of the cases. In 1, 2, and 3 one canuse the Krull-Remark-Schmidt theorem [9, 5.18.11]. Indeed, O ⊗ H∗

is artiniian if so is O: since H is finite dimensional O ⊗H∗ is even anartiniian O-module. Thus the O ⊗ H∗-module Un must have a finitelength which implies the uniqueness of the decomposition.


We can use a stronger version of the Krull-Remark-Schmidt theoremin 4 and 5 [9, 5.19.18]. Un is injective O ⊗ H∗-module in this case;so are its direct summands. The endomorphism ring of an injectiveindecomposable module is local and we can use the Krull-Remark-Schmidt theorem. ✷Now we can state and prove the local trivialization theorem.

Theorem 10. The germ extension is cleft for generic χ ∈ Spec O.If O is finitely generated then the n-jet extension is cleft for everyχ ∈ Spec O and n <∞.

Proof. The second statement follows from Corollary 9.2 since then-jets of functions O(χ)/m

nχ is a local artiniian ring (even finite dimen-

sional) in the case of finitely generated O.The first statement can be reduced to a component of the spectrum:

let I be a minimal prime ideal of O. We replace the extension U ⊇ Owith U ⊗O O/I ⊇ O/I. Thus, we can assume O is a domain withoutloss of generality. This argument becomes sloppy unless O is finitelygenerated. Otherwise, no minimal prime ideal can exist. However,the term generic is unclear in this case. Thus, we can make up thedefinition of a generic point and think that the theorem still holds. Forexample, generic could mean generic in closure of each point: it entitlesus to replace O with O/I where I is an arbitrary prime ideal.We should use the field fractions Q(O) to comprehend the first state-

ment. Geometrically, it is a field of rational functions on the spec-trum. By Corollary 9.3 the extension U ⊗O Q(O) ⊇ Q(O) is cleft. Letγ : H −→ U ⊗O Q(O) be a splitting map. The algebra U ⊗O Q(O) canbe realized as a generalized ring of quotients of U by the multiplicativeset O \ {0}. Let hi be a base of H . Let V be the complement of zeroesof the denominators of γ(hi). It is obvious that U(χ) ⊇ O(χ) is cleft foreach χ ∈ V ✷

4. Pasting.

We do algebraic geometry of H-Galois extensions on affine schemeson the level of rings till Section 8 where we sheafify the enterprise. AfterSection 2, given an H-Galois extension of an affine scheme X and anopen cover ∪Vi of X , we can restrict the extension to each set Vi. Nowwe are interested in the inverse problem: given H-Galois extensionson each Vi, compatible on intersections, we want to patch an H-Galoisextension on X .We will need the following fact. We consider two sets T , I and two

maps ∗, ⋆ : T → I. We assume we have an H-extension Ui ⊇ Oi for

10 DMITRIY RUMYNIN

each i ∈ I and an H-comodule algebra morphism φt : Ut∗ −→ Ut⋆ foreach t ∈ T .

Lemma 11. Let O be an inverse limit of algebras Oi and U be aninverse image of Ui. Then U ⊇ O is an H-extension. If all Ui possesscentral invariants then so does U .

Proof. We may think that U = {(xi) ∈∏

Ui | φt(xt∗) = xt⋆}. It isclearly a subalgebra. Thus, it has a natural algebra structure.Clearly, φt(Ot∗) ⊆ Ot⋆ and, therefore, O is well-defined subalgebra

of U . It is central provided so are Oi.It is simpler to write a formula forH∗-action rather than H-coaction:

h · (xi) = (h · xi). It is well-defined since φt are homomorphisms of H-comodule algebras. The formula for the coaction may be obtained bydualizing. It is obvious that O is a subalgebra of invariants of U underthis action. ✷We see no reason why a limit of H-Galois extensions should be an

H-Galois extension. However, it holds for a special kind of limits.Let us be given a commutative k-algebra O, a finite set of elements{fi} of O such that

∑

iOfi = O, and an H-Galois extension withcentral invariants Ui ⊇ Oi for each i where Oi denotes the localizationOf−1

i . Let Oij be the localization Of−1i f−1

j and Oijk be the localization

Of−1i f−1

j f−1k . Our datum gives rise toH-Galois extensions Uij = Ui⊗Oi

Oij ⊇ Oij. We assume there are given isomorphisms of H-comodulealgebras φi,j : Uij → Uji such that φi,j = φ−1

j,i and φ′j,k ◦ φ

′i,j = φ′

i,k

for each i, j, k where φ′i,j is an induced isomorphism from Ui ⊗Oi

Oijk

to Uj ⊗OjOijk. We also require that φi,j restricted to Oij is a natural

isomorphism with Oji. The datum (fi, Ui, φi,j) will be called an H-structure on the algebra O.We prove that an H-structure on O is generated by a unique H-

Galois extension. In other words, we prove that the functor GALH islocal [16]. Sometimes, a local functor is called a sheaf. We will needthe following technical lemma.

Lemma 12. Let σ : N → M be a homomorphism of O-modules. σ isinjective (surjective, isomorphism) if and only if σ is locally injective(locally surjective, local isomorphism).

Locally injective means it becomes injective after localization to someneighborhood of every point of the spectrum. To prove the lemma oneshould look at the support of the kernel of σ. It must have an emptysupport since σ is locally injective but the only module with trivialsupport is a zero module. Indeed, the annulator of a non-zero module


B is contained in a maximal ideal I. Thus, supp B = Z(AnnOB) ⊇Z(I) = {I}. This lemma may be also found in [22, Corollary 4.7].

Theorem 13. For each H-structure on an algebra O there exists aunique H-Galois extension U ⊇ O such that U ⊗O Oi

∼= Ui.

Proof. Oi and Oij form a directed system of algebras which inverseimage is O. By Lemma 11 the inverse limit U of Ui is an H-extension ofO. By [22, Exercise 4.C.13] U is a projective O-module which locallylooks like Ui. To be precise, the natural maps ψi : U ⊗O Oi → Uiare isomorphisms. It may be better understood in terms of vectorbundles: Ui define vector bundles on open sets of the covering of thespectrum which can be pasted to a vector bundle on the spectrumwhich is presented by the projective module U .We are ready to show that U ⊇ O is H-Galois. The canonical

map can : U ⊗O U → U ⊗ H is an O-module map. Moreover, thelocalization U ⊗O Uf

−1i is naturally isomorphic to Ui⊗O Ui ∼= Ui⊗Oi

Uiand U ⊗ Hf−1

i is naturally isomorphic to Ui ⊗ H . The localizationcanf−1

i of the canonical map becomes the canonical map of Ui underthese isomorphisms. Thus, the canonical map is locally isomorphismand, therefore, an isomorphism by Lemma 12.Let U be another extension with given properties. Then there is a

map of H-comodule algebras π : U −→ U . In particular, π is a mapof projective O-modules. Restricted to each open set of the covering,π becomes an isomorphism. By Lemma 12 π must be an isomorphism.✷

Corollary 14. GALH is a sheaf.

A natural question is whether GalH is a subsheaf of GALH . Weexpect that the answer is no. Otherwise, one must show that if each Uiis cleft then so is U . Thus, given splittings γi : H → Ui, one has to gluea splitting γ : H → U . It can be done if γi agree on “intersections” Uijbut it is unclear how to do it without any restrictions on behavior onintersections.

5. Frobenius form.

A Frobenius manifold has a Riemannian metric as a part of its struc-ture. Similarly, Hopf-Galois extensions carry a canonical (up to ascalar) non-degenerate associative form which can fail to be symmetric.Chosen Λ, a non-zero left integral of H∗, we construct an O-bilinearform 〈, 〉 : U × U −→ O by 〈x, y〉 = Λ · (xy) = x0y0Λ(x1y1) for eachx, y ∈ U .

Lemma 15. [18, 1.7.5] The for 〈, 〉 is non-degenerate.

12 DMITRIY RUMYNIN

Non-degeneracy means that the map ζ : U −→ Hom(UO, O) givenby ζ(u)(v) = 〈u, v〉 is an isomorphism. The following corollary is im-mediate from Lemma 15, Theorem 6, and the definition of a Frobeniusalgebra.

Corollary 16. The algebras Uχ are Frobenius Kχ-algebras.

The question when this form is symmetric is quite interesting. Thefollowing theorem is a sufficient condition for symmetricity. It is aslight generalization of [29, Theorem 17].

Theorem 17. Let H be unimodular with the antipode of order 2. Theform 〈, 〉 is symmetric if one of the following holds:1. U ⊇ O is cleft.2. O is semiprime.

Proof. Let us treat the cleft case first. We assume that U is iso-morphic to a twisted product Oσ[H ]. We notice that h1Λ(h2) = Λ(h)1for each h ∈ H . Indeed, the equality α(h1Λ(h2)) = α(Λ(h)1) holdsfor each α ∈ H∗ since α(h1Λ(h2)) = α(h1)Λ(h2) = αΛ(h) = α(1)Λ(h).It was proved in [26] that H is unimodular with the antipode of or-der 2 if and only if Λ(xy) = Λ(yx) for each x, y ∈ H . Therefore,〈a ⊗ h, b ⊗ g〉 = abh1g1Λ(h2g2) = abΛ(hg) = baΛ(gh) = 〈b ⊗ g, a⊗ h〉for each b⊗ g, a⊗ h ∈ U .The element 〈u, v〉 − 〈v, u〉 belongs to every maximal ideal for each

u, v ∈ U by Theorem 10 and the consideration above. If O is semiprimethe intersection of maximal ideals (i.e. the radical) is zero. Thus, theform is symmetric. ✷The following question seems to be of interest. Assume that the

ground field k is formally real, i.e. zero cannot be presented as a sumof squares. Under which conditions is the form positive definite. OverR it is a question of having a Riemannian structure.

Proposition 18. Let k be formally real. If the form is positive definitethen H is cosemisimple.

Proof. 0 < 〈1, 1〉 = Λ(1)1 Thus, Λ(1) 6= 0 and H∗ is cosemisimpleby the Sweedler criterion. ✷We may also notice that a formally real field has zero characteristic.

Thus, H must be as well semisimple by the Larson-Radford theorem.The form allows us to introduce two more features. The map C :

U ⊗O U ⊗O U → k given by C(u⊗ v ⊗ w) = 〈uv, w〉 encodes the mul-tiplication. There is also the Nakayama automorphism [25] defined by〈u, v〉 = 〈v,Nak(u)〉. It measures the failure of the form to be sym-metric. In particular, Nak is inner if and only if U admits a symmetricassociative bilinear form.


6. Connections.

The study of connections is an important part of modern geometry.We don’t have any particular reason to study connections on Hopf-Galois extensions but we hope that it may be useful in the furtherdevelopment of the subject. We wish to have an existence and unique-ness theorem for a some kind of connections on Hopf-Galois extensionssimilar to the existence and uniqueness of the torsion-free Riemannianconnection on the tangent bundle of a Riemannian manifold. We don’tsucceed but we collect some facts in this direction.We study bilinear pairings ∇ : L × U −→ U where L is the Lie

algebra of derivations of O (vector fields on the spectrum). We denotethe result of the pairing by ∇Xu with X being a vector field. ∇ iscalled a connection if

∇X(au) = X(a)u+ a∇Xu and ∇aXu = a∇Xu.

for each a ∈ O, X ∈ L(O), and u ∈ U . A connection ∇ is called aFrobenius connection if

X〈u, v〉 = 〈∇Xu, v〉+ 〈u,∇Xv〉.

A connection ∇ is called multiplicative if

∇X(uv) = (∇Xu)v + u∇Xv.

A connection ∇ is called equivariant if

∇Xu0 ⊗ u1 = (∇Xu)0 ⊗ (∇Xu)1

A connection ∇ is called Nakayama if

Nak−1(∇XNak(u)) = ∇Xu.

Roughly speaking Frobenius connection is the one along which theform is covariant constant. Indeed, being Frobenius is equivalent to∇X〈, 〉 = 0. A multiplicative connection may be thought of as a wayto extent derivation of O to derivations of U . The properties we havejust defined are not independent.

Proposition 19. A multiplicative equivariant connection is Frobenius.A Frobenius connection is Nakayama.

Proof. For each u, v ∈ U we have X(〈u, v〉) = ∇X(u0v0Λ(u1v1)) =

∇X(u0)v0Λ(u1v1) + u0∇X(v0)Λ(u1v1) + u0v0X(Λ(u1v1)) =

= (∇Xu)0v0Λ((∇Xu)1v1)+u0(∇Xv0)Λ(u1(∇Xv)1) = 〈∇Xu, v〉+〈u,∇Xv〉

which proves the first claim. To show the second one it suffices tonotice the following sequence of equalities for all u, v ∈ U and X ∈ L

〈∇Xu, v〉 = X(〈u, v〉)−〈u,∇Xv〉 = X(〈v,Nak(u)〉)−〈∇Xv,Nak(u)〉 =

14 DMITRIY RUMYNIN

= 〈v,∇XNak(u)〉 = 〈Nak−1(∇XNak(u)), v〉 ✷

The next three propositions deal with the issue of existence of aconnection with certain properties.

Proposition 20. A connection always exists.

Proof. U is direct summand of a free O-module On. We identify Uwith the submodule of On. Let π : On → U be a projection. Let ei bea base of On. The free module admits a trivial connection making eicovariant constants. We want to restrict this connection to U . Givenu = uiei ∈ U , we define ∇Xu = π(X(ui)ei). The map ∇ is obviouslya connection: for instance, ∇Xau = π(X(aui)ei) = π(X(a)uiei) +π(aX(ui)ei) = X(a)π(uiei) + aπ(X(ui)ei) = X(a)u + a∇Xu for X ∈L, a ∈ O, u ∈ U . ✷

Proposition 21. If the characteristic of k is not 2 and there exists aNakayama connection then there exists a Frobenius connection.

Proof. Let ∇ be a Nakayama connection. Let us define T by〈TXu, v〉 = X(〈u, v〉)−〈∇Xu, v〉−〈u,∇Xv〉. Then 〈u, TXv〉 = 〈TXv,Nak(u)〉 =X(〈v,Nak(u)〉)−〈∇Xv,Nak(u)〉−〈v,∇XNak(u)〉 = X(〈u, v〉)−〈u,∇Xv〉−〈Nak−1(∇XNak(u)), v〉.It is easy to show that ∇ = ∇ + 1

2T is a connection: 〈∇aXbu, v〉 =

〈∇aXbu, v〉+12[aX(〈bu, v〉)−〈∇aXbu, v〉− 〈bu,∇aXv〉] = ab〈∇Xu, v〉+

aX(b)〈u, v〉+ 12[aX(b)〈u, v〉+abX(〈u, v〉)−aX(b)〈u, v〉−ab〈∇Xu, v〉−

ab〈u,∇Xv〉] == ab〈∇Xu, v〉+ aX(b)〈u, v〉+ 12ab〈TXu, v〉).

∇ is Frobenius: 〈∇Xu, v〉 + 〈u, ∇Xv〉 = 〈∇Xu, v〉 +12[X(〈u, v〉) −

〈∇Xu, v〉−〈u,∇Xv〉]+〈u,∇Xv〉+12[X(〈u, v〉)−〈∇Xu, v〉−〈u,∇Xv〉]. ✷

The next lemma describes another trick which may be performedwith connections. It may be restated that a certain affine hyperplaneof H∗ is acting on the space of connections.

Lemma 22. Let h be an element of H∗ such that h(1) = 1 and ∇ bea connection. The pairing h ·∇ defined by (h ·∇)Xu = h1 · (∇XSh2 ·u)is a connection.

Proof. It is clear that (h·∇)fXu = h1 ·(∇fXSh2 ·u) = h1 ·(f∇XSh2 ·u) = (h1 · f)h2 · (∇XSh3 · u) = f(h · ∇)Xu for each X ∈ L, u ∈ U .It suffices to check (h ·∇)Xfu = h1 · [∇X(Sh2 ·fu)] = h1 · [∇X((Sh3 ·

f)(Sh2 ·u))] = h1 · [∇Xf(Sh2 ·u)] = h1 · [X(f)(Sh2 ·u)+f∇X(Sh2 ·u)] =h1 · [X(f)(Sh2 · u)] + h1 · [f∇X(Sh2 · u] = h(1)X(f)u+ f(h · ∇)Xu =X(f)u+ f(h · ∇)Xu. ✷

We remind that H being cosemisimple is equivalent to Λ(1) 6= 0for some integral Λ. Without loss of generality we may assume that


Λ(1) = 1. This allows us to integrate connections: given ∇, we obtainΛ · ∇. We also remind that in zero characteristic a cosemisimple Hopfalgebra is involutive and unimodular (and even semisimple) [19, 20].

Proposition 23. If H is cosemisimple then Λ · ∇ is an equivariantconnection for any connection ∇. If ∇ is multiplicative then so is Λ ·∇provided H is unimodular and involutive.

Proof. We utilize [27, Formula 3] with a = S−1(h) ∈ H∗ to provethe first statement:

hΛ1 ⊗ Λ2 = Λ1 ⊗ S−1(h)Λ2.

Let us apply Id⊗ S to this:

hΛ1 ⊗ S(Λ2) = Λ1 ⊗ S(Λ2)h.

Now, h · [(Λ · ∇)Xu] = hΛ1 · [∇X(S(Λ2) · u)] = Λ1 · [∇X(S(Λ2)h · u)] =(Λ · ∇)Xh · u which is reformulation of the equivariance condition inthe language of H∗-action.H being unimodular and involutive is equivalent to Λ being cocom-

mutative. It easily implies that Λ1⊗Λ2⊗Λ3⊗Λ4 = Λ4⊗Λ1⊗Λ2⊗Λ3.If f ∈ O then (Λ · ∇)X(f) = Λ1 · ∇X(SΛ2 · f) = Λ(1)X(f) = X(f).H∗ is involutive since so is H . We are ready to check that Λ · ∇ ismultiplicative provided so is ∇:

(Λ · ∇)X(uv) = Λ1 · ∇X(SΛ2 · uv) = Λ1 · ∇X [(SΛ3 · u)(SΛ2 · v)] =

= [Λ1 · ∇X(SΛ4 · u)](Λ2SΛ3 · v) + Λ1SΛ4 · u[Λ2 · ∇X(SΛ3 · v)] =

= Λ1 · [∇X(SΛ2 · u)]v + Λ2S−1Λ1 · u[Λ3 · ∇X(SΛ4 · v)] =

= [(Λ · ∇)Xu]v + u(Λ · ∇)Xv. ✷

Now we would like to address an issue of uniqueness of connection.Let us assume that we are given two connections ∇1 and ∇2. LetD = ∇1 − ∇2. It is straightforward to see that DX is an endomor-phism of the O-module U . Thus, two connections differ by an ele-ment of HomO(L,EndOU). According to [24, 8.3.3], EndOU ∼= U#H∗,the smash product, as algebras. In particular, HomO(L,EndOU) ∼=HomO(L, U#H

∗) ∼= HomO(L, U)n where n is the dimension of H . We

have just proved the following proposition.

Proposition 24. The space of connections is a HomO(L, U)n-torsor.

If the connections ∇1 and ∇2 are equivariant then the differenceD = ∇1 −∇2 is an element of HomO(L,EndO⊗H∗U) and the space ofequivariant connections is a torsor over this group. It allows a niceinterpretation if U is cleft: let us assume U ∼= Oσ[H ] is given. Now,EndO⊗H∗U ∼= Comod(H,U). Right H-comodule maps from H to U (or

16 DMITRIY RUMYNIN

integrals in the terminology of [6, 7]) were actively studied. We thinkthat the fact we have just noticed is worth writing as a proposition.

Proposition 25. Given an isomorphism U ∼= Oσ[H ], the space ofequivariant connections becomes a HomO(L,Comod(H,U))-torsor.

Adding other special properties in this spirit will further refine thespace of connections. For example, if ∇1 and ∇2 are multiplicativeconnections then ∇1

X −∇2X is a differentiation of U for each X . Such

process may eventually lead to an interesting uniqueness theorem.The last proposition is just a reformulation of Frobenius property

involving the map C defined in Section 5.

Proposition 26. We assume that ∇ is Frobenius. Given X, ∇X is aderivation of U if and only if ∇X · C = 0

Proof. Being a derivative means the equality

〈∇X(uv), w〉 = 〈∇X(u)v, w〉+ 〈u∇X(v), w〉

for each u, v, w ∈ U . Since the connection is Frobenius it may berewritten

X(C(u⊗ v ⊗ w))− 〈uv,∇Xw〉 = 〈∇X(u)v, w〉+ 〈u∇X(v), w〉

By the definition,

∇X ·C(u⊗v⊗w) = X(C(u⊗v⊗w))−C(∇Xu⊗v⊗w)−C(u⊗∇Xv⊗w)−C(u⊗v⊗∇Xw)

This proves the proposition. ✷

Corollary 27. A Frobenius connection is multiplicative if and only if∇ · C = 0.

It may be also interesting to investigate a significance of flatness. Aconnection is flat if the curvature form RX,Y u = [∇X ,∇Y ]u−∇[X,Y ]uis zero which is equivalent to ∇ defining a representation of Lie algebraL on U .

7. Miyashita-Ulbrich action.

Miyashita-Ulbrich action was introduced in [7]. If U ⊆ O is anH-Galois extension with not necessarily central O we can define theMiyashita-Ulbrich action for any algebra morphism α : U −→ B. Wedenote BO and BU the centralizer of α(O) and α(U) in B. Miyashita-Ulbrich action is a right H-action on BO defined by the formula c

α←

x =∑

α(ai)cα(bi) for any x ∈ H so that∑

ai⊗bi = can−1(1⊗x). It hasa property that for c ∈ BO and b ∈ U the formula cα(b) = α(b0)(c

α←

b1) holds. Furthermore, the invariants of the Miyashita-Ulbrich actionis precisely BU [7].


Keeping in mind that U ⊇ O is an extension with central invariantswe consider the identity map from U to U as α. This defines a right H-action on U such that the center of Uχ is the subalgebra of invariants.

We use a shorthand notation ← rather thanId← for this action. Using

antipode one can get a left Miyashita-Ulbrich as well. By doing soone obtains a left action of the quantum double D(H) in the case ofcocommutative H [5].Miyashita-Ulbrich action seems to be an important piece of structure

in our set-up. We think of Uχ as a deformation of right H-module. Onemay also think of Uχ as a deformation of an algebra structure: it wasdone in [29].Let Eχ = EndkUχ. Then Eχ carries a structure of H-H-bimodule

by hφh′(u) = φ(u← h)← h′. We are interested in the first Hochschildcohomology HH1(H,Eχ) because normalized first Hochschild cocycleswith coefficients in Eχ constitute infinitesimals of deformations of mod-ule structure. Indeed, letD = k[ǫ]/(ǫ2) be the ring of dual numbers. Aninfinitesimal deformation is a right H-module structure Uχ[ǫ]⊗D H →Uχ[ǫ] of the form:

a⊗ h 7→ a← h + µ(a, h)ǫ

for a ∈ Uχ, h ∈ H . The following is the associativity condition:

a← hh′ + µ(a, hh′)ǫ = a← h← h′ + µ(a, h)← h′ǫ+ µ(a← h, h′)ǫ.

It may be rewritten as µ(a, hh′) = µ(a, h) ← h′ + µ(a ← h, h′) whichis the Hochschild 1-cocycle condition for the map µ : H → Eχ givenby µ(h)(a) = µ(a, h). The unitary condition µ(a, 1) = 0 correspondsto the normalization condition on the cocycle.We should point out that this consideration never uses specific fea-

tures of the situation: it works for any algebra H and a family of rightH-modules Uχ. The following theorem is an adaptation of [11, theo-rem 3.2] for module deformations. One may probably prove a strongerversion of this theorem utilizing other ideas of [11].

Theorem 28. We assume HH1(H,Eχ) = 0 for some χ. If C is a curvein SpecO smooth at the point χ then there exists an open neighborhoodW of χ in C such that Uη⊗L ∼= Uχ⊗L as H-modules under Miyashita-Ulbrich action for each η ∈ W and some field extension L ⊇ k.

Proof. Let t be a local parameter on C at the point χ. It gives us ageneric point β(t) of C such that β(0) = χ. The generic point may bethought of as an element of hom(O,K[[t]]) where K is the field O/χ.By Lemma 5, U ⊗O K[[t]] is an H-Galois extension with central in-

variants. Therefore, it experiences the Miyashita-Ulbrich action which

18 DMITRIY RUMYNIN

may be written as a formal deformation of that of Uχ:

φ : Uχ[[t]]⊗k[[t]] H −→ Uχ[[t]].

φ may be given through a bunch of φn : Uχ × H → Uχ such thatφ(a ⊗ h) =

∑

n φn(a, h)tn. Clearly, φ0(a ⊗ h) = a ← h. The non-

zero φn with the smallest positive n is a first normalized Hochschild1-cocycle with coefficients in Eχ by the argument similar to the oneabout infinitesimals.But every cocycle is a coboundary! Thus, there exists Θ ∈ Eχ such

that φn(a, h) = Θ(a ← h) − Θ(a) ← h. We consider an H-moduleisomorphism Fn : Uχ[[t]] → Uχ[[t]] given by Fn(a) = a + Θ(a)tn. It isclear that F−1

n (a) = a−Θ(a)tn + o(tn). Let us compute

F−1n (Fn(a)h) = F−1

n (a← h+Θ(a)← htn + φn(a, h)tn + o(tn)) =

a← h+[−Θ(a← h)+Θ(a)← h+φn(a, h)]tn+o(tn)) = a← h+o(tn).

Thus, Fn “kills” the tn-term of the deformation. We may go on definingFn+1, Fn+2, . . . in the similar fashion. The product F = · · ·Fn+1 ◦ Fn,though infinite, is well-defined because the coefficient at tk is computedin the finite product Fk ◦ · · · ◦ Fn. It is clear that

F−1(F (a)h) = a← h

and, therefore, F performs an H-module isomorphism between Uχ[[t]]and the trivial deformation Uχ⊗KK[[t]]. But Uχ[[t]] may be specializedto Uη for η from some open subset W of C. Thus, Uη⊗O/ηK((t)) mustbe isomorphic to Uχ ⊗K K((t)) as right H-modules. ✷Theorem 28 gives a fine tool for calculations. Let L be a semisimple

classical Lie algebra of dimension n and rank r. We assume that k

is algebraically closed of good characteristic (i.e. the quotient of theroot lattice by any sublattice generated by a root subsystem has nop-torsion). The list of bad primes is 2 for Bn, Cn, Dn, 2 and 3 for E6,E7, F4, G2, and 2, 3, and 5 for E8. One should consult [29, 3.2] or [10]for the definition of reduced enveloping algebras. Roughly speaking,they are algebras Uχ as U = U(L) and O = O(L) ∼= O(L∗(1)). Weconsider a line αχ with α ∈ k and regular semisimple χ. U0 is justrestricted enveloping algebra. By [10, Corollary 3.6] Uαχ for α 6= 0 is adirect sum of pr copies of the algebra of pn−r×pn−r-matrices. Since thedimension of the center does not change with a field extension (by anelementary argument as in [29, Theorem 26]) we obtain the followingcorollary of Theorem 28.

Corollary 29. If HH1(u(L),Endku(L)) = 0 where u(L) is the moduleover itself under the right adjoint action then the center of u(L) hasdimension pr.


It follows from [17] that the dimension of the center is at least pr. Tobe precise one needs slightly stronger restriction on p: p has to be goodand should not divide n + 1 if L has An as a direct summand. To thebest of our knowledge, it is unknown whether the dimension is preciselypr even for sl2. We don’t know how to compute HH1(u(L),Endku(L))but it looks as a possible way to put hands on the center. Not onlycould it settle the question about the center of u(L) but it would alsoimply that the centers of Uχ for other χ were pr-dimensional. Indeed, asone degenerates an algebra to another one the dimension of the centercannot decrease: thus, u(L) should hit the maximal dimension of thecenter among Uχ.

8. H-torsors on schema.

An H-extension of a k-ringed space (X,R) (i.e. a topological ringwith a sheaf of algebras) is a sheaf of H-comodule algebras U on X(i.e. H coacts on U(V ) for each open set V and restriction maps areH-equivariant) such that R is a subsheaf of U and R(V ) = U(V )H foreach open set V . An H-extension is called H-Galois if U(V )H ⊆ U(V )is H-Galois for each V .This definition is not very helpful. The problem is that we can-

not glue such objects from local data. To check that we have an H-extension we must look at each open set. For instance, if X is an affinescheme and R is a sheaf of algebraic functions O then a Hopf-Galoisextension of the algebra O(X) does not necessarily give a Hopf-Galoisextension of (X,O). However, if we restrict our attention to H-Galoisextensions with central invariants then it suffices to work with opensets from some affine covering.We are interested only in a scheme S and the sheaf of regular func-

tions O. We define an H-torsor on S in the spirit of [13]. The idea is todefine Hopf-Galois extensions of the algebras functions over some affinecover and then to identify those which are isomorphic locally. Then weproceed to prove that H-torsors are the same as H-Galois extensionsof the sheaf of regular functions with central invariants on each openset.Let S be a scheme over k. The following datum will be called an

H-structure: a set of indices I, an affine open covering ∪i∈IAi ⊇ S, acollection of H-Galois extensions with central invariants Ui ⊇ O(Ai),and a collection of H-comodule algebra isomorphisms φi,j : Ui ⊗O(Ai)

O(Ai∩Aj) −→ Uj⊗O(Aj)O(Ai∩Aj) such that φi,j = φ−1j,i and φ

′j,k◦φ

′i,j =

φ′i,k for each i, j, k where prime means restriction to Ai ∩Aj ∩Ak.

20 DMITRIY RUMYNIN

Two H-structures (I, Ai, Ui, φi,j) and (I ′, A′i, U

′i , φ

′i,j) are called iso-

morphic if there exists the following datum:1. functions Φ : I → I ′ and Ψ : I ′ → I such that Ai = A′

Φ(i) and

A′i = AΨ(i);2. H-comodule algebra isomorphisms Φi : Ui → U ′

Φ(i), Ψi : U′i →

UΨ(i) such that the following diagrams are commutative.

Ui ⊗O(Ai) O(Ai ∩ Aj)

Uj ⊗O(Aj) O(Ai ∩Aj)

UΨ(i) ⊗O(AΨ(i)) O(Aψ(i) ∩AΨ(j))

UΨ(j) ⊗O(AΨ(j)) O(AΨ(i) ∩AΨ(j))

U ′Φ(i) ⊗O(A′

Φ(i)) O(A

′Φ(i) ∩ A

′Φ(j))

U ′Φ(j) ⊗O(A′

Φ(j)) O(A

′Φ(i) ∩A

′Φ(j))

U ′i ⊗O(A′

i) O(A

′i ∩ A

′j)

U ′j ⊗O(A′

j) O(A

′i ∩ A

′j)

✲

✲

✛

✛

❄

❄

❄

❄

φi,j

φΨ(i),Ψ(j)

φ′Φ(i),Φ(j)

φ′i,j

Φi ⊗ Id

Φj ⊗ Id

Ψi ⊗ Id

Ψj ⊗ Id

Given an H-structure (I, Ai, Ui, φi,j) and a refinement ∪i∈J Ai, wedefine the refinement of the H-structure: the set of indices is {(i, j) ∈I × J | Ai ⊇ Aj}; Ai,j = Ai; Ui,j = Ui ⊗O(Ai) O(Aj); and φ(i,j),(k,n) =φi,k ⊗O(Ai∩Ak) IdO(Aj∩An)

TwoH-structures (I, Ai, Ui, φi,j) and (I ′, A′i, U

′i , φ

′i,j) are called equiv-

alent if there exists a common refinement ∪Ai of the two coverings suchthat the refinements of the H-structures are isomorphic.It is straightforward to check that being equivalent is indeed an

equivalence relation. H-torsor on S is an equivalence class of H-data.

Theorem 30. The notion of H-torsor is equivalent to the notion ofH-Galois extension with central invariants of (S,O).

References

[1] A. A. Balinskii, S. P. Novikov, Poisson brackets of hydrodynamic type,Frobenius algebras and Lie algebras, Soviet Math. Dokl., 32 (1985), 228-231.

[2] R. J. Blattner, S. Montgomery, Crossed products and Galois extensions ofHopf algebras, Pacific J Math., 137 (1989), 37-54.

[3] S. U. Chase, M. Sweedler, Hopf algebras and Galois theory, LNM 97,Springer, Berlin, 1969.


[4] M. Cohen, D. Fischman, Semisimple extensions and elements of trace 1,J.of Algebra, 149 (1992), 419-437.

[5] M. Cohen, S. Westreich, From supersymmetry to quantum commutativity,J. of Algebra, 168 (1994), 1-27

[6] Y. Doi, Algebras with total integrals, Comm. in algebra, 13 (1985), 2137-2159.

[7] Y. Doi, M. Takeuchi, Hopf-Galois Extensions of algebras, the Miyashita-Ulbrich action, and Azumaya algebras, Journal of Algebra, 121 (1989), 488-516.

[8] M. Durdevic, Quantum principal bundles as Hopf-Galois extensions, q-algpreprint

[9] C. Faith, Algebra: Rings, modules, and categories, Springer, 1973.[10] E. M. Friedlander, B. Parshall, Modular representation theory of Lie alge-

bras, Amer. J. Math., 110(1988), 1055-1094.[11] M. Gerstenhaber, S. D. Schack, Relative Hochschild cohomology, rigid alge-

bras and the Bockstein, J. Pure Appl. Algebra, Sbornik e USSR 43(1986),53-74.

[12] C. Greither, B. Pareigis , Hopf-Galois theory for separable field extensions,J Algebra, 106 (1987), 239-258.

[13] A. Grothendieck, A general theory of fibre spaces with structure sheaf, NSFResearch project on geometry of function space, Rep. n. 4, 1955.

[14] J. Gubeladze, Anderson’s conjecture and the maximal class of monoids overwhich projective modules are free, Math, 63(1), 1989, 165-180.

[15] N. Hitchin, Frobenius manifolds, Notes by D. Calderbank.[16] J. Jantzen, Representations of algebraic groups, Academic Press, Orlando,

1987.[17] V. Kac, B. Weisfeiler, Coadjoint action of a semisimple algebraic group

and the center of the enveloping algebra in characteristic p, Indag. Math.,38(1976), 136-151.

[18] H. F. Kreimer, M. Takeuchi, Hopf algebras and Galois extensions of algebras,Indiana Univ. Math. J., 30 (1981), 675-692.

[19] R. G. Larson, D. E. Radford, Semisimple cosemisimple Hopf algebras, Amer.J. Math., 109, (1987), 187-195

[20] R. G. Larson, D. E. Radford, Finite-dimensional cosemisimple Hopf algebrasin characteristic 0 are semisimple, J. of Algebra, 117 (1988), 267-289.

[21] R. G. Larson, M. E. Sweedler, An associative orthogonal bilinear form forHopf algebras, Amer. J. Math., 91 (1969), 75-93.

[22] B. R. McDonald, Linear algebra over commutative rings, Marcel Dekker,1984.

[23] D. McDuff, D. Salamon, J-holomorphic curves and quantum cohomology,AMS 1994, Second edition 1995.

[24] S. Montgomery, Hopf algebras and their actions on rings, CBMS RegionalConf. Ser. in Math., no. 82, 1993.

[25] T. Nakayama, On Frobeniusean algebras 2, Annals in math., 42 (1941),1-21.

[26] U. Oberst, H.-J. Schneider, Uber untergruppen endlicher algebraischenGruppen, Manuscripta Math., 8(1973), 217-241.

[27] D. E. Radford, The trace functions and Hopf algebras, J. Algebra, 163

(1994), 583-622.

22 DMITRIY RUMYNIN

[28] A. N. Rudakov, Local universal algebras and reduced representations of Liealgebras, Math. USSR Izvestija, 14 (1980), 169-174.

[29] D. Rumynin, Hopf-Galois extensions with central invariants, submitted toJ. of Algebra.

[30] H.-J. Schneider, Representation theory of Hopf-Galois Extensions, Israel J.Math, 72 (1990), 196-231.

[31] J. P. Serre, Faisceaux algebriques coherents, Ann. of math (2) (1955), 6,197-278

[32] M. E. Sweedler, Hopf algebras, Benjamin, New-York, 1969[33] M. E. Sweedler, Cohomology of algebras over Hopf algebras, Trans AMS,

127(1968), 205-239.

Department of Mathematics & Statistics, LGRT, UMass, Amherst,

MA, 01003.

E-mail address : [email protected]

arxiv:q-alg/9707022v1 16 jul 1997 · 1. introduction let h be a ﬁnite dimensional hopf algebra...

Documents