Contemporary Mathematics A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space Matthew Ando and Jack Morava I believe in the fundamental interconnectedness of all things. _Dirk Gently [Ada88 ] 1.Introduction Let T denote the circle group, and, if X is a compact smooth manifold, let L* *X def=C1 (T; X) denote its free loop space. The group T acts on LX, and the fixed point manifold is ag* *ain X, considered as the subspace of constant loops. In the 1980's, Witten showed that the fixed-point f* *ormula in ordinary equivariant cohomology, applied to the free loop space LX of a spin manifold X, yields the * *index of the Dirac operator (i.e. the ^A-genus) of X_a fundamentally K-theoretic quantity [Ati85]. He also * *applied the fixed-point theorem in equivariant K-theory to a Dirac-like operator on LX to obtain the el* *liptic genus and "Witten genus" of X [Wit88 ]_quantities associated with elliptic cohomology. Among homotopy theorists, these developments generated considerable exciteme* *nt. The chromatic pro- gram gives organizes the structure of finite stable homotopy types, locally at * *a prime p, into layers indexed by nonnegative integers. The nth layer is detected by a family of cohomology theor* *ies En; rational cohomology, K-theory, and elliptic cohomology are detecting theories for the first three la* *yers [Mor85 , DHS88 , HS98]. The geometry and analysis related to rational cohomology and K-theory are re* *asonably well-understood, but for n 2 and for elliptic cohomology in particular, very little is known. W* *itten's work provides a major suggestion: for n = 1 and n = 2 his analysis gives a correspondence analysis underlying Enanalysis underlying En-1 applied to X $ applied to LX. (1.0* *.1) This paper represents our attempt to understand why Witten's procedure appea* *rs to connect the chro- matic layers in the manner of (1.0.1). To do this we consider very generally th* *e fixed-point formula attached to a complex-oriented theory E with formal group law F. We recall that for n > * *0, such a theory detects chromatic layer n if the formal group law F has height n. Our first result is that the fixed-point formula of a suitable equivariant e* *xtension of E (Borel cohomology is fine, as is the usual equivariant K-theory) applied to the free loop space y* *ields a formula which is identical to the Riemann-Roch formula for the quotient F=(^q) of the formal group law F b* *y a free cyclic subgroup (^q) (compare formulae (3.2.2)and (4.2.3)). ___________ Much of the work for this paper was carried out at Johns Hopkins University,* * where the first author was supported by an NSF Postdoctoral Fellowship. Supported by the NSF. cO0000 (copyr* *ight holder) 1 2 MATTHEW ANDO AND JACK MORAVA The quotient F=(^q) is not a formal group, so to understand its structure, w* *e work p-locally and study its p-torsion subgroup F=(^q)[p1 ]. We construct a group Tate(F) with a canonic* *al map Tate(F) ! F=(^q); which induces an isomorphism of torsion subgroups in a suitable setting. Our se* *cond result is that the group Tate(F)[p1 ] is a p-divisible group, fitting into an extension F[p1 ] ! Tate(F) ! Qp=Zp of p-divisible groups. If the height of F it n, the height of Tate(F)[p1 ] is n* * + 1, but itsetale quotient has height 1. In a sense we make precise in x5.3, it is the universal such extensio* *n. Thus the fixed-point formula on the free loop space interpolates between the* * chromatic layers in the same way that p-divisible groups of height n+1 withetale quotient of height 1 interp* *olate between formal groups of height n and formal groups of height n+1. This is discussed in more detail, fro* *m the homotopy-theoretic point of view, in our earlier paper [AMS98 ] with Hal Sadofsky; this paper is a kind* * of continuation, concerned with analytic aspects of these phenomena. We show that Witten's construction i* *n rational cohomology produces K-theoretic genera because of the exponential exact sequence 0 ! Z ! C ! Cx ! 1 (1.0* *.2) expressing the multiplicative group (K-theory) as the quotient of the additive * *group (ordinary cohomology) by a free cyclic subgroup; while his work in K-theory produces elliptic genera * *because of the Tate curve 0 ! qZ ! Cx ! Cx=qZ ! 1 (1.0* *.3) (where q is a complex number with |q| < 1), expressing the elliptic curve Cx=qZ* * as the quotient of the multiplicative group by a free cyclic subgroup. These analytic quotients have already been put to good use in equivariant to* *pology. Grojnowski con- structs from equivariant ordinary cohomology a complex T-equivariant elliptic c* *ohomology using the elliptic curve C= which is the quotient of the complex plane by a lattice; and Rosu uses* * Grojnowski's functor to give a striking conceptual proof of the rigidity of the elliptic genus. Groj* *nowski's ideas applied to the multiplicative sequence (1.0.3)give a construction of complex T-equivariant ell* *iptic cohomology based on equivariant K-theory; details will appear elsewhere. Completing this circle, R* *osu has used the quotient (1.0.2)to give a construction of complex equivariant K-theory [Gro94 , Ros99, R* *K99 ]. Several of the formulae in this paper involve formal infinite products; see * *for example (3.2.2)and (4.2.3). On the fixed-point formula side, the source of these is the Euler class of the * *normal bundle of X in LX (3.1.2). From this point of view, the problem is that the bundle is infinite-d* *imensional, so it does not have a Thom spectrum in the usual sense. However, has a highly nontrivial circle ac* *tion, which defines a locally finite-dimensional filtration by eigenspaces. Following the program sketched in* * [CJS95 ], we construct from this filtration a Thom pro-spectrum, whose Thom class is the infinite product. 1.1. Formal group schemes. In this paper (especially in section 5) we shall * *consider formal schemes in the sense of [Str99, Dem72 ]. A formal scheme is a filtered colimit of affin* *e schemes. For example the "formal line" ^A1def=colimspecZ[x]=xn n is a formal scheme. Note that an affine scheme is a formal scheme in a trivial * *way. An important feature of this category which we shall use is that it has finite products. For example, ^A1x ^A1= colimspecZ[x]=(xn) Z[y]=(ym ) : In particular a formal group scheme means an abelian group in the category of f* *ormal schemes. A formal group scheme whose underlying formal scheme is isomorphic to the formal scheme * *^A1is called a commutative one-dimensional formal Lie group. We shall simply call it a formal group. The first reason for considering formal schemes is that formal groups are no* *t quite groups in the category of affine schemes, because a group law F(s; t) = s + t + . .2.R[[s; t]] RIEMANN-ROCH AND THE FREE LOOP SPACE * * 3 over a ring R gives a diagonal R[[s]] -!R[[s; t]] ~=R[[s]]^ R[[t]] only to the completed tensor product. The second reason for considering formal schemes is that, if G is an affine * *group scheme, then its torsion subgroup Gtorsis a formal scheme (the colimit of the affine schemes G[N] of tor* *sion of order N), but not in general a scheme. If X is a formal scheme over R, and S is an R-algebra, then XS will denote t* *he resulting formal scheme over S. 2. The umkehr homomorphism and an ungraded analogue 2.1. Let E be a complex-oriented multiplicative cohomology theory with form* *al group law F, and let h: X ! Y be a proper complex-oriented map of smooth finite-dimensional conn* *ected manifolds, of fiber dimension d = dimX - dimY . The Pontrjagin-Thom collapse associates to th* *ese data an "umkehr" homomorphism [Qui71] h!: E*(X) ! E*-d(Y ): We will be concerned with similar homomorphisms in certain infinite-dimensional* * contexts. In order to do so, we systematically eliminate the shift of -d in the degree by restricting ou* *r attention to even periodic cohomology theories E. The examples show (3.3)that this amounts to measuring qu* *antities relative to the vacuum. 2.2. Even periodic ring theories. Let E be a cohomology theory. If X is a sp* *ace, then E*(X) will denote its unreduced cohomology; if A is a spectrum, then E*(A) will denote its* * cohomology in the usual sense. These notations are related by the isomorphism E*(X) ~=E*(1 X+), where X* *+ denotes the union of X and a disjoint basepoint. The reduced cohomology of X will be denoted eE(* *X). Let * denote the one-point space. A cohomology theory E with commutative multiplication is even if Eodd(*) = 0* *. It is periodic if E2(*) contains a unit of E*(*). If E is an even periodic theory, then we write E(X) f* *or E0(X) and E for E0(*). We sometimes write XE = specE(X) for the spectrum, in the sense of commutative alg* *ebra, of the commutative ring E(X). A space X is even if H*(X) is a free abelian group, concentrated in even deg* *rees. In that case the natural map colimFE ! XE; (2.2* *.1) where F is the filtered system of maps of finite CW complexes to X, is an isomo* *rphism. This gives XE the structure of a formal scheme. The functor X 7! XE from even spaces to formal sc* *hemes over E preserves finite products and coproducts: if X and Y are two even spaces, then (X x Y )E ~=XE x YE ~=specE(X)^ E(Y ): Here ^ refers to the completion of the tensor product with respect to the topol* *ogy defined by the filtrations of E(X) and E(Y ). 2.3. Orientations and coordinates. Let P def=CP1 be the classifying space fo* *r complex line bundles. Let m:P x P ! P be the map classifying the tensor product of line bundles. It i* *nduces a map PE x PE mE--!PE; which makes PE a formal group scheme over E. Of course it is a formal group: le* *t i:S2 ! P be the map classifying the Hopf bundle. A choice of element x 2 eE(P) such that v = i*x 2 * *eE(S2) ~=E-2(*) is a unit is called a coordinate on PE. There is then an isomorphism E(P) ~=E[[x]]; 4 MATTHEW ANDO AND JACK MORAVA which determines a formal group law F over E by the formula F(x; y) = m*x 2 E(P x P) ~=E[[x; y]]: Any even-periodic cohomology theory E is complex-orientable. An orientation on* * E is a multiplicative natural transformation MU ! E: These correspond bijectively with elements u 2 eE2(P) such that i*u = 2(1); (2.3* *.1) where is the suspension isomorphism [Ada74 ]. A coordinate x thus determines a* *n orientation u = v-1x. Definition 2.3.2.We shall use the notation (E; x; F) to denote an even perio* *dic cohomology theory E with coordinate x and group law F. We shall call such a triple a parametrized t* *heory. 2.4. Thom isomorphism. An orientation u 2 eE2(P) gives the usual Thom classe* *s and characteristic classes for complex vector bundles. If k is an integer, let k_denote the trivia* *l complex vector bundle of rank k. If X is a connected space and V is a complex vector bundle of rank d over X,* * then we write XV def=1 (P(V 1_)=P(V )) for the suspension spectrum of its Thom space, with bottom cell in degree 2d. W* *e write ffVusualfor the Thom isomorphism ffVusual:E*(X) ~=E*+2d(XV ): In the same way, a coordinate x 2 eE(P) gives rise to a Thom isomorphism ffV :E(X) ~=E(XV ): If v = i*x 2 eE(S2) is the associated orientation, the isomorphisms ffusualand * *ff are related by the formula ffV = vrankVffVusual: Remark 2.4.1.One effect of condition (2.3.1)is that ffd_usualcoincides with * *the suspension isomorphism ffd_usual= 2d:E*(X) ~=E*+2d(Xd_): The Thom isomorphism ff defined by a coordinate chooses v 2 eE(S2) ~=E(*1_) as * *ff1_. Thus ffusualmay be viewed as a composition of Thom isomorphisms d_)-1 ffV ffVusual:E(Xd_) (ff----!E(X) --!E(XV ): If i :1 X+ ! XV denotes the zero section, then we write eusual(Vd)ef=i*ffVusual(1) 2 E2d(X) e(V )def=i*ffV(1) 2 E(X) for the usual and degree-zero Euler classes of V ; these are related by the for* *mula e(V ) = vrankVeusual(V ): If U(n) denotes the unitary group and T is its maximal torus of diagonal mat* *rices, then the map E(BU(n)) ! E(BT) ~=E((BT)n) is the inclusion of the ring of invariants under the action of the Weyl group W* *. The coordinate gives an isomorphism E(BT) ~=E((BT)n) ~=E[[x1; :::; xn]]; RIEMANN-ROCH AND THE FREE LOOP SPACE * * 5 with W acting as the permutation group n on the xi's. Thus we can define degree* *-zero Chern classes ciin E(BU(n)) by the formula Xn nY cizn-i= (z + xi): (2.4* *.2) i=0 i=0 If F is the group law resulting from the coordinate x, then we call the cithe "* *F-Chern classes". Returning to the map X h-!Y; as in (2.1), we can now define an umkehr map E(X) hF--!E(Y ); using the degree-zero Thom isomorphism ff. We write F to indicate the dependenc* *e on the coordinate. Definition 2.4.3.If X is any manifold, we denote by pX the map X X p--!*: If E is an even periodic theory with group law F, then its F-genus is the eleme* *nt pXF(1) of E. 2.5. The Riemann-Roch formula. The Riemann-Roch formula compares the umkehr * *homomor- phisms hF and hG of two coordinates with formal group laws F and G, related by * *an isomorphism : F ! G: The book of Dyer [Dye69 ] is a standard reference. Proposition 2.5.1.If h:X ! Y is a proper complex-oriented map of fiber dimen* *sion 2d, then 2 3 dY x hG(u) = hF 4u . ___j_5; (2.5* *.2) j=1(xj) where the xiare the terms in the factorization Yd zd+ c1zd-1+ : :+:cd = (z + xi) j=1 of the total F-Chern class of the formal inverse of the normal bundle of h. * * |___| Remark 2.5.3.Changing the coordinate by a unit u 2 E multiplies the umkehr h* *omomorphism by ud; by such a renormalization, we can always assume that is a strict isomorphism. 2.6. The fixed-point formula. Notations for circle actions. Let T denote the circle group R=Z. If X is a T* *-space then we write XT def=ET xX T for the Borel construction and XT for the fixed-point set. Let T* = Hom[T; Cx] * *be the character group of T; we will also write ^T= T*- {1} for the set of nontrivial irreducible represe* *ntations. For k 2 T*, let C(k) be the associated one-dimensional complex representation. There is then an asso* *ciated complex line bundle C(k)Tover BT. It is convenient to choose an an isomorphism T* ~=Z; this determines, in par* *ticular, an isomorphism BT ~=CP1 . For k 2 Z we have C(k) = C(1)k and C(k)T= C(1)kT. If ^q2 E(BT) is t* *he Euler class of C(1)T, then the Euler class of C(k)Tis [k](^q). 6 MATTHEW ANDO AND JACK MORAVA Equivariant cohomology. Definition 2.6.1.Let (E; x; F) be a parametrized theory. A T-equivariant coh* *omology theory ET is an extension of (E; x; F) if (1)There is a natural transformation E(X=T) ! ET(X); which is an isomorphism if T acts freely on X. In particular the coefficie* *nt ring ET(*) is an algebra over E(*), and so it is 2-periodic. (2)There is a natural forgetful transformation ET(X) ! E(X): If X is a trivial T-space then the composition E(X) ! ET(X) ! E(X) is the identity. (3)ET has Thom classes and so Euler classes for complex T-vector bundles, whi* *ch are multiplicative and natural under pull-back. If V=X is such a bundle, then we write eT(V ) 2 E* *T(X) for its (degree-zero) Euler class. These are compatible with the Thom isomorphism in E in the se* *nse that, if the T-action on V=X is trivial, then eT(V ) = e(V ): (4)If L1 and L2 are complex T-line bundles, then eT(L1 L2) = eT(L1) +F eT(L2): Definition 2.6.2.If ET is equivariantly complex oriented as above, a homomor* *phism ET ! ^ETof multiplicative T-equivariant cohomology theories is a suitable localization if (1)E^T(*) is flat over ET(*), 1.When k 6= 0, eT(C(k)) maps to a unit of ^ET(*), and (2)The fixed-point formula (2.6.4)holds for ^ET. In order to state the fixed-point formula, we need the following observation* * of [AS68 ]. Lemma 2.6.3.Let ET be a suitable theory. Let S be a compact manifold with tr* *ivial T-action, and let V be a complex T-vector bundle over S. If the fixed-point bundle V Tis zero, th* *en eT(V ) is a unit of ET(S). Proof. Recall [Seg68] that the natural map M V (k) C(k) -!V k2^T is an isomorphism, where V (k) def=Hom[C(k); V ] is the evident vector bundle o* *ver S with trivial T-action. By applying the ordinary splitting principle to V (k), we are reduced to the ca* *se that V = L C(k), where L is a complex line bundle over S with trivial T-action. If ET is suitable then* * the Euler class of V is eT(L C(k)) = e(L) +F eT(C(k)): Since S is a compact manifold, e(L) is nilpotent in E(S), so eT(V ) is a unit o* *f ET(S)_because eT(C(k)) is a unit of ET(*). * * |__| Now suppose that M is a compact almost-complex manifold with a compatible T-* *action. Let j :S ! M denote the inclusion of the fixed-point set; it is a complex-oriented equivaria* *nt map, with T-equivariant normal bundle . The fixed-point formula which we require in Definition 2.6.2 is* * the equation * pMF(u) = pSF _j_u_e: (2.6* *.4) T() RIEMANN-ROCH AND THE FREE LOOP SPACE * * 7 By Lemma 2.6.3 eT() is a unit of ET(S), so this localization theorem is a corol* *lary to the projection formula jFj*(x) = x . eT() for the umkehr of the inclusion of the fixed-point set. Example 2.6.5.The Borel extension EBorel(X) def=E(XT) of an even periodic ring theory has Thom classes eT(V ) = e(VT) for complex T-v* *ector bundles, and the localization defined by inverting the multiplicative subset generated by eT(C(k* *)); k 6= 0 will be suitable. Example 2.6.6.Let KT denote the usual equivariant K-theory. Then KT = Z[q; * *q-1], where q is the representation C(1), considered as a vector bundle over a point. The Euler* * class of a line bundle is eT(L) = 1 - L, so ^q= 1 - q. The group law is multiplicative: Gm (x; y) = x + y - xy: (2.6* *.7) We have [k](^q) = 1 - qk, and consequently ^K(X) def=KT(X) KT Z((q)) is suitable. Example 2.6.8.If H*T(X) def=H*(XT; Q[v; v-1]) is Borel cohomology with two-periodic rational coefficients, and ^q= e(C(1)T), * *then HT(*) ~=Q[[^q]], and e(C(k)T) = k^q: The rational Tate cohomology ^H*(X) def=H*T(X)[^q-1] is suitable. Example 2.6.9.More generally, two-periodic ^K(n)T (with n finite positive) i* *s suitable: if [p]K(n)(X) = Xpn and k = k0ps with (k0; p) = 1 then ns pns [k](^q) = [k0](^qp) = k0^q + . .2.Fp((^q)) has invertible leading term. Integral lifts of K(n) behave similarly; the Cohen* * ring [AMS98 ] of Fp((^q)) defines a completion of the Borel-Tate localization. These coefficient rings have natural topologies, which are relevant to the c* *onvergence of infinite products in Corollary 6.2.3. 3. Application to the free loop space Let ^ETbe a suitable localization (2.6.2)of an equivariantly complex oriente* *d cohomology theory, let X be a compact complex-oriented manifold, and let LX be its free loop space. Since L* *X is not finite-dimensional, the existence of an umkehr homomorphism pLXFis not clear. However, T acts on LX* * by rotations with fixed set X of constant loops, and Witten discovered that the fixed-point formula (2.* *6.4)for the F-genus pLXF(1) of LX continues to yield interesting formulae. In this section we review his ca* *lculation. 3.1. The normal bundle to the constant loops and its Euler class. One approx* *imates the space C1 (S1; C) by the sub-vector space of Laurent polynomials M C[T*] ~= C(k) ,! C1 (S1; C): k2T* The tangent space of LX is C1 (S1; TX). If p 2 X is considered as a constant lo* *op, then the tangent space to LX at p is the T-space TLXp ~=C1 (S1; TXp). It is a T-bundle with a Laurent * *polynomial approximation TXp C[T*] 8 MATTHEW ANDO AND JACK MORAVA Thus the normal bundle of the inclusion of X in LX has approximation M ' TX C(k): (3.1* *.1) k2^T If Yd zd+ c1zd-1+ : :+:cd = (z + xi) j=1 is the formal factorization of the total F-Chern class of TX, then Yd Y eT() = (xj+F [k]F(^q)); (3.1* *.2) j=1k6=0 where ^q= eT(C(1)). 3.2. The fixed point formula. Applying (2.6.4) to the inclusion X -!LX yields the formula 2 3 Yd Y 1 pLXF(1) = pXF4 ___________5: (3.2* *.1) j=1k6=0xj+F [k]F(^q) Equation (3.2.1)requires some interpretive legerdemain. For example, the lea* *ding coefficient of Y (x +F [k]F(^q)) k6=0 Q is the objectionable expression k6=0(k^q); but, as physicists say, this quant* *ity is not directly `observable'. For this reason, we consider the the renormalized formal product Y (x +F [k](^q)) F(x; ^q) def=x __________: k6=0 [k](^q) In section 6 below we provide a natural setting for such formal products. The fixed point formula suggests that we define the equivariant F-genus of L* *X to be 2 3 Yd x p"LXF(1) def=pXF4 ___j____5: (3.2* *.2) j=1F(xj; ^q) 3.3. Examples. The additive group law. When F is the additive group law F becomes Y x Ga(x; ^q)= x (__ + 1) k6=0k^q Y x2 = x (1 - ____22): k>0 k ^q This is the Weierstrass product for ss-1^qsin^q-1ssx, so for the theory ^Hof (2* *.6.8), formula (3.2.2) gives 2 3 dY x =2 "pLXF(1) = (2ssi=^q)d4 ___j____5 [X] : j=1sinh(xj=2) RIEMANN-ROCH AND THE FREE LOOP SPACE * * 9 This is just the ^A-genus of X, up to a normalization depending on the dimensio* *n of X. In [Ati85], Atiyah rewrites the formal product Y x (x + k^q) k6=0 as !2 Y Y x2 x k __2- ^q2 k>0 k>0 k Q and invokes zeta-function renormalization [Den92 ] to replace ( k>0k)2 with 2s* *s, yielding Y x2 2ssx (__2- ^q2) ; k>0k specializing ^qto i then gives the classical expression. From our point of view* * it's natural to think of the Chern class ^qof C(1) as the holomorphic one-form z-1dz on the complex projecti* *ve line, and thus to identify ^qwith its period 2ssi with respect to the equator of CP1 as in x2.1 of [Del89]* *: the `Betti realization' of the Tate motive Z(n) is (2ssi)nZ C. The multiplicative group law. In the case of the equivariant K-theory ^Kof e* *xample (2.6.6), the Euler class of a line bundle L is eT(L) = 1 - L. Writing q for the generator C(1) of * *T*, the Euler class of C(1) is ^q= eT(C(1)) = 1 - q: The multiplicative group law (2.6.7)gives eT(L) +Gm [k](^q) = 1 - qkL; and so the formal product Gm(x; ^q) becomes Y (1 - Lqk)(1 - Lq-k) Gm(x; ^q)= (1 - L) ________________k-k ! k>0 (1 - q )(1 - q ) Y Y (1 - Lqk)(1 - L-1qk) (3.3* *.1) = L (1 - L) ________________k2: k>0 k>0 (1 - q ) Aside for the powers of L, this is essentially the product expansion for the We* *ierstrass oe function Y (1 - qkL)(1 - qkL-1) oe(L; q) = (1 - L) ________________k22 Z[L; L-1][[q]] k>0 (1 - q ) (see for example [MT91 ] or p. 412 of [Sil94]). The infinite factor is objectio* *nable: in the product (3.2.2) defining the hypothetical ^K-genus of LX this factor contributes an infinite po* *wer of topTX, but if c1X = 0 (e.g. if X is Calabi-Yau) and we are careful with the product, we can replace G* *m with . The resulting invariant is the Witten genus [Wit88 , AHS98 ]. Segal [Seg88] replaces the form* *al product Y (1 - qkL)(1 - q-kL) k6=0 which arises in the multiplicative case with Y Y ( q-kL) (1 - qkL)(1 - qkL-1): k>0 k>0 He eliminates the infinitePproduct of L's by assuming topTX trivial and he uses* * zeta-function renormaliza- tion to replace q- k with q-1=12. 10 MATTHEW ANDO AND JACK MORAVA 4.A Riemann-Roch formula for the quotient of a formal group by a free subg* *roup The starting point for this paper was the discovery that the formal products* * (3.2.1)and (3.2.2)which arise in applying the fixed point formula to study the F-genus of the free loop* * space are precisely the same as those obtained from the Riemann-Roch theorem for the quotient of F by a free cy* *clic subgroup. We explain this in section 4.2, after briefly reviewing finite quotients of formal groups,* * following [Lub67 , And95 ]. 4.1. The quotient of a formal group by finite subgroup. In this section, we * *assume that F is a formal group law over a complete local domain R of characteristic 0 and residue* * characteristic p > 0. If A is a complete local R-algebra, the group law F defines a new abelian group stru* *cture on the maximal ideal mA of A. We will refer to (mA; +F) as the group F(A) of A-valued points of F. If H is a finite subgroup of F(R), then Lubin shows that there is a formal g* *roup law F=H over R, determined by the requirement that the power series Y fH (x) def= (x +F h) 2 R[[x]] (4.1* *.1) h2H is a homomorphism of group laws F -fH-!F=H; in other words there is an equation F=H(fH (x); fH (y)) = fH (F(x; y)): The main point is that the power series fH is constructed so the kernel of fH a* *pplied to F(R) is the subgroup H. Q The coefficient f0H(0) = h6=0h of the linear term of fH (x) is not a unit o* *f R, and so fH is an isomorphism of formal group laws only over R[f0H(0)-1]: Over this ring, we might as well re* *place fH with the strict isomorphism Y x +F h fH (x) gH (x) def=x ______= _____0; h6=0 h fH (0) and define G to be the formal group law -1 -1 G(x; y) = gH F(gH (x); gH (y)) over R[f0H(0)-1]; then F=H and G are related by the isomorphism t(x) = f0H(0)x: If F is the group law and R the ring of coefficients of a parametrized theor* *y, then the Riemann-Roch formula (2.5.2)for a compact complex-oriented manifold X is the equation 2 3 dY x pXG(u) = pXF4u ___j__5 (4.1* *.2) j=1gH (xj) over R[f0H(0)-1]. 4.2. The case of a free cyclic subgroup. Now suppose that (E; x; F) is a par* *ametrized theory. Let R be the even periodic theory defined by the formula R(X) def=E(BT x X): The projection BT ! * gives a natural transformation E(X) ! R(X). In particular* * the coordinate x 2 E(P) gives a coordinate x 2 R(P). The group law F is just the group law F, considere* *d over the E-algebra R. What extra structure is available over R? A character 2 T* gives a map BT !* * P, and so an R-valued point u() of PE. As varies through the group of characters, these points assem* *ble into a homomorphism of groups u:T* ! F(R): (4.2* *.1) RIEMANN-ROCH AND THE FREE LOOP SPACE * * 11 It is easy to see that this is the inclusion of a subgroup. As usual we write ^* *q= u(1) for the generator of T* using the isomorphism T* ~=Z. We write (^q) for the subgroup u(T*) generated by* * ^q. The analogue of Lubin's formula (4.1.1)is Y f(^q)(x) = (x +F [k](^q)); (4.2* *.2) k2Z and the resulting Riemann-Roch formula 2 3 Yd Y 1 pXF=(^q)(u) = pXF4u ___________5 j=106=kxj+F [k]F(^q) has right-hand side identical to the formal product (3.2.1)arising from the fix* *ed-point formula. Of course this does not avoid the problems of applying the fixed-point formu* *la to the free loop space. From this point of view, the trouble is thatQthe quotient object F=(^q) isn't a* * formal group; for example the coefficient of x is the product (4.2.2)is k6=0(k^q) This particular problem is* * fixed by using Y x +F [k]F(^q) g(^q)(x)= x __________ 06=k2T*[k]F(^q) = F(x; ^q): The Riemann-Roch formula in this case is 2 3 dY x pXG(u) = pXF4u ____j___5 (4.2* *.3) j=1F(xj; ^q) with right-hand-side identical to the renormalized genus (3.2.2). However, expanding such Weierstrass products as formal power series is still* * highly nontrivial, as the examples above have shown. 5.The structure of F=(^q) Suppose now that F is a formal group law over a complete local ring E with r* *esidue field k of characteristic p > 0, and that F is its pullback over the power series algebra R = E[[^q]]. T* *he coordinate defines a homomorphism u:Z! F(R) (5.0* *.4) u(n)= [n]F(^q); but there is no reason to expect the cokernel of this homomorphism to be a form* *al group. However, F=(^q) certainly makes sense as a group-valued functor of complete local R-algebras. In section 5.1, we show by construction that the torsion subgroup of F=(^q) * *has a natural approximation by a representable functor. We construct a formal group scheme Tate(F) over R t* *ogether with a natural transformation Tate(F)tors! F=(^q)tors of group-valued functors, which is an isomorphism if ^qhas infinite order in F.* * The formal group scheme Tate(F)torsis our model for F=(^q). Because E has finite residue characteristic, we work p-locally: the formal g* *roup scheme Tate(F)[p1 ] is a p-divisible group in the sense of [Tat67, Dem72 ]. The p-torsion subgroup G[p* *1 ] of a formal group G of finite height is a connected p-divisible group, but Tate(F)[p1 ] is not; its ma* *ximaletale quotient is a constant height-one p-divisible group Qp=Zp, and its connected component is just the p-d* *ivisible group F[p1 ] of F. In other words, there is an extension F[p1 ] -!Tate(F)[p1 ] -!Qp=Zp 12 MATTHEW ANDO AND JACK MORAVA of p-divisible groups over R. We will see in (5.3.1)that it is in fact the univ* *ersal example of such an extension, and it follows that if En is the ring which classifies lifts of a formal group * *G of height n over an algebraically closed field k [LT66 ], then R = En[[^q]] represents the functor which classifi* *es lifts of a p-divisible group of height n + 1 with connected component G[p1 ]. Thus ^qmay be viewed as a "Serre-* *Tate parameter" in the sense of [Kat81]. 5.1. A model for the torsion subgroup of F=(^q). One difficulty in represent* *ing the quotient F=(^q) over E[[^q]] is that the subgroup (^q) does not act universally freely on F: co* *nsider, for example, the spe- cialization ^q= 0. It turns out that, as long as one restricts to the torsion s* *ubgroup of F=(^q), this is the only obstruction. By freeing up the action, we are able to construct a represen* *table functor whose torsion subgroup coincides with that of F=(^q) whenever ^qis of infinite order. Following [KM85 , x8.7], let Tate(F) be the scheme over R = E[[^q]] defined* * by the disjoint union [ Tate(F) = FR x {a}: a2Q\[0;1) If A is a complete local R-algebra, then Tate(F)(A) = {pairs(g; a) withg 2 F(A) anda 2 Q \ [0; 1):} This has a group structure given by ( (g; a) . (h; b) = (g +F h; a + b) if a + b < 1 (g +F h -F ^q; a + bi-f1)a + b 1. By construction, Tate(F)(A) is the quotient in the exact sequence 0 -! Z -! F(A) x Q -!Tate(F)(A) -!0 n 7! ([n](^q); n) (5.1* *.1) (x; a)7! (x -F [[(a)](^q); ](a)); where [(a) and ](a) are the integral and fractional parts of the rational numbe* *r a. Equivalently, Tate(F) is the pushout of Z ! Q ! Q=Z along the homomorphism u: Z ----! Q ----! Q=Z ? ? ? u?y ?y ?y= FR ----! Tate(F)----! Q=Z: Thus Tate(F) is a kind of homotopy quotient of F by Z. Proposition 5.1.2.Projection onto the first factor in the construction above* * defines a natural trans- formation Tate(F)tors! (F=(^q))tors; which is an isomorphism if ^qis not in F(A)tors. Proof. Let A be a complete local R algebra, and suppose that ^qis not torsio* *n in F(A). We see from (5.1.1) that there is an isomorphism Tate(F)(A) ~=F=(^q)(A) x Q which is compatible with the projection and clearly induces an isomorphism Tate(F)tors(A) ~=(F=(^q))tors(A): * *|___| Remark 5.1.3.The Proposition holds in any context in which ^qis not torsion.* * Another source of examples is p-adic fields. Let S = E[1_p]((^q)). Let L be a complete nonarchime* *dean field with norm k - k. Given a continuous map S ! L, the group law F defines a group structure on the * *set {v 2 L|kvk < 1}; and RIEMANN-ROCH AND THE FREE LOOP SPACE * * 13 we write F(L) for this group. The subgroup generated by ^qis necessarily free, * *and the argument shows that there is an isomorphism of groups Tate(F)tors(L) ~=(F=(^q))tors(L): Remark 5.1.4.If the ring E is the ring of coefficients in an even periodic r* *ing theory, then the preceding construction could be carried out with R = E(BT) and the homomorphism u (4.2.1)* *. The result is an extension PE ! Tate(PE) ! (T* Q)=T* of group schemes over R. Proposition 5.1.2 becomes an isomorphism Tate(PE)tors~=((PE)R=T*)tors: 5.2. Notation. If A is an abelian group, then AT = specTA is the resulting c* *onstant formal group scheme over T. The category of test rings is the category of Artin local E-alge* *bras with residue field k. Definition 5.2.1.If G is a formal group scheme over a ring T, and A is an ab* *elian group, let Hom[A; G] be the group of homomorphisms AT -!G of formal group schemes over T. Similarly, let Ext[A; G] be the set of isomorph* *isms classes of extensions of formal group schemes G -!X -!AT: If G is a formal group scheme over E, let Hom_[A; G] and Ext_[A; G] be the func* *tors from test rings to groups such that Hom_[A; G](T) def=Hom[AT; GT] and Ext_[A; G](T) def=Ext[AT; GT]: Now if G is a formal group over E, pulling back over G defines a natural poi* *nt :G -!G x G = GG and hence a homomorphism u:ZG -!GG: It is clear that this gives an isomorphism G ~=Hom_[Z; G]. Equivalently, if F is a formal group law over E, then R = E[[^q]] pro-repres* *ents the functor Hom_[Z; F] on the category of test rings, with universal example u(n) = [n]F(^q) (5.0.4). * *Similarly, if E is the ring of coefficients of an even periodic ring theory, then R = E(BT) pro-represents the* * functor Hom_[T*; PE], with the homomorphism u of (4.2.1)as the universal example. 5.3. Universal properties. A universal extension. A continuous homomorphism of E-algebras from R to a t* *est algebra T defines an extension FT -!(Tate(F))T -!(Q=Z)T and hence an extension F[p1 ]T -!(Tate(F)[p1 ])T -!(Z[1_p]=Z)T : of torsion subgroups. Lemma 5.3.1.The ring R pro-represents the functor Ext_[Qp=Zp; F], with Tate(* *F) as the universal ex- ample. 14 MATTHEW ANDO AND JACK MORAVA Proof. In the exact sequence Hom_[Q; F] -!Hom_[Z; F] -!Ext_[Q=Z; F] -!Ext_[Q; F]; the first and last terms are zero because p acts nilpotently on F and as an iso* *morphism on Q. |___| A universal p-divisible group. Any p-divisible group over a field k is natu* *rally an extension 0-! -!et (5.3* *.2) of a connected group by anetale group. If the residue field k is algebraically * *closed, then the sequence (5.3.2) has a canonical splitting [Dem72 , p. 34]. We will be interested in the case wh* *en ethas height one, which is to say that it is isomorphic to the constant group scheme (Qp=Zp)k. Tate showed [Tat67] that the functor G 7! G[p1 ] is an equivalence between t* *he categories of formal groups of finite height and connected p-divisible groups. Let's fix a one-dime* *nsional formal group 0 of height n over the algebraically closed field k and define to be the product ex* *tension 0! ! (T* Qp=Zp)k : Definition 5.3.3.If G is a p-divisible group over k, and if T is a test ring* *, then a lift of G to R is a pair (F; ffi) consisting of a p-divisible group F over R and an isomorphism Fk ffi-!~G = of p-divisible groups over k. An equivalence of lifts (F; ffi) and (F0; ffi0) i* *s an isomorphism f :F ! F0 such that ffi = ffi0fk: The set of isomorphism classes of lifts of G to R will be denoted LiftsG(R). As* * R varies, LiftsG(R) defines a functor from test rings to sets. Lubin and Tate construct a formal power series algebra En over the Witt ring* * of k which pro-represents the functor Lifts0. There is an even periodic cohomology theory with En as ring* * of coefficients, and the universal lift F of 0 as formal group. Theorem 5.3.4.The ring R = En(BT) pro-represents Lifts, with universal examp* *le Tate(F)[p1 ]. Proof. Let T be a test ring. Suppose that (H; ffl) is a lift of . Then (H0; * *ffl0) is a lift of 0. According to [LT66 ], there is a unique pair (f; a) consisting of a map f :En ! T and an * *isomorphism a:(FT; fflunivT) ~=(H0; ffl0) of lifts of 0. On the other hand, ffletinduces an isomorphism Het= (Het(k))T ~=(et(k))T = (T* Qp=Zp)T: Assembling these gives an extension FT ! H ! (T* Qp=Zp)T which defines an isomorphism Lifts~=Ext_[F; T* Qp=Zp] : R pro-represents the right-hand side by Lemma 5.3.1. * * |___| Remark 5.3.5.The analogous result for the ordinary multiplicative group Gm i* *s described in [KM85 , x8.8]. Closely related examples occur in [AMS98 ], which are motivated by pur* *ely homotopy-theoretic questions about Mahowald's root invariant. RIEMANN-ROCH AND THE FREE LOOP SPACE * * 15 5.4. Examples. If L 2 A((q)) is a unit, the formal product defining the Weie* *rstrass function of3.3 defines an element oe(L; q) 2 A((q)). The functional equation oe(qL; q) = (-L)-* *1oe(L; q) then implies that the modified product oe[L; r] def=q-[(r)([(r)+1)=2(-L)[(r)oe(L; q) (where [(r) is the integral part of r 2 Q) satisfies the identity oe[qL; r +1] * *= oe[L; r] and can thus be regarded as a function from a localization of Tate(Gm ) to the Z((q))-line. Since oe(L; * *0) = 1 - L we can think of the modified function as a deformation of the usual coordinate at the identity of t* *he multiplicative group. Similarly, if we regard ^qas an element of the locally compact field C, the * *modified sine function s[x; r] def=ss-1^qsin^q-1(ssx - r) satisfies the identity s[x + ^q; r + 1] = s[x; r] and so can be interpreted as * *a function from Tate(Ga)(A) to A. It is also a deformation of the usual coordinate on the additive group, in that* * s[x; 0] ! 0 as ^q! 1. In general, however, there seems to be no reason to expect that a coordinate* * on F will extend to a coordinate on Tate(F): our construction yields a group object, but not a group * *law. In the two examples above, we do have (something like) coordinates, which define interesting genera* *: ordinary cohomology leads to the ^A-genus, suitably normalized, if r = 0; but if r 6= 0 2 Q=Z, Cauchy's t* *heorem (applied to a small circle C around the origin) yields 2 3 _1_Z pX 4j=dY___^q-1ssxjz_5 z-d-1dz = (___ss_)dO(X) : 2ssiC Ga j=1sin^q-1ss(xjz + r^q) ^qsinssr The K-theory genus extends similarly, but the resulting function is just (-1)dq* *-d[(r)([(r)+1)=2times the standard elliptic genus. Any multiplicative cohomology theory E can be described as taking values in * *a category of sheaves over specE(*), and the Borel extension of such a theory takes values in sheaves* * over spec(E(*)[[^q]]). The construction in 5.1 of the Tate group as a disjoint union of copies of such aff* *ines implies that a theory E with formal group F has a natural extension to an equivariant theory taking val* *ues in a category of sheaves over the group object Tate(F). Similarly, a suitable localization of the Borel * *extension defines an equivariant theory taking values in sheaves over a suitable localization of Tate(F). This r* *esembles (but is easier than) the constructions of [GKV , Gro94, RK99 ], for here we're only patching togeth* *er Borel extensions.] 6.Prospectra and equivariant Thom complexes Cohen, Jones, and Segal show that pro-objects in the category of spectra are* * the appropriate context in which to study Thom complexes, and so umkehr maps, for semi-infinite vector bun* *dles. In this section we observe that the ideas of this paper fit very naturally into their framework. 6.1. Thom prospectra. If V and W are complex vector bundles over a space X, * *we extend the notation for Thom isomorphisms in 2.4.1 by writing ffVWdef=ffV O (ffW )-1:E(XW ) ! E(XV ) : `Desuspending' by V W then gives a homomorphism ff-W-V:E(X-V ) ! E(X-W ) : The inclusions of a filtered vector bundle V : 0 = V0 V1 . . . define maps iVn: XVn-1! XVn of Thom spectra, which desuspend to define a pro-object Vn X-V def={. .!.X-Vn i--!X-Vn-1! : :}: 16 MATTHEW ANDO AND JACK MORAVA in the category of spectra. The E-cohomology of X-V is the colimit E(X-V) def=colimnE(X-Vn); as in the appendix to [CJS95 ]. Lemma 6.1.1.On cohomology, the homomorphism induced by iVn is e(Vn=Vn-1)ff-V* *n-Vn-1. |___| Example 6.1.2.For any integer n, let M In = C(k); n|k|>0 thus I = colimnIn is a filtered T-vector bundle over a point. More generally, i* *f V is a complex vector bundle over a space X, let IV def=V I be the corresponding filtered T-vector bundle. In this notation the Laurent app* *roximation (3.1.1)to the formal normal bundle of the constant loops in LX is ITX. We write jn for the map jn = iInV:X-InV ! X-In-1V: Example 6.1.3.If Vn is the sum of n copies of C(1), considered as a T-vector* * bundle over a point, then the Borel construction on V is the Thom prospectrum for CP11constructed in [CJS* *95 ]. 6.2. Equivariant cohomology. Suppose now that ^ETis a suitable extension of * *an equivariantly ori- ented theory ET, and let X be a finite complex with trivial T-action, as above.* * As n varies, the Thom isomorphisms ffInV:ET(X) ET-!~(X-InV) = are not compatible with the maps jn; but this can be cured over ^ETby a suitabl* *e renormalization. By (2.6.3) the class Y un(V ) def=e(V In) = eT(V C(k)): 0<|k|n is a unit of ^ET(X), and the homomorphism !n(V ):E^T(X) -!^ET(X-InV) defined by the formula !n(V ) = un(V )ff-InV is an isomorphism. Theorem 6.2.1.The diagram ^ET(X) _______ E^T(X) ? ? !n-1(V?)y ?y!n(V ) E^T(X-In-1V)-jn---!^ET(X-InV) commutes; in particular, the maps !n(V ) assemble into a "Thom isomorphism" ~= -IV !V :^ET(X) -!E^T(X ) : * *|___| If V and W are two complex vector bundles over X, then (as with the usual Th* *om isomorphism) we define ~= -IV !VW:^ET(X-IW ) -!E^T(X ): RIEMANN-ROCH AND THE FREE LOOP SPACE * * 17 Corollary 6.2.2.If the vector bundle V has rank d, then the relative isomorp* *hism !Vd:^ET(X-Id_) ! ^ET(X-IV) is given by the formula 0 1 Yd Y x + [k](^q) !Vd= @ _j_F_____A ff-IV-Id_; j=1k6=0 [k](^q) where the xj are the terms in the formal factorization (2.4.2)of the total F-Ch* *ern class of V . |___| The terms in this product are well-defined at any finite stage, if not in th* *e limit. If ET is ^Hor ^K(and c1(V ) = 0), then the infinite products converge and we have the Corollary 6.2.3.If the vector bundle V has rank d, then the diagram -Id_ E^T(Xd_)ffd_----!^ET(X-Id_) ? ? ^ffVd_?y ?y!Vd_ -IV E^T(XV )ffV----!^ET(X-IV) : commutes; where ^ffVd_def=F(V_)cffVd_: d(V ) More precisely, the assertion is that these examples are naturally topologiz* *ed, and that at any finite_ stage the diagram commutes modulo error terms which converge in this topology t* *o zero as n grows. |__| In other words, in a suitable theory with group law F the isomorphism !Vd_is* * very much like a Thom isomorphism ^ffVd_for a theory with F replaced by its extension Tate(F). From t* *his point of view, the fact that Witten's formula for the ff-genus of LX equals the ^ff-genus of X can be i* *nterpreted as saying that the inclusion j :X ! LX behaves as if there is a cohomological analogue j*:E^T(X-ITX ) ! ^ET(LX) of the Thom collapse map, with an associated umkehr jF = j*!TX satisfying the p* *rojection formula jFj*(x) = x . eT() : Perhaps the intuition underlying the physicists' interest in elliptic cohomolog* *y is that a reasonable equivariant theory applied to the geometrical object X-ITX captures more information than t* *hat theory does, when applied directly to X. Thus the equivariant K-theory of this formal neighborhoo* *d of X is the Tate elliptic cohomology of X. This seems to be related to the recent construction (by Kontse* *vich and others) of new invariants for singular complex algebraic varieties, by considering them as var* *ieties over the Laurent series field C((q)). 6.3. Polarizations. In the preceding account, the role of the rational param* *eter in the construction of Tate(F) is geometrically unmotivated, because we have ignored some issues conne* *cted with the polarization [CJS95 ] of the loopspace. Such a structure is an equivalence class of splittings of the tangent bundle* * of LX into a sum of positive- and negative-frequency components: if X is complex-oriented (e.g. symplectic, w* *ith a choice of compatible almost-complex structure), then the composition LX ! L(BU) = U x BU ! U=SO of the map induced by the classifying map for the tangent bundle of X with the * *projection to a classifying space for such splittings defines a canonical polarization. 18 MATTHEW ANDO AND JACK MORAVA The universal cover of the free loopspace. If X is simply-connected, the fun* *damental group of its free loopspace will be isomorphic to H2(X; Z), which will be nontrivial in general. * *There is thus good reason to consider the simply-connected cover gLXof the loopspace: this can be defined* * as the space of smooth maps of a two-disk to X, modulo the relation which identifies two maps if their* * restrictions to the boundary circle agree, and if furthermore their difference, regarded as an element of th* *e deck-transformation group ss2(X) = H2(X; Z), is null-homotopic. The circle acts on gLXby rotating loops, * *as does the fundamental group of LX, and in general the fixed-point set gLXT~=H2(X; Z) x X will have many components. The choice of a basepoint defines a lift of the cano* *nical polarization to a map gLX! R x SU=SO which restricts to a locally constant map H2(X; Z) x X ! R x SU=SO ! R defined by evaluating ff 2 H2(X; Z) on the first Chern class of X. We think of * *the polarization as defining the zero-frequency modes in the Fourier decomposition of small loops near a fix* *ed-point component, so that shifting by fi 2 H2(X; Z) gives an isomorphism T T TLgXx;ff~=TLgXx;ff+fi C() : T-equivariant Picard groups and orientations. We can regard the cohomology o* *f a space Y as a sheaf of rings over the zero-dimensional scheme (ss0Y )Z def=specH0(Y; Z) : Similarly, the set of equivalence classes of T-line bundles over a T-space X is* * naturally isomorphic to H2(YT; Z), which can be interpreted as a sheaf of groups over (ss0Y )Z, the fib* *er above component Yibeing the constant group scheme coker[H2T(*; Z) ! H2T(Yi)] ; when the circle action on X is trivial, this is just a complicated way of index* *ing the summands of H2(Y; Z). An orientation on a complex-oriented cohomology theory E with formal group F* * defines a natural homomorphism from the Picard group of line bundles over Y to the group of E(Y )* *-valued points of F. This suggests that a complex orientation for the Tate extension of an equivariantly * *complex-oriented theory should be defined as a natural transformation from the Picard group of T-line bundles * *over X to Tate(F)(E(Y )), both regarded as schemes over (ss0Y )Z; by restriction such a transformation de* *fines a map ss0Y ! Q=Z : In other words, such a generalized orientation assigns to a component of Y and * *a T-line bundle over it, a characteristic class in the cohomology of the component, together with an r 2 Q* * depending on the component, which shifts by an integer when the bundle is twisted by a character. T When Y is gLX, the polarization defines a very natural map of this sort, whi* *ch sends ff to . In more general situations, e.g. in Givental's work [Vo99 ] on the quantum coh* *omology of a symplectic manifold (X; !) the evaluation map ff 7! plays a similar role; but in t* *his generalization, ! no longer needs to be an integral class. 7. Concluding remarks The parts of this paper which deal principally with the fixed point formula * *on the free loop space are formulated in terms of a general equivariant cohomology theory ET, but those wh* *ich relate to the quotient F=(^q) use only the group law F of the theory R(X) = E(BT x X), and so essentia* *lly use the Borel theory EBorel. We do this because in that case we can be more specific about our const* *ructions. In good cases one can hope to do better. RIEMANN-ROCH AND THE FREE LOOP SPACE * * 19 Specifically, suppose that the multiplication homomorphism T x T ! T induces* * a map ET ! ETxT and that one has an isomorphism ETxT(*) ~=ET(*) ET(*). The upshot is a group s* *cheme G = specET(*) E over E, such that the formal group PE associated to E(P) is the completion of G* *. Indeed [GKV , Gro94], one expects in general that there is an abelian group scheme G which is the mor* *e fundamental object in T-equivariant E-theory, with ET(*) as structure sheaf. In any case, as remarked in x5.2, over GG = G x G there is a natural map ZG ! GG; (7.0* *.1) and the natural map GG ! Hom[Z; G] is an isomorphism. One could consider the gr* *oup Tate(G) over G, fitting into a short exact sequence GG -!Tate(G) -!Q=Z: The group Tate(PE) considered in the main text would then arise from Tate(G) by* * completing in both copies of G. This is the situation in K-theory. Over the ring KT = Z[q; q-1] one has a ho* *momorphism u:Z ! GmKT n 7! qn and this gives rise to a group Tate(Gm ) over KT. Over the completion KT ! K(BT* *) = Z[[^q]] one recovers the group Tate(Gm ) considered in the main text. On the other hand, the group * *Tate(Gm )torsbecomes isomorphic to the torsion subgroup of the classical Tate curve Tate(q) already * *over the suitable localization K^= Z((q)) of (2.6.6), where u is the inclusion of a sub-groupscheme. See [KM85* * , x8.8] for details. Interestingly enough, in the case of K-theory this is the solution to the pr* *oblem of the infinite products. We have purposefully written the equivariant Euler class of L C(k) as e(L) +F * *[k]F(^q), as in any finite situation the formula 1 - L1L2= (1 - L1) + (1 - L2) - (1 - L1)(1 - L2) makes it possible to calculate the Euler class of a vector bundle using the for* *mal multiplicative group, i.e. the right-hand side. However, in order to calculate the infinite product Gm(x; * *^q) (3.3.1), one is forced to use the left-hand side. Thus from this point of view the renormalization is han* *dled by knowing about the global multiplicative group, instead of merely its formal completion. It seems reasonable to hope that the rich theory of T-equivariant ring spect* *ra will provide additional examples of such ET. 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