manuscripta mathematictl a
Volume 55 1986 Editores M. Barner, Freiburg S. Hildebrandt, Bonn A. Prestel, Konstanz H. Br6zis, Paris E. Hlawka, Wien P. Roquette, Heidelberg P. M. Cohn, London T. Kato, Berkeley J. Tits, Paris A. Dold, Heidelberg H. Kraft, Basel
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Springer-Verlag Berlin Heidelberg New York Tokyo Printers: Brühlsche Universitätsdruckerei, Giessen Printed in Germany — © Springer-Verlag Berlin Heidelberg 1986 Springer-Verlag GmbH & Co. KG, 1000 Berlin 33
Springer-Verlag Berlin Heidelberg New York Tokyo Index
Afflerbach, L.: The Sub-Lattice Structure of Linear Congruential Random Number Generators. 455 Bear, D.: Wild Hereditary Artin Algebras and Linear Methods 69 Baldes, A.: Degenerate Elliptic Operators Diagonal Systems and Variational Integrals 467 Balibrea, F., Vera, G.: On the Sublinear Functional Associated to a Family of Invariant Means.. 101 Bauer, H.: A Class of Means and Related Inequalities 199 Bracho, J.: Cyclic Crystallizations of Spheres 213 Brandt, R., Stichtenoth, H.: Die Automorphismengruppen hyperelliptischer Kurven 83 Brion, M.: Quelques proprietes des espaces homogenes spheriques 191 Büch, J.: Bifurkation von Minimalflächen und elementare Katastrophen 269 Defant, A., Govaerts, W.: Tensor Products and Spaces of Vector-Valued Continuous Functions . 433 Degen, W.: Die zweifachen Bluteischen Kegelschnittflächen 9 Elias, J.: A Sharp Bound for the Minimal Number of Generators of Perfect Height Two Ideals .. 93 Forstneric, F.: Some Totally Real Embeddings of Three-Manifolds 1 Frey, G., Rück, H.-G.: The Strong Lefschetz Principle in Algebraic Geometry 385 Garcia, A.: Weights of Weierstrass Points in Double Coverings of Curves of Genus One or Two . 419 Govaerts, W., see Defant, A 433 Grüter, M.: Eine Bemerkung zur Regularität stationärer Punkte von konform invarianten Varia• tionsintegralen 451 Haggenmüller, R., Pareigis, B.: Hopf Algebra Forms of the Multiplicative Group and Other Groups 121 Hayashi, N.: Classical Solutions of Nonlinear Schrödlinger Equations 171 Heath, P. R.: A Pullback Theorem for Locally-Equiconnected Spaces 233 Herzog, J., Waldi, R.: Cotangent Functors of Curve Singularities 307 Klein, C.: Arzela-Ascoli's Theorem for Riemann-Integrable Functions on Compact Spaces 403 Koltsaki, P., Stamatakis, S.: Über eine von J. Hoschek erzeugte Klasse von Strahlensystemen... 359 Kuhlmann, F.-V., Pank, M., Roquette, P.: Immediate and Purely Wild Extensions of Valued Fields 39 Manolache, N.: On the Normal Bündle to Abelian Surfaces Embedded in P4 (€) 111 Michel, J.: Randregularität des 6 -Problems für die Halbkugel in Cn 239 Pälfy, P. P.: On Partial Ordering of Chief Factors in Solvable Groups 219 Pank, M., see Kuhlmann, F.-V., et al 39 Pareigis, B., see Haggenmüller, R 121 Roquette, P., see Kuhlmann, F.-V., et al 39 Rück, H.-G., see Frey, G 385 Sander, W.: Weighted-Additive Deviations with the Sum Property 373 Sanders, H.: Cohen-Macaulay Properties of the KoszufHomology 343 Stamatakis, S., see Koltsaki, P 359 Stichtenoth, H., see Brandt, R 83 Takase, K.: On the Trace Formula of the Hecke Operators and the Special Values of the Second L-Functions Attached to the Hilbert Modular Forms 137 Vera, G., see Balibrea, F 101 Waldi, R., see Herzog, J 307
Covered by Zentralblatt für Mathematik and Current Mathematical Publications manuscripta math. 55, 121 - 136 (1986) manuscripta mathematica © Springer-Verlag 1986
HOPF ALGEBRA FORMS OF THE MULTIPLICATIVE GROUP AND OTHER GROUPS
Rudolf Haggenmüller and Bodo Pareigis
The multiplicative group functor, which associates with each k-algebra its group of units, is affine with Hopf algebra k[x,x~ ]. The purpose of this gaper is to determine explicitly all Hopf alge• bra forms of k[x,x ] with only minor restrictions on k ( 2 not a zero-divisor and Pic(2)(k) = 0 ). We also describe explicitly (by generators and relations) the Hopf algebra forms of kCß , kC4 and kC$ , where Cn is the cyclic group of order n . Some of our results could be drawn from [l,III §5.3.3] where a similar result as ours is indicated (and left as an exercise). We prefer however a less technical approach, in particular we do not use the extended theory of algebraic groups and functor sheaves.
The principal tool of this note is the theory of faithfully flat descent which is used to prove that the Hopf algebra forms of kG with finitely generated group G (with finite automorphism group F) are in one-to-one correspondence with the F-Galois extensions of k. The progress in recent years in describing the quadratic extensions of k and the explicit construction of the correspondence allow us to compute the forms of kG for all groups with Aut(G) = in terms of generators and relations.
Consider the functor C: k-AZg^ —>• Gl , the CA.KcZt $uncto>i9 defined by C(A) = {(a,b)multiplication observe that C is represented
121 HAGGENMÜLLER - PAREIGIS by the k-algebra H = k[c,s]/(s2+c2-1) . This must be a Hopf alge• bra and is called the tK^igOYiomQÄHA,c aJLgohKOL* The coalgebra structure of H is given by A(c) = c»c-s«s A(s) =c«s+s«c e(c) = 1 e(s) = 0 . The antipode is S(c) = c , S(s) = -s .
Obviously c and s play the role of cos and sin resp. and the diagonal map reflects the summation formulas for cos and sin: cos(x + y) = cos(x)cos(y) - sin(x)sin(y) sin(x + y) - cos(x)sin(y) + sin(x)cos(y) . e gives the value at 0° and S is the reflection on the x-axis. The geometric meaning of the group structure on C(A) is the addi- tion of the corresponding angles with the x-axis for the points (a,b) resp. (c,d) .
Let us now ask for a QKOULp-ZJJlQ, QJtWQ.vit e f 1 in H , i.e. an element e with A(e) = e e e , e(e) = 1 . A little calculation shows that such an element exists if and only if there is ick with i2 = -1 and all the group-like elements are then of the form (c + is)n , n € TL . Observe that e * = c - is , if e = c + is . The diagonal map on e reflects the summation formula for the exponential function exp(x + y) = exp(x)exp(y) . If furthermore 2 is invertible in k then we get an isomorphism of Hopf algebras k[c,s]/(s2 + c2 - 1) ^ kS (where U2L corresponds to e = c + is) because of c = -j(e + e *) , s = "^"(e - e *) . The affine k-group represented by k2Z - k[x,x is the mLUL&pLLccLtive, gKOUp or the group of units. Hence we have the following:
If A is a k-algebra over a commutative ring k with 2 ^€ k and iek , then the circle group C(A) is isomorphic to the multiplicative group U(A) . Moreover if H = k[c,s]/(s2 + c2 - 1) is defined over a field k with 2^0 and if K = k[i] , then H K - k2Z K as Hopf algebras. So we have two Hopf algebras H
122 HAGGENMÜLLER - PAREIGIS and kZ which after faithfully flat extension of the base ring become isomorphic. We say that H is a K-jJoim of kZ , the multi- plicative group.
To describe all such forms we want to use the theory of faith• fully flat descent. We sketch the main ideas following [3]. Let us discuss the general notions in two specific cases. Consider a direct- ed graph 6 such as F-gaJL resp. hop$ (see diagram below), which comes equipped with functors for all vertices i € {o,l,2} of the graph and all Lck-A&j^: >• L-Mod . F^1: L-Mod — In our examples we take
FJJ(N) = L F^(N) = N F^(N) = N eL N . Observe that these functors are in general no additive functors. For q: L —>• M in k-AZg there are coherent natural isomorphisms L M ^
We define categories 6 (F-gaZ resp. hopA ) in the following Li Lt Li way. Objects will be (K,K) in F-gal^ resp. (H,H) in hop^ with K, HeL-Mod and K: F-gal—>• L-Möd resp. H: kop& —>• L-Mod graph maps such that tf(i) = F^(H) . Morphisms are L-module homo- morphisms which are "natural transformations" with respect to the graph maps.
Let F be a finite group with elements fc F then the above Situation is represented by the diagram on the next page where the last part shows the properties of the functors F^ with respect to change of basis.
If q: L —>• M is a change of basis morphism and
f: (A ,AT ) —>• (B ,B ) is a morphism in 4 then q*(f) denotes Li Li Li Li LI the morphism f M obtained by this change of basis.
123 4: F-gal:
0^1^-2
rL F^(N) = N • • N (i factors) CO (—i o M T . PS < w l-J
kopin: w o o — < F% M) * F% • M) (KM,y: N (K »L M) «M (K ©L M) iL j L I? CD 12 CD it cp (similar) q q F.(N) • M T F.(N) •_ M (K,K)oM: L • M —> K • M (K • K) • M iL J L HAGGENMÜLLER - PAREIGIS called Amitsur complex. For a functor G: k-A£g^ —>- GK we define a 1-cocycle d2(q>) = d3(cp)d1(cp) . 1 cp is homologous to A> if cp = d2(TT)\pd1 (TT)" for some TTgG(L) . The pointed set of classes of 1-cocycles is H*(L/k,G) . For (B,B) € 6^ we shall use the functor Avut(B9B) for G , where Aa£(B,8)(L) = AutL(BL,BL) , the group of automorphisms of (BL,BL) in -6L (?-QCÜL^ resp. hopf^) . Now we can formulate the descent theorem. THEOREM 1. LQA L be a faXAkfuJULvi flat k-aJ^eMa, loX 6 6e an adml66lblt AttucÄLLKt and lat (B,B) In 6k . Tnen #ie/te 16 a blitction boJwazn tha 6it of L~fotim6 S(L/k,(B,ß)) o£ (B,B) and H1(L/k,Au^:(B,B)) . Tne blitction JJ6 c\ivQ,n In tkd following way: leX the. da* 6 of (c,C) be an L-foKm of (B,B) with. l6omoKpkl6m o): (cL,CL) « (BL,BL) , Inen dj(üj)"1 d*(a>) cp: B « L e L C • L • L >• B » L e L ^ #ie a660(U.ate.d 1-cocycZt. If a l-cocLfd.& cp€AutL^L(B • L • L) 4.6 cjivtn, tkdn loX c 6e tlm £QuuaJ&Lz B • d2 B 9 L B » L • L Te.n60H.lw with L lnduc&6 an l6omon.pkl6m a): c * L ^ B • L and thdKQ, J6 a anlerne 6-6ln.uctu.ie. C , aacn ^na^: tu: (CL,CL) —»- (BL,BL) ^ an 6^-l6omofipkl6m. Proofs of this theorem can be found in [2,3]. Now we can describe the L-forms of the Hopf algebra kG by 1 H (L/k,f/opt{-AuX( G)) , where Hopf-Aut( G)(L) = Hopf-AutL(LG) for any commutative k-algebra L . 125 HAGGENMÜLLER - PAREIGIS The category F-gaZ^ contains in particular all Galois extensions of L with group F and kopf^ contains all Hopf algebras over L. A change of basis q: L —> M induces functors Ä. —>• L n F and (F-galL —> ~ga£M kopfL —> kopin which also preserve the Galois extensions and Hopf algebras). If M is a faithfully flat extension of L then only F-Galois extensions resp. bialgebras over L can be lifted to F-Galois extensions resp. bialgebras over M , i.e. these properties are preserved and reflected by faithfully flat base extensions. Let L be a faithfully flat k-algebra. An isomorphism class (C,C) in -6^ is called an L-^ötm of an object (B,B) in 4^ , if after base ring extension the object (CL»^L^ ~ • L = (C 9 L,C • L) is isomorphic to (B,B) e L in 4^ . The set of L- forms of (B,B) will be denoted by 5(L/k,(B,B)) . (C,C) is called a fownri of (B,B) if there exists a faithfully flat k-algebra L such that (C,C) is an L-form of (B,B) . The set of all forms is denoted by £(B,B) . If we simply write lim for the direct limit taken over all faithfully flat extensions of k then we have for our special cases F-Gql(K.K) = lim F-Gal(L/k,(K.K)) resp. Hont(H.H) = lim HopX(L/k.(H.H)) . Our aim is to describe Hopf(kG) where G is a finitely gene- rated group (with automorphism group F = ) and k is a commuta- tive ring (where 2 is not a zero-divisor and Pic^2)(k) = 0 ), i.e. we want to describe all bialgebras H over k which after faithfully flat ring extension L become isomorphic to kG L ^ LG . The trigonometric algebra was a first example with G = TL . Each L€k-A£g defines a cosimplicial object k > L ~T> L €> L $- L • L • L d2 * 126 HAGGENMÜLLER - PAREIGIS Observe that the use of the graph actually allows arbitrary bialgebras (C,C) to become L-forms of a Hopf algebra (H,W) , i.e. to be in Hopf(H9H) . If we had used instead the graph for köpf then only Hopf algebras (C,C) would have been admitted as L-forms of the Hopf algebra (H,tf) . But in both cases the forms are described by H*(L/k9Hopf-Aut( • H)) , since each bialgebra automorphism of L • H is automatically a Hopf algebra automorphism [8, Lemma 4.0.4] . Thus we have REMARK: A blaJLatbKa. fonm of a Hopf aZaibKCL it> a Hopf aZatbKa. Let G be a group and x€ LG be a group-like element, i.e. A(x) = x • x and e(x) = l.If x= £ _ a g , then x is group-like g€G g iff a =0 for almost all g€G g 5 a 'a = 0 if g 4= h g n 1 (*) a *a = a -.8 8 8 A S 1 VG G • Let V(LG) denote the set of all group-like elements of LG . THEOREM 2. LoX G be a fLvüJtoJLvi atmnoLtzd anoup. Tkan Hopf-Aut^(LG) - V(LF) wk Proof: Let J aff € V(LF) . Then the L-linear map cp with cp(g) - IftF aff(g) for all gcG is a Hopf algebra automorphism. In fact we have 127 HAGGENMÜLLER - PAREIGIS = If aff(g)f(g') (by (*)) - If aff(gg') = cp(gg') and cp(l) - If aff(l) = If af » 1 , hence cp is an algebra homomorphism. (cp • cp)(A(g)) = cp(g) » cp(g) « If aff(g) • If. af,f(g) = If aff(g) • f(g) - If afA(f(g» - A(If aff(g» = Acp(g) and £cp(g) = e(If aff(g)) = lf af = 1 = e(g) give that cp is a Hopf algebra homomorphism. But with ^(g) = V -1 lf aff (g) we get 1 _1 = If af Iff af!ff (g) - If afg = g hence 41 = • Hopf-Aut^(LG) . Now let a^f define the identity on LG . Define a(g,h) : = l {af|f€ F* f(g) = h} for g,h€ G . Because of l aff(g) = g for all geG we have a(g,h) = 6^ ^ . Use the fact that the a^ are orthogonal idempotents to get V =ng a(g,f(g))=ng6g)fl(g) = 6fVd, so J,f aff = idcV(LF) which shows V(LF) c Hopf-AutL(LG) . Before we continue we remark the following. For any group G and Hopf algebra homomorphism cp: LG —>• LG let cp(g) s [ , a , g1 g g >g for g«G . Since g is group-like so is cp(g) hence the 128 HAGGENMÜLLER - PAREIGIS coefficients fag' gl^'€ ^ satisfy (*) for each g . Now let cp: LG —>• LG be a Hopf algebra automorphism with inverse s a enera vp . Assume that Si»**'>ßn * g ting System for the group G . Then cp is completely described by its action on the g^ . Let co(g.) = y a.# x. . with a..eL. x..€G . Since the (a,.). are L ij ij IJ ij IJ'J orthogonal idempotents we can refine this set by 1 = ^i^j aij^ = I b, where all the b, are products IT. a... . Then the b. satis- K K. 1 1J £ K s fy (*) and the a„ are sums of certain ^jc' • Hence cp(g.) = \ b^y^k ^or certaiti group elements . If g fc G and n„ n n„ n 1 r 1 r g = g ... gj[ then cp(g) = cp(gi ) ... cp(g. ) 1 r 1 r Taking the product of the sum expressions for the l bkfkf^(g) shows f f£(g) = g for all geG and all k . By symmetry we get the result cp = £ b^f^* ^(LF) . We wish to acknowledge that the argument given above as well as the following example were kindly communicated to us by Pere Menal. The example shows that the theorem does not hold for infinitely generated abelian groups. If G = e e = e r an( series of idempotents e^,e2>... with £ « ^ f° = J * e.+e. for i+i, then i 1 J LG ^TT g^^TT + (i - ei>«iri>€ LG i=l i=l is a Hopf algebra automorphism but not in V(L(Aut(G))) . 129 HAGGENMÜLLER - PAREIGIS We now introduce some facts about Galois extensions of commutative rings. Let F be a finite group. A commutative k-algebra K is called an F-GcüLoi6 e.xtzn&lon of k if 1) F is a subgroup of Aut^CK) , 2) K is a finitely generated projective k-module, 3) F c End (K) is a free generating System over K . F The k-algebra E^ = Map(F,k) , the set of maps with algebra structure induced by the ring structure on k , is called the tKX.vi.oJi F-GaZoX6 Utte.nA4.on> where F acts by (fa)(f) = a(f"1ft) , f,f'€ F, F k k atMap(F,k) . E^ has the k-basis v^ with v^(f') = 6^ ^, . For Galois extensions there is the following PROPOSITION 3. Gal-Autk(Ek) - V(kF) . For a proof see [3, Prop. 2.14]. COROLLARY 4. let G be. a. iX.vuX.QZvi azne,n.ated QKOUP wXXh ilnXXe. automoKphX6m aioup F = Gr-Aut(G) . Thm thtKZ t6 a bXj&cXion beXwttn the, Hopf, alaebKa. yjowd tlovA(kG) o& kG und the. [potnteA) 62t o£ F-GaloX6 extenbXonA Gal(k,F) o£_ k . Proof: We first observe that each F-Galois extension K over k is faithfully flat. Furthermore there is a K-isomorphism F F ü): K • K - ER - Efc • K of F-Galois extensions of K defined by id(a • b)(f) = f *(a)b , where K • K is a K-algebra by a«(b « c) = b « ac and F acts on K e K by f(a e b) = f(a) « b . So every F F-Galois extension K of k is a K-form of E and Gal(k,F) = F F-GaKE^) . By Theorem 1 together with Proposition 3 we have F 1 F-GaKL/k.E. ) - H (L/k,V(-F)) and Theorems 1 and 2 give 1 H (L/k,V(-F)) = HopX(L/k,kG) . By going to the limit we get If F is a commutative group then all the objects mentioned in the proof above are abelian groups and all morphisms are group homo• morphisms. 130 HAGGENMÜLLER - PAREIGIS We want to make the isomorphism of the Corollary explicit, so that we can construct the Hopf algebra form associated with an F-Galois extension of k . For that purpose we first construct a 1-cocycle for a given F-Galois extension K of k . Then we use this 1-cocycle to construct the corresponding form of kG . In bet- F ween we have to identify Gal-Autk(Ek) - V(kF) and V(kF) - Hopf-Autk(kG) . In general if the class of (C,C) is a K-form of (B,B) with -1 isomorphism Cü: C • K - B • K then the 1-cocycle cp = dj(ü)) dj(co) is given by the commutative diagram B » K e K B » K ® , (K 9 K) d*(co) »-1 • >i 1 \ C 9 K (K • K) C » K • K C 9 K • (K 9 IC) d*0) ü) • 1 | 2 B • K • K B e K4 «, (K • K) . 2 F Let (B,B) be the trivial F-Galois extension E and K some F F (K-)form (C,C) of it. Then co:K«K-E-E, «K was given by -1 r -1 K 0)(a 9 b)(f) = f (a)b or equivalently 0)(a e b) = £f f (a)b»vf . F The corresponding 1-cocycle cp € Gal-AutKoK(Ek • K • K) describes an element £ af-l^ *n * cp(ve ) = l afvf . -1 K v v Let co (ve) = 1 a. • b,«K • K , i.e. I f(ai)bi = 6f e , then by the diagram above 9(ve ) = Zf LI (f (a.) • b.)vf f a so the corresponding element in V(K 9 KF) is i|) = Ifli ( ( i) • b±)£. Consider vp as an element of Hopf-AutR R(K • KG) . Then by the 131 HAGGENMÜLLER - PAREIGIS construction given in Theorem 1 the associated K-form of kG is the equalizer H in H y KG 5 K • KG . **1 From *(Ig 1 • cgg) = [f I. Ig (f(a£) • b.cg)f(g) )g = Ig(If>i f<«i) • Vf-l(g) we have )g H = \l cgg € KG | l (cg * l)g = Ig (Ifj. f(a.) • Vf"l(g) where g«G, f«F = Gr-Aut(G) . p We claim now that H = (KG) , the subset of fixed elements in KG under the diagonal action of F given by f(ag) = f(a)f(g) . Let l Cgg be in H and f € F . Then f(I cgg) = I V(f • l)(cg • l)f(g) = Vi (f • l)(f(a.) • b.cf,.1(g))f(g) = Vl (ff(a.).b.c(ffl)_1(h))h = * Chh • hence H c (KG) . Let £ c g € KG satisfy l f(c )f(g) = l c g for all feF , then by applying f to the group elements we get I f(cg)g = £ cgf ^"(g) . Furthermore observe that ^ f(a^cg) is fixed under all feF hence an element in k . The inverse map of _1 y\) = l aff in V(K • KF) * Hopf-AutRdK(K • KG) is l aff since f £^ a^f«^t a^,f ^ = 1 by the orthogonality of the a^ . üsing all this we get (I a^'Sd (cg « l)g) = I (f(a.)cg « b.Jf'^g) = l (f(a.cg) « b.)g = l (1 « f(a.)b.f(cg))g = I (1 « cg)g . So we have proved the following THEOREM 5. Lot G be. CL &JYUJteJLij QZnZKdtzd Qloup U)i£k JAJflitd owtomofipklm <\n.owp F = Gr-Aut(G) . Thm thtte. AM_ a bijacjLLon 132 HAGGENMÜLLER - PAREIGIS be&ueen Gal(k,F) and HovA(kG) wkich a6&o(Ucut£6 with zack F-Galoi6 dxtm^ion K OJL k the. Hop6 alc\zbn.a H = { l cgg€ KG | I f(cg)f(g) = I cgg ioK all f € F } . fan.th 0): H • K - KG , Cü(h « a) = ah . This Theorem can be favorably applied in the Situation F = C2 , the cyclic group with two elements, because in this case the C2~Galois extensions or "quadratic extensions" of k are well known. So the groups G which are of interest are C^, C^, C^, and TL » We will give a complete description of all forms of kG for these groups with minor restrictions on k . Assume in the following that 2 is not a zero divisor in k and that Pic^^Ck) = 0 . Then all quadratic extensions of k are free [7] and can be described as K = k[x]/(x2-ax-b) where a2+4b = u is a unit in k . Their non-trivial automorphism is f(x) = a - x . If 2 is invertible in k , then a can be chosen zero thus K = k[x]/(x2-b) with b a unit in k . In this case two such extensions are isomorphic iff b*bf is a Square in k . For the general equivalence of two such extensions we refer to [3,4,5,7]. THEOREM 6. a) The. Hopfi alaehKa o£_ kZ , the, multtpltcatlvz an.owp9an.e. H = k[c,s]/(s2-asc-bc2+u) . b) The, Hopj alaehKa jornm oj kC3 aie. H = k[c,s]/(s2-asc-bc2+u, (c+l)(c-2), (c+l)(s-a)) . c) Thd Hop& alazbn.a Aonm* oj kc4 an.e. H = k[c,s ]/(s2-asc-bc2+u, c(ac - 2s)) . d) The, ttopj alGjdbKa joMi* o{\ kC6 ane, H = k[c,s]/(s2-asc-bc2+u, (c-2) (c-1) (c+1) (c+2) , (c-1) (c+1) (sc-2a)). In alZ ca6£6 a, b, uck 6atlj>jy a2+4b = u and u li> a unit oj k • Tfeeac {\0wn6 and 6pltt by K = k[x]/(x2-ax-b) . The, \iopj alaebna 6tn.ucXun.t £4 de,jine,d by 133 HAGGENMÜLLER - PAREIGIS A(c) = u (a2+2b)c e c - a(c 0 s + s e c) + 2s © s) A(s) = u (-abc 9 c + 2b(c e s + s • c) + as e s) e(c) = 2 , e(s) = a , S(c) = c , S(s) = ac - s . In the special case of 2c U(k) , ae k can be taken zero. If we replace c by 2c1 , s by 2bs' and u by 4b then the forms of kZ are H = k[c',s']/(c*2-bs*2-1) .For b = -1 this is the trigonometric algebra discussed in the beginning of this note, for b = 1 this is isomorphic to kZZ. . If k = IR , the field of reals then there are precisely two quadratic extensions of IR , the complex numbers and IRx IR . Hence these are the only two possible forms of IRZ . If k = Q , the field of rational numbers, then there are infinitely many forms of Q2Z namely H = k[c',s* ]/(c,2+ds,2-l) where d runs through all positive squarefree natural numbers or d = -1 . Proof of the Theorem: Let G = TL and Y.TL = kttjt"1] . Let 7 a.t1 be an element of a form H . If a. - a. + B.x€ K , then L 1 . . 111' 1 1 l (ou + ßix)t = l f(a. + Bix)t' implies a_i + B_.x = a + ß^(a-x) , hence 1 1 1 _i H - {#aot° + £i>o ^(t + t' ) + ß.Cxt + (a - x)t ) } . 1 1 1 _1 Define c. := t + t" , si : = xt + (a - x)t for i ^ 0 . Then H is generated by the c. and s. as a k-module. Observe c =2 y 1 1 o 2 sQ = a and a + 4beU(k) . Define furthermore c := c^ and s := s^ . Then the following relations hold: c.-c = c.,. + c. . , s.•c = s.,- + s. . , for i ^ 1 . 1 l+l l-l ' 1 l+l 1-1 ' Since c , s e k this shows byJ induction that c and s are o ' o k-algebra generators of H . They satisfy the following relation s2 - asc - bc2 + u = 0 which is easily checked in k2Z . So there is an epimorphism k[c,s]/(s2-asc-bc2+u) —>• H . This map is injec- tive iff it is injective after tensoring with K . But in the Situation K[c,s]/(s2 -asc-bc24-u) —> H « K - K[t,t there is an inverse homomorphism t»—>• (a - 2x) *((a - x)c - s) . By (a - 2x) 134 HAGGENMÜLLER - PAREIGIS u it is clear that (a - 2x) is invertible. Furthermore (a - 2x)"1((a - x)c - s)-(a - 2x)~1(-xc + s) = 1 shows that this map is weil defined and maps t * to (a - 2x) (-xc + s) . Now it is easy to see that K[c,s]/I —• H e K —>• K[c,s]/I is the identity hence the given map is an isomorphism. The coalgebra structure on H is induced by that of kZ and is expressed by the given formulas. For the Hopf algebra forms of kC^, kC^, and kC^ take those of kZ and express the relation tn = 1 (n = 3, 4, or 6) in terms of c and s . 2 -1 Case n = 3: t = t iff C2 = c and S2 = ac - s iff (c + l)(c - 2) = 0 and (c + l)(s - a) = 0 . To show that c2 = c 2-1 2-2 and s = ac - s implies t = t observe that xt + (a-x)t = -1 2-2 -1 xt + (a-x)t implies t-t = t -tas coefficients of x . 2-1 2-1 Then 2t = 2t implies t = t since 2 is not a zero divisor in KC3 . Case n = 4: t2 = t"2 iff ((a-x)c - s)2 = (-xc + s)2 iff c(ac - 2s)(a - 2x) = 0 iff c(ac - 2s) = 0 . 4 -2 Case n = 6: t = t iff c, = c0 and s. = ac« - s» iff 2 2 2 4 2 4 2 2 (c - l)(c -4) = 0 and (c - l)(sc - 2a) = 0 . REMARK: There is an interesting example of a separable field extension which is (Hopf-)Galois with the Hopf algebra H = Q[c,s ]/(c2+s2-1 ,sc) , which is a form of QC^ . The extension is Q(u):Q with u = , which is definitely not Galois in the ordinary Galois theory. The Operation of H on Q(u) is given by 2 c(l) = 1 c(|i) - 0 c(u ) = -p2 c(p3) = 0 3 M 2 3 s(l) = 0 S( ) = -p s(p ) = 0 s(u ) = p . 135 HAGGENMÜLLER - PAREIGIS This Operation satisfies 1. c2(a) + s2(a) = a for a€ K 2. c(ab) = c(a)c(b) - s(a)s(b) s(ab) = s(a)c(b) + c(a)s(b) for a, beK , 3. c(l) = 1 s(l) = 0 . A straightforward computation shows that Q(u) is Galois over Q with the Hopf algebra H and this Operation. Another example of such a Galois extension is Q(u):Q with u = 3/2 and the Hopf algebra H = «j[c ,s ]/(3s2+c2-1, (2c-KL)s) , which is a form of QC^ with coalgebra structure A(c) = c*c-3ses , A(s) = c«s + s«c, e(c) = 1 , e(s) = 0 . The Operation is defined by 2 2 c(l) = 1 c(u) = -|u c(M ) = yi s(D = 0 s(\i) = |p s(u2) = -|p2 . In a separate paper we will determine all separable field extensions which are Galois with a Hopf algebra H [9] . REFERENCES [l] DEMAZÜRE, M. and GABRIEL, P.: "Groupes algebriques" (Tome I). North Holland - Amsterdam (1970). [2] GROTHENDIECK, A.: Technique de descente I. Seminaire Bourbaki, Exp. 190 (1959/60). [3] HAGGENMÜLLER, R.:"Über Invarianten separabler Galoiserweiterungen kommutativer Ringe" Dissertation, Universität München, (1979). [4] HAGGENMÜLLER, R.: Diskriminanten und Picard-Invarianten freier quadratischer Erweiterungen. ManuAcn.4.pta matk. 36 (1981), 83-103. [5] KITAMURA, K.: On the free quadratic extensions of a commutative ring. ÖAaka JMoXk. 10 (1973), 15-20. [e] KNUS, M.A. and 0JANGUREN, M.:,fTheorie de la Descente et Algebres d'Azumaya." Springer LN 389 (1974) . [7] SMALL, C: The group of quadratic extensions. J. PuA£ Appl. Alg. 2 (1972), 83-105, 395. [8] SWEEDLER, M.: "Hopf Algebras" W.A.Benjamin - New York (1969) . [9] GREITHER, C. and PAREIGIS, B.: Hopf Galois Theory of Separable Field Extensions. To appear in J. AZg. Mathematisches Institut der Univ. Theresienstraße 39 D-8OOO München 2 Federal Republic of Germany (Received August 2, 1985) 136