On the Chow rings of classifying spaces for classical groups

On the Chow Rings of Classifying Spaces for Classical Groups.

LUISALBERTOMOLINAROJAS(*) - ANGELOVISTOLI(**)

ABSTRACT- We show how the stratification method, introduced by Vezzosi in his study ofPGL3, provides a unified approach to the known computations of the Chow rings of theclassifying spaces ofGLn,SLn,Spn,OnandSOn.

1. Introduction.

To algebraic topologists, the cohomology of classifying spaces of linear algebraic groups (or, equivalently, of compact Lie groups) has been an important object of study for a long time. Recently, B. Totaro ([Tot99]) has introduced an algebraic analogue of this cohomology, the Chow ring of the classifying space of a linear algebraic groupG, denoted by A_G. If H_G de- notes theintegral cohomology of theclassifying spaceof G, there is a natural ring homomorphism A_G!H_G, which is, in general, neither sur- jective not injective.

Remarkably, the Chow ring A_G seems to be smaller and easier to control than H_G, while still containg a lot of information: for example, ifGis a finiteabelian group A_Gis the symmetric algebra on the group of char- acters, while H_Gis much larger (unlessGis cyclic). This is truly surprising to someone who is familiar with the theory of Chow rings of smooth pro- jective algebraic varieties, because these tends to be infinitely more complicated and less computable than their cohomology.

Rationally, the situation is very well understood. If G is a connected algebraic group, then the homomorphism A_GQ!H_GQ is an iso- morphism, and both rings coincidewith thering of invariants under the

(*) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Roma Tre, Largo San Leonardo Murialdo 1, I-00146 Roma, Italy; e-mail: molina@mat.uniroma3.it

(**) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Bologna, Piazza di Porta San Donato 5, I-40126 Bologna, Italy; e-mail: vistoli@dm.unibo.it

Weyl group in the symmetric algebra of the ring of characters of a maximal torus; this is classical, due to Leray and Borel, in the case of cohomology, and to Edidin and Graham ([EG97]) for theChow ring. Furthermore, this ring of invariants is always a polynomial ring, as was shown by Chevalley.

With integral coefficients, the situation is much more subtle.

TheChow ring A_Ghas been computed for the classical groups GLn, SLn, Sp_n, On or SOn, but not for thePGLn series. The results are as follows.

Each of the groups above comes with a tautological representation, of di- mensionn(or 2n, in thecaseof Sp_n). Every representationV of an alge- braic groupGhas Chern classes c_i(V)2Aⁱ_G. WhenGis a classical group, we denote the Chern classes of the tautological representation simply byc_i.

Burt Totaro ([Tot99]) and R. Pandharipande ([Pan98]) described A_G when GˆGLn, SLn, Sp_n, On and SOn when n is odd. Wewill usethe following notation: ifRis a ring,t₁;. . .;t_nare elements ofR,f₁;. . .;f_rare polynomials inZ[x₁;. . .;x_n], wewrite

RˆZ[t1;. . .;tn]= f1(t1;. . .;tn);. . .;fr(t1;. . .;tn)

to indicatethetheringRis generated byt1;. . .;tn, and the kernel of the evaluation mapZ[x1;. . .;xn]!Rsendingx_itot_iis generated byf1;. . .;fr. When there are nof_ithis means thatRis a polynomial ring in thet_i.

First thecaseof thespecial groups.

THEOREM[B. Totaro].

(1) A_GLnˆZ[c₁;. . .;c_n].

(2) A_SLnˆZ[c₂;. . .;c_n].

(3) A_SpnˆZ[c₂;c₄;. . .;c_2n].

The first two cases follow very easily from the well known description via generators and relations of the Chow ring of a Grassmannian.

In all threecases, theChow ring is isomorphic to thecohomology ring.

THEOREM[R. Pandharipande, B. Totaro]

(1) A_On ˆZ[c1;. . .;cn]=(2codd).

(2) If n is odd, thenA_SOnˆZ[c2;. . .;cn]=(2codd).

Thenotation 2coddmeans ``all the elements 2ciforiodd''.

In these cases the Chow ring is not isomorphic to the cohomology ring:

thering H_On was computed independently by Brown (see [Bro82]) and Feshbach ([Fes83]); the result is considerably more involved. The fact that these formulae are so simple is another manifestation of the tamer nature of A_G, as opposed to H_G.

Whennis odd, then On'SOnm₂, and this allows to obtain theresult for SOn from that for On. When nis even this fails, and the situation is more complicated. Even rationally, the Chern classes of the tautological representation do not generate the Chow ring, or the cohomology. It is well known that when nˆ2m the tautological representation has an Euler class em2H^2m_SOn, whosesquareis ( 1)^mcn: this class, together with the even Chern classesc2,c4;. . .;cn 2generate A_SOnQˆH_SOnQ. Totaro noticed that whennˆ4 theclass e2 is not in theimageof A_SOn; shortly afterwards, Edidin and Graham ([EG95]) constructed a classy_m2A^m_SOn, whoseimagein H_SOnis, rationally, 2^{m 1}e_m.

Subsequently, Pandharipande computed A_SO4: heshowed that it is generated byc2,c3,c4andy2, and gave the relations (his description of the classy2is different, but equivalent to that of Edidin and Graham). Finally, in her Ph.D. thesis Rebecca Field obtained the general result ([Fie04]), which is as follows.

THEOREM[R. Field]. When nˆ2m is even, then

A_SOnˆZ[c₂;. . .;c_n;y_m]= y²_m ( 1)^m2^{n 2}c_n; 2c_odd; y_mc_odd : Furthermore there are many results, due to Totaro himself ([Tot99]), to P. Guillot ([Gui04a] and [Gui04c]), and to N. Yagita ([Yag02]), for finite groups. Chow rings of classifiying spaces of exceptional groups have been studied by Yagita (see [Yag04], [Yag05]) and Guillot ([Gui04b]).

As we already mentioned, the PGL_n series is much harder (this is an example of a universal phenomenon, that of all the classical groups, these are the ones giving rise to the deepest problems). Fornˆ2 wehavethat PGL₂ˆSO₃, and for this group everything is well understood. The co- homology withZ=3Zcoefficients of the classifying space of PGL₃has been computed in [KMS75], and that of PGL_n with Z=2Z coefficients when n2 (mod 4) in [KM75] and [Tod87]; furthermore, several results on the cohomology of PGLpwithZ=pZcoefficients were proved in [VV03]. To our knowledge, not much else was known about the cohomology of the classi- fying spaceof PGL_n withZ=pZcoefficients, whenpdividesn.

Concerning the Chow ring, for nˆ3 there is a difficult paper of G. Vezzosi ([Vez00]), where he describes A_PGL3almost completely. Here is his basic idea. The fundamental tool is the equivariant intersection theory that Edidin and Graham ([EG98]) have forged starting from Totaro's idea.

Vezzosi stratifies the adjoint representationsl₃of PGL₃by typeof Jordan canonical form, compute the Chow ring of each stratum, and then get generators for A_PGL3using the localization sequence for equivariant Chow

groups. To get relations he restricts to appropriate subgroups of PGL3. His technique has been refined and improved by the second author in [Vis05], where he studies the Chow ring and the cohomology of the clas- sifying spaceof PGL_p, wherepis an odd prime.

Thepurposeof this articleis to show how this stratification method provides a unified approach to all the known results on the Chow ring of classical groups. Consider a classical group G with its tautological representation V. Then one stratifies V in strata in which thestabi- lizers are, up to an extension by a unipotent group, smaller classical group. Using thelocalization sequence for equivariant Chow groups this gives generators for the Chow rings, with relations that come out naturally. To show that therelations suffice, one restricts to appro- priatesubgroups ofG: a maximal torus first, to show that therelations sufficeup to torsion, then to somefinitesubgroup to handletorsion.

This turns out to bereasonably straightforward for all theclassical groups, except for PGL_n. From our point of view, this is due to the fact that the natural representation to use for PGLn, which is theadjoint representation, has a much more complicated orbit structure then in thecases of theother groups.

For thecases of Sp_n and O_n, Totaro's proofs, based on his very inter- esting and important [Tot99, Proposition 14.2], are much simpler. In the caseof SOnfor evenn, Totaro's method, as implemented by Field, does not seem easier than the stratification method. In the case of PGLn, the stratification method provides the best known results; but there is also a very recent preprint of Kameko and Yagita where they also compute the additivestructureof A_PGLnand H_PGLn, with completely different methods, using the Adams spectral sequence for Brown-Peterson cohomology ([KN05]).

A few words about the future (¹). Despite its elementary nature, the stratification methods is powerful; also, as was pointed out to the second author by N. Yagita, it might yield interesting results when applied to more general cohomology theories. However, it seems clear that to pro- ceed much further one will eventually need to introduce substantial amounts of homological machinery, as provided by the theory of motivic cohomology of Voevodsky and Morel. Thus, the way is indicated by the work of N. Yagita (see for example [Yag03] and [Yag05]).

(¹) The authors are well aware of the risks involved in making predictions, as people always play Chesterton's game ``Cheat the Prophet''.

Acknowledgments. ± Pierre Guillot pointed out an error in the first version of this paper. We thank him heartily. We are also grateful to the referee, who did an extremely careful and competent job.

The second author would like to acknowledge the very interesting discussions he has had with Nobuaki Yagita and Andrzej Weber on the subject of this paper.

2. Preliminaries and notation.

In this section we recall some definitions and notations, and state some tecnical results that will be used throughout this paper.

All schemes and algebraic spaces are assumed to be of finite type over a fixed fieldk. All algebraic groups will be linear algebraic group schemes overk, and all representations will bek-representations. IfXis a scheme overkandk⁰is an extension ofk, wesetX_k^0defˆ X_SpeckSpeck⁰.

Thenotation for algebraic groups will bestandard: thusGmwill bethe multiplicativegroup overk, andm_nthegroup schemeofn^throots of 1.

LetGa linear algebraic group overk, andX a smooth scheme overk with aG-action.

Edidin and Graham ([EG98]), expanding on the idea of Totaro, have defined theG-equivariant Chow ring ofX, denoted A_G(X), as follows. For eachi0, choose a representationVofGwith an open susbschemeUV on whichGacts freely (in which case we call (V;U) agood pair for G), and such that thecodimension ofVnU is greater thani. Theaction ofGon XUis also free, and the quotient (XU)=Gexists as a smooth algebraic space; then Edidin and Graham define

Aⁱ_G(X)^defˆAⁱ (XU)=G

;

where the right hand term is the usual Chow group of rational equivalence classes of cycles of codimensioni. This is easily seen to be independent of thegood pair (V;U) chosen. Then one sets

A_G(X)^defˆM

i0

Aⁱ_G(X):

IfGacts freely onX, then there is a quotientX=Gas an algebraic space of finitetypeover k, and theprojectionX!X=GmakesXinto a G-torsor over X=G; and in this casethering A_G(X) is canonically isomorphic to A(X=G).

Totaro's definition of the Chow ring of a classifying space is a particular caseof this, as

A_GˆA_G(Speck):

The formal properties of ordinary Chow rings extend to equivariant Chow rings. We recall briefly the properties that we need, which will be used without comments in the paper, referring to [EG98] for the details.

Iff:X!Yis an equivariant morphism of smoothG-schemes there is an induced ring homomorphism f:A_G(Y)!A_G(X), making A_G into a contravariant functor from smooth G-schemes to graded commutative rings. Furthermore, iff is proper there is an induced homomorphism of groupsf:A_G(X)!A_G(Y); the projection formula holds. This means that if h2A_G(Y) andj2A_G(X), then

f(jfh)ˆfjh;

in other words,fis a homomorphism of A_G(Y)-modules.

There is also a functoriality in the group: ifH!Gis a homomorphism of algebraic groups, the action ofGonXinduces an action ofHonX, and there is homomorphism of graded rings A_G(X)!A_H(X). When H is a subroup ofGwewill refer to this as arestriction homomorphism.

IfHis a subgroup ofG, then there is anH-equivariant embeddingX intoXG=H, defined in set-theoretic terms by sendingxinto (x;1). Then thecompositeof therestriction homomorphism A_G(XG=H)!

!A_H(XG=H) with thepullback A_H(XG=H)!A_H(X) is an iso- morphism.

Of paramount importance is the localization sequence; ifYis a closed G-invariant subscheme ofX, and wedenotebyi:Y,!Xandj:XnY,!X the inclusions, then the sequence

A_G(Y) !ⁱ A_G(X) !^j A_G(XnY)!0 is exact.

Furthermore, if E is a G-equivariant vector bundle on X, there are Chern classes c_i(E)2Aⁱ_G(X), enjoying the usual properties. Also, the pullback A_G(X)!A_G(E) is an isomorphism.

In particular, since the equivariant vector bundles over Speckarethe representations of G, we get Chern classes c_i(V)2Aⁱ_G for every re- presentation ofG; and thepullback A_G!A_G(V) is an isomorphism.

We also need other easy properties of equivariant Chow rings, for which we do not have a suitable reference.

LEMMA2.1. Let G a linear algebraic group, X a smooth G-scheme, H a normal algebraic subgroup G. Suppose that the action of H on X is free with quotient X=H. Then there is canonical isomorphism of graded rings

A_G(X)'A_G=H(X=H):

PROOF. Le t (V;U) bea good pair forG, such that thecodimension of VnUis greater theni. Then

Aⁱ_G(X)ˆAⁱ (XU)=G

ˆAⁱ ((XU)=H)=(G=H)

ˆAⁱ_G=H (XU)=H :

Now, thequotient (XV)=His aG=H-equivariant vector bundle over X=H, (XU)=His an open subscheme of (XV)=Hwhose complement has codimension larger thani. This yields isomorphisms

Aⁱ_G=H (XU)=H

'Aⁱ_G=H (XV)=H 'Aⁱ_G=H(X=H):

Theresulting isomorphisms Aⁱ_G(X)'Aⁱ_G=H(X=H) yield the desired ring isomorphism A_G(X)'A_G=H(X=H).

LEMMA2.2. Let G be an affine linear group acting on a smooth scheme X, E!X an equivariant vector bundle of rank r. Call E₀E the com- plement of the zero section of E. Then the pullback homomorphism A_G(X)!A_G(E0)is surjective, and its kernel is generated by the top Chern classcr(E)2A^r_G(X).

PROOF. Call s:X!E the zero-section. Then the statement follows immediately from the exactness of the localization sequence

A_G(X) ^s! A_G(E)!A_G(E0)!0;

from thefact that thepullbacks:A_G(E)!A_G(X) is an isomorphism, and from the self-intersection formula, which implies that the composite A_G(X) ^s! A_G(E) ^s! A_G(X) is multiplication by cr(E).

LEMMA 2.3. Let H a linear algebraic group with an isomorphism f:H'Aⁿ_k of varieties, such that for any field extension kk⁰ and any h2H(k⁰), the automorphism ofAⁿ_k0that corresponds underfto the action of h on H_k⁰ by left multiplication is affine. Furthermore, let G be a linear

algebraic group acting on H via group automorphisms, corresponding to linear automorphisms ofAⁿ_k underf.

If G acts on a smooth scheme X, form the semidirect product G^j H, and let G^j H act on X via the projection G^j H!G. Then the homo- morphism

A_G(X)!A_GHl (X)

induced by the projection G^j H!G is an isomorphism.

PROOF. Le t (V;U) (resp. (V⁰;U⁰)) bea good pair forG^j H(resp.G).

ThenG^j Hacts onU⁰via theprojectionG^j H!G: it follows thatG^j H acts onXHUU⁰, and sincetheaction ofG^j HonHis transitive, and thestabilizer of theorigin isG, there is an isomorphism

(XHUU⁰)=(G^j H)ˆ(X(G^jH)=HUU⁰)=(G^j H) '(XUU⁰)=G:

Look at thefollowing commutativediagram:

Notethatp₁ is an affine bundle: in fact, it is a fiber bundle with fiber isomorphic to Aⁿ, and structuregroup G^j Hthat acts onAⁿ by affine transformations, sincetheaction ofGonHis affineand theaction ofHon itself is affine. It follows from [Gro58a, p. 35] thatp₁is an isomorphism. On theother hand, sinceUU⁰ is an open set of VV⁰ on which G acts freely,p₂ is theidentity on theequivariant Chow ring A_G(X), up to a de- gree that can be made arbitrarily large: so we have a commutative triangle

where the horizontal arrow is exactly the map induced by the projection G^j H!G.

Here is another auxiliary result: it is well known (see for instance [Vez00]) that A_mn'Z[j]=(nj), where j is thefirst Chern class of the

character given by the inclusionm_n,!Gm. IfGis an algebraic group, we will denote by j2A_Gmn theimageof junder the map A_mn!A_Gmn in- duced by the projection Gm_n !m_n. Using theprojectionGm_n!G, wecan consider A_Gmnas an A_G-algebra. Then A_Gmnadmits thefollowing description:

LEMMA2.4. As anA_G-algebra,A_Gmnis generated by the elementj, and the kernel of the evaluation map A_G[x]!A_Gmn sending x intoj is the ideal(nx). In other words,

A_GmnˆA_G[j]=(nj):

PROOF. See [Vis05, Lemma 4.3].

3. The special groups:GLn,SLnandSp_n.

Let us fix a fieldk: wewriteGL_n, SL_nand Sp_n for thecorresponding algebraic groups overk.

These groups are always much easier to study: they are special, in the sense that every eÂtaleprincipal bundleis Zariski locally trivial (this ter- minology is due to Grothendieck, see [Gro58b]). For GLnand Sp_ntheidea works in a very similar way: let us work out Sp_n, which is marginally harder. We proceed by induction onn, thecasenˆ0 being trivial.

Consider V ˆA²ⁿ, the tautological representation of Sp_n, with its symplectic formh:VV !kgiven in coordinates by

h(x1;. . .;x2n;y1;. . .;y2n)ˆx1yn‡1‡. . .‡xny2n xn‡1y1 . . . x2nyn: Denote bye₁;. . .;e_2n thecanonical basis ofV.

Theorbit structureofVis very simple: there are two orbits, the origin and its complementU^defˆ Vn f0g. Consider the subspace

V⁰ˆ he1;. . .;en 1;en‡1;. . .;e2n 1i;

therestriction of h to V⁰ is a non-degenerate symplectic form, and V ˆV⁰ he_n;e_2ni. This induces an embedding Sp_{n 1},!Sp_n, identifying Sp_{n 1}with thestabilizer of thepair (e_n;e_2n).

Let G the stabilizer of the element en: then we have that Sp_{n 1}GSp_n. Thefirst inclusion admits a splitting: ifA2G, thenA stabilizes the orthogonal complement h ien ^?. It follows that A induces a linear endomorphism on the quotient h ie_n ^?=h i 'e_n V⁰, and this en-

domorphism is easily seen to preserve the symplectic formhj_V0, so it is an element of Sp_{n 1}. Thus wehavea projectionG!Sp_{n 1}: le tHits kernel, so thatGˆSp_{n 1}^j H.

ThestructureofHis as follows; thematrices inHareexactly thosefor which there are scalarsa₁;. . .;a_{2n 1} such that

Ae_iˆ

e_i a_i‡nen if i ˆ 1; . . . ; n 1

e_n if i ˆ n

e_i‡a_{i n}e_n if i ˆ n‡1; . . . ; 2n 1

a1e2‡. . .‡a2n 1e2n 1‡e2n if i ˆ 2n:

8>

>>

><

>>

>>

:

This yields an isomorphism of varieties H'A^{2n 1}. It is not hard to see that the conditions of Lemma 2.3 are satisfied for the action of Sp_{n 1}onH;

hence the embedding Sp_{n 1}G induces an isomorphism of rings A_G'A_Spn1, so thecomposite

A_Spn(U)!A_Spn1(U)!A_{Spn 1}(en)ˆA_Spn1

is an isomorphism. The restriction of the representationVto Sp_{n 1}is the direct sum ofV⁰ and of a trivial 2-dimensional representation: hence the Chern classes c_iˆc_i(V) restrict to the c_i(V⁰). From theinduction hy- pothesis, we conclude that A_Spn(U) is generated by the images of c2;. . .;c2n 2.

From Lemma 2.2 we conclude that every class in A_Spncan bewritten as a polynomial in c₂;. . .;c_{2n 2}, plus a multipleof c_2n. By induction on the degree we conclude thatc₂;. . .;c_2n generate A_Spn.

To prove their algebraic independence, let us restrict to A_Tn, where Tn'Gⁿ_m is thestandard maximal torus in Sp_n, consisting of diagonal matrices with entries (t1;. . .;tn;t₁¹;. . .;t_n¹). Then A_Tn is thepolynomial

ring Z[j₁;. . .;j_n], where j_i is thefirst Chern class of the1-dimensional

representation given by thei^thprojectionT_n!G_m. Then the total Chern class of therestriction ofV_n toT_n is

(1‡j₁). . .(1‡j_n)(1 j₁). . .(1 j_n)ˆ(1 j²₁). . .(1 j²_n);

hence the restriction of c_2i is the i^th elementary symmetric function of j²₁;. . .; j²_n. This proves the independence of thec2i.

As we mentioned, the argument for GLnis very similar. For SLn, one can proceed similarly, but it is easier to use the fact that, if GL_n acts freely on an algebraic variety U, theinduced morphism U=SL_n!

!U=GL_n makes U=SL_n into a principal G_m-bundleon U=GL_n, asso- ciated with the determinant homomorphismdet:GLn!Gm. He nce , by

Lemma 2.2, wehavean isomorphism A_SLn 'A_GLn=(c1), which gives us what wewant.

REMARK 3.1. All these arguments work with cohomology, when kˆC. The localization sequence in cohomology does not quite work in the sameway, as therestriction homomorphism from thecohomology of the total space to that of an open subset is not necessarily surjective. However, ifYis a smooth closed subvariety of a smooth complex algebraic varietyX, of purecodimensiond, then there is an exact sequence

!Hⁱ_G^2d(Y)!Hⁱ_G(X)!Hⁱ_G(XnY)!Hⁱ_G^2d‡1(Y)! : Hence if we know that either the pullback H_G(X)!H_G(XnY) is surjec- tive, or the pushforward H_G(Y)!H_G(X) is injective, we can conclude that we have an exact sequence

0!H_G(Y)!H_G(X)!H_G(XnY)!0;

and this is sufficient to mimic the arguments above and give the result for cohomology.

REMARK 3.2. These results can also be proved very simply from a result of Edidin and Graham (see [EG97]): ifGis a special algebraic group, T a maximal torus andW theWeyl group, thenatural restriction homo- morphism A_G!(A_T)^W is an isomorphism.

4. The Chow ring of the classifying space ofOn.

Let us fix a fieldkof characteristic different from 2. IfV ˆkⁿ is ann- dimensional vector space, we define a quadratic form q:V !k in the standard form

q(x1;. . .;xn)ˆx1xm‡1‡. . .‡xmx2m

whennˆ2m, and

q(x1;. . .;xn)ˆx1xm‡1‡. . .‡xmx2m‡x²_2m‡1

when nˆ2m‡1. Wewill denoteby O_n the algebraic group of linear transformations preserving this quadratic form.

THEOREM4.1. [R. Pandharipande, B. Totaro].

A_OnˆZ[c1;. . .;cn]=(2codd):

REMARK 4.2. Let V⁰ beanother n-dimensional vector space overk, with a non-degenerate quadratic form q⁰:V⁰!k. Wecan associatewith this another algebraic group O(q⁰), which will not beisomorphic to O_nˆO(q), in general, unlesskis algebraically closed.

However, one can show that there is an isomorphism of Chow rings A_On 'A_O(q0), such that theclasses ci(V) in theleft hand sidecorrespond to theclasses ci(V⁰) in theright hand side. Theprinciplethat allows to prove this has been known for a long time ([Gir71, Remarque 1.6.7]): it is the existence of a bitorsor I!Speck. This is the scheme representing the functor of isomorphisms of (V;q) with (V⁰;q⁰). OnIthere is a left action of O(q⁰) and right action of O_n, by composition. These two actions commute, and make I into a torsor under both groups (because (V;q) and (V⁰;q⁰) become isomorphic after a base extension).

In general, assume thatGandG⁰are algebraic groups over a fieldk(in fact, any algebraic space will do as a base), and I!Speck is a (G⁰;G)- bitorsor: that is, onIthere is a right action ofGand left action ofG⁰, and this makesIinto a torsor under both groups. IfXis ak-algebraic space on whichG⁰acts on the left, then we can produce ak-algebraic spaceI^GX on whichGacts on theleft, by dividing theproductI_SpeckXby theright action ofG, defined by the usual formula (i;x)gˆ(ig;g ¹x). Theleft action of G⁰ is by multiplication on thefirst component: thequotients GnX and G⁰n(I^GX) arecanonically isomorphic.

This operation gives an equivalence of the category of G-algebraic spaces with the category of G⁰-algebraic spaces. When applied to re- presentations, it yields representations, and gives an equivalence of the category of representations of G and of G⁰. Furthermore, given a re- presentationVofG, with an open subsetUVon whichGacts freely, we get a representation V⁰ˆI^GV with an open subset U⁰ˆI^GU on whichG⁰acts freely, so that the quotientsGnUandG⁰nU⁰areisomorphic.

In Totaro's construction this gives an isomorphism of A_Gwith A_G0. So, in particular, the result that we have stated for O_n also holds for O(q⁰) for any other non-degeneraten-dimensional quadratic formq⁰, and wehave

A_O(q0)ˆZ[c₁;. . .;c_n]=(2c_odd):

The proof of the Theorem will be split into two parts: first we show that thec_igenerate A_On, then that ideal of relations is generated by the given ones.

For the first part we proceed by induction onn.

Fornˆ1,q(x)ˆx²₁, and O1ˆm₂, so

A_O1ˆA_m2 'Z[c₁]=(2c₁):

For n>1, let Bˆ fv2Aⁿ jq(v)6ˆ0g, and set Qˆq ¹(1). Then q:B!Gmis a fibration, with fibers isomorphic toQ. This fibration is not trivial, but it becomes trivial after an eÂtale base change. Set

Be ˆ f(t;v)2GmBjt²ˆq(v)g;

and consider the cartesian diagram

wherethefirst column is projection onto thefirst factor, and thetop row is defined by the formula (t;v)7!tv.

There are obvious commuting actions of m₂ and On onB, thefirst de-e fined by e(t;v)ˆ(et;v), and thesecond byM(t;v)ˆ(t;Mv). Thequo- tientB=me ₂is isomorphic toB, and theinduced action of O_non thequotient coincides with the given action onB. From Lemma 2.1, we obtain an iso- morphism

A_On(B)'A_m2On(B):e

Then there is an isomorphism ofG_m-schemesBe 'G_mQdefined by theformula (t;v)7!(t;v=t). Thegiven actions ofm₂and of On onBe induce commuting actions on GmQ given by e(t;v)ˆ(et;ev) for e2m₂ and M(t;v)ˆ(t;Mv) forM2On. These define an action ofm₂OnonGmQ, and A_On(B) is isomorphic to A_m2On(G_mQ).

This action of m₂O_n on G_mQ extends uniquely to an action of m₂OnonA¹Q, defined by the same formulae. This action is defined by two separate action onA¹andQ, and theaction onA¹is linear, defined by thenon-trivial character ofm₂through theprojectionm₂On!m₂. Callj the first Chern class of this representation. From Lemma 2.2, we have an isomorphism

A_m2On(GmQ)'A_m2On(Q)=(j):

…4:1†

To investigate A_m2On(Q) wewill also usean orthogonal basise⁰₁;. . .;e⁰_n ofV, in whichqhas theform

q(x1e⁰₁‡. . .‡xne⁰_n)ˆx²₁‡. . .‡x²_m x²_m‡1 . . . x²_n

whennˆ2m, and

q(x₁e⁰₁‡. . .‡x_ne⁰_n)ˆx²₁‡. . .‡x²_m‡1 x²_m‡2 . . . x²_n whennˆ2m‡1.

Now, theaction ofm₂OnonQis transitive; letHthestabilizer of the pointe⁰₁2Q. ThestructureofHis as follows. SetV^0defˆhe⁰₂;. . .;e⁰_ni, so that Vis theorthogonal sumhe⁰₁i V⁰, and callq⁰therestriction ofqtoV⁰. Then thegroup O_q⁰ of linear automorphisms of V⁰ preserving q⁰ is naturally embedded into On, as thestabilizer ofe⁰₁. Noticethat in an appropriate basisq⁰has thestandard form

q⁰(x1;. . .;xn 1)ˆx1xm‡1‡. . .‡xmx2m

whennˆ2m‡1, and theoppositeof thestandard form q⁰(x1;. . .;xn 1)ˆ (x1xm‡. . .‡xm 1x2m 2‡x²_{2m 1})

when nˆ2m; in both cases the orthogonal group O(q⁰) is isomorphic to On 1, and weidentify it with On 1.

Thestabilizer ofe⁰₁ inm₂O_n is thegroupm₂O_{n 1}, embedded into m₂O_n with theinjectivehomomorphism

(e;M)7!(e;eM):

It follows that

A_m2On(Q)'A_m2On (m₂On)=(m₂On 1) 'A_m2On1:

Weobtain a chain of isomorphisms

A_On(B)'A_m2On(Q)=(j) A_{m2On 1}=(j):

Finally, from Lemma 2.1 we get an isomorphism A_m2On1=(j)'A_On1[j]=(j)

'A_On1:

ThecompositeA_On!A_On(U)!A_On1 is thepullback induced by theem- bedding O_{n 1}O_n.

Therestriction ofV to On 1 is thedirect sum ofV⁰ and a trivial 1-di- mensional representation, hence the restriction A_On !A_On1carriesciinto c_i(V⁰). Therefore, by induction hypothesis, the images ofc1;. . .;cn 1gen- erate A_On(B).

Next, we claim that the restriction homomorphism A_On(Aⁿn f0g)!

!A_On(B) is an isomorphism. To see this, set Cˆ fv2Aⁿn f0g jq(v)ˆ0g

with its reduced scheme structure, and consider the fundamental exact sequence

A_On(C) !ⁱ A_On(Aⁿn f0g)!A_On(B)!0:

We need to show that i is thezero map. In fact, q:Aⁿn f0g !A¹ is smooth, sincethecharacteristic of thebasefield is not 2, so C is the scheme-theoretic inverse image of f0g. The map q:Aⁿn f0g !A¹ is O_n-equivariant, if we let O_n act trivially on A¹; and thefundamental class [0]2A_On(A¹) equals zero. Sincetheinverseimageof [0] in A_On(Aⁿn f0g) is [C], wecan concludethat

[C]ˆ02A_On(Aⁿn f0g):

Next we show that the pullbacki:A_On(Aⁿn f0g)!A_On(C) is surje ctive : in this case, for every a2A_On(Cn f0g), wehaveaˆib for some b2A_On(Aⁿn f0g), so

i(a)ˆii(b)ˆ[C]bˆ0

by theprojection formula, and i is the zero map, as claimed.

To show surjectivity, notice that the action of O_nonCis transitive. Let us investigate the stabilizer G of e₁2C. Se t nˆ2m or nˆ2m‡1, as usual. If wedefine

V⁰ˆ he₂;. . .;e_m;e_m‡2;. . .;e_ni

then the restriction ofqtoV⁰has thestandard form, andVis theorthogonal sumV⁰ he1;em‡1i. This gives an embedding On 2On, identifying On 2

with thestabilizer of thepair (e₁;e_m‡1).

An analysis very similar to that we have carried out for the stabilizer of a vector under Sp_nleads to the conclusion that the stabilizer Gofe1 is a semidirect product On 2^j H, whereHis isomorphic toA^{n 1} as a variety, the action of an element ofHon itself is given by an affine map, and the action of O_{n 2} on H is linear: by Lemma 2.3, the embedding O_{n 2}G induces an isomorphism of rings A_G'A_{On 2}, so thecomposite

A_On(C)!A_On2(C)!A_On2(e₁)ˆA_On2

is an isomorphism. But thec_irestrict in A_On2 to theChern classes ofV⁰: hence, by inductions hypothesis, they generate A_On2. Hence the pullback

A_On !A_On(C) is surjective, as claimed. This ends the proof that the ci

generate A_On. Let us investigate the relations.

Thequadratic formqinduces an isomorphism V 'V^_ of representa- tions of O_n, hence for eachiwehavec_i(V)ˆ( 1)ⁱc_i(V). This shows that 2c_iˆ0 wheniis odd.

To show that these generate the ideal of relations among the ci,

let JZ[x1;. . .;xn] be the ideal generated by 2x1, 2x3;. . .. Le t

P2Z[x1;. . .;xn] be a homogeneous polynomial such thatP(c1;. . .;cn)ˆ

ˆ02A_On: we need to check thatPis inJ. By modifyingPby an element of J, wemay assumethatPis of theformQ‡R, whereQis a polynomial in the evenx_i, whileRis a polynomial in which every monomial contains some xiwithiodd, and all of whose coefficients are either 0 or 1.

LetTm'G^m_mbethestandard torus in On: the embeddingTmOnsends (t₁;. . .;t_m) into thediagonal matrix with entries (t₁;. . .;t_m;t₁¹;. . .;t_m¹) if nˆ2m, and (t₁;. . .;t_m;t₁¹;. . .;t_m¹;1) if nˆ2m‡1. Then A_Tmˆ

ˆZ[j₁;. . .;j_m], where j_i is thefirst Chern class of thei^th projection

x_i:Tn !Gm. Therestriction of V to Tn splits as r^defˆx₁‡. . .‡x_m‡

‡x₁¹‡. . .‡x_m¹whenmis even, andr‡1 whennis odd. Hence the total

Chern class of the restriction ofV toT_nis

(1‡j₁). . .(1‡j_m)(1 j₁). . .(1 j_m)ˆ(1 j²₁). . .(1 j²_m);

and this means that the restrictions of thec_iis 0 wheniis odd, whilec_2j restricts to thej^thsymmetric function of j²₁;. . .; j²_m. Hence the restric- tions of even Chern classes are algebraically independent. In the decom- position 0ˆP(c1;. . .;cn)ˆQ(c2;. . .;c2m)‡R(c1;. . .;cn) thesummand R(c1;. . .;cn) restricts to 0, soQ(c2;. . .;c2m) also restricts to 0. This implies thatQˆ0. So wehavethatPhas coefficients that are either 0 or 1.

Now takea basise⁰₁;. . .;e⁰_nofVin whichqhas a diagonal form. Consider thesubgroupmⁿ_nO_n consisting of linear transformations that take each e⁰_iintoe⁰_ior e⁰_i. If wecallh_ithefirst Chern class of thecharacter obtained composing thei^thprojectionmⁿ_n!m_n with the embeddingm_n,!Gm, then by Lemma 2.4 we have

A_mⁿ_nˆZ[h₁;. . .;h_n]=(2h₁;. . .;2h_n):

There is a natural ring homomorphism from A_mn

ninto thepolymomial ring F₂[y₁;. . .;y_n] that sends eachh_iintoy_i. Therestriction ofVtomⁿ_nhas total Chern class (1‡h₁). . .(1‡h_n); hence the image ofc_iinF₂[y₁;. . .;y_n] is the i^thelementary symmetric polynomials_iin they_i. Thes_iarealgebraically independent inF2[y1;. . .;yn], theimageof 0ˆP(c1;. . .;cn) isP(s1;. . .;sn), and P has coefficients that either 0 or 1. This implies that Pˆ0, and completes the proof of the theorem.

5. The Chow ring of the classifying space ofSOn.

Let kbe a field of characteristic different from 2, setV ˆkⁿ, and let q:V !kbe the same quadratic form as in the previous section. Consider thesubgroup SO_n O_n of orthogonal linear transformations of determi- nant 1.

Ifnis odd, A_SOn can be easily computed from A_On, as was noticed in [Pan98] and [Tot99].

THEOREM5.1. [R. Pandharipande, B. Totaro]. If n is odd, then A_SOn ˆZ[c2;. . .;cn]=(2c_odd):

PROOF. When is nodd there is an isomorphism O_n'm_nSO_n; the determinant characterdet:On!m_n (whosefirst Chern class in A_On isc1) corresponds to the projection m_nSOn!m_n. Then from Lemma 2.4 we get that

A_SOn'A_On=(c1) and theconclusion follows.

5.1 ±The Edidin-Graham construction

From now on weshall assumethatnis even, and writenˆ2m.

In this case, A_SOnis not generated by the Chern classes of the standard representation, not even rationally. This can be seen easily fornˆ2. We havethat SO2consists of matrices of the form

t 0 0 t ¹

and so is isomorphic toGm. Then

A_SO2ˆA_Gm ˆZ[j];

wherejis the first Chern class of the tautological representationLˆA¹, on whichG_macts via multiplication. HenceVˆLL^_, so c₂(V)ˆ j².

For generaln, thevector spaceVwill still split as thedirect sum of two totally isotropic subspaces, one dual to the other: however, whenn>2 this splitting is not unique, and the totally isotropic subspaces are not invariant under the action of SOn, so V is not a direct sum of two nontrivial re- presentations (and V is in fact irreducible). Still, in topology V has an

Euler classem2H^2m_SOn, whosesquareis ( 1)^mcm. Let us recall Edidin and Graham's construction of an algebraic multiple ofem(see [EG95]).

In what follows we will use the classical conventions for projectiviza- tions and Grassmannians; those seem a little more natural in intersection theory than Grothendieck's. So, ifWis a vector space, we denote byP(W) the vector space of lines inW, and byG(r;W) theGrassmannian of sub- spaces of dimensionr; and similarly for vector bundles.

Denote by I(m;V) thesmooth subvariety of G(m;V) consisting of maximal totally isotropic subspaces of V. It is well known that O_n acts transitively on I(m;V), and that I(m;V) has two connected components, each of which is an orbit under the action of SO_n. Let us choose one of the orbits, for example, the one containing the subspace he1;. . .;emi. Every totally isotropic subspaceof dimensionm 1 ofV is contained in exactly two maximal totally isotropic subspaces, one in each connected component.

There is a well known equivalence of categories between O_n-torsors and vector bundles of ranknwith a non-degenerate quadratic form. IfEis a vector bundle on a schemeX with a non-degenerate quadratic form, this corresponds to a On-torsorp:P!X, the torsor of isometries betweenE and VX; with this torsor wecan associatea m₂-torsor (that is, an eÂtaledoublecover) P=SO_n!X via the determinant homomorphism det:O_n!m₂. This cover can be described geometrically as follows.

Consider the subschemeI(m;E) of totally isotropic subbundles in the relative Grassmannian G(m;E)!X ; theprojection I(m;E)!X is proper and smooth, and each of its geometric fibers has two connected components. Let I(m;E)!eI(m;E)!X be the Stein factorization; then eI(m;E)!X is an eÂtale double cover, and is precisely the double cover P=SOn !X. This can be seen as follows.

OnPwe have, by definition, an isometry ofpEwithVP. InVP wehavea maximal totally isotropic subbundlehe1;. . .;emi P, so we get a maximal totally isotropic subbundleof pE. This defines a morphism P!I(m;E) over X; thecompositeP!I(m;E)!eI(m;E) induces the desired isomorphismP=SOn'eI(m;E).

Hence, giving a reduction of structure group of P!X to SOn is equivalent to assigning a sectionX!eI(m;E). This yields an equivalence of thegroupoid of SO_n-torsors onX with the groupoid of vector bundles E!X of rank n with a non-degenerate quadratic form, and a section X!eI(m;E). We shall refer to such a structure as an SO_n-structure on E.

Furthermore, given an SOn-structureonE, iff:T!Xis a morphism of algebraic varieties, andLis a totally isotropic subbundleoffEof rank m, wesay that L is admissible if theimageof T under the morphism

T!I(m;X) corresponding toLis contained in the inverse image of the given embeddingXeI(m;E).

Here is the construction of Edidin and Graham ([EG95, Section 6]).

We will follow their notation. LetEbea vector bundleof ranknwith an SO_n-structure on a smooth algebraic variety X. For e ach iˆ1;. . .;m consider the flag variety fi:Qi!X of totally isotropic flags

L1L2. . .Lm iE, with e achLs of ranks. For eachi, de note by

L1L2. . .L_{m i}f_iEtheuniversal flag on Q_i. Therestriction of

thequadratic form to L^?_{m i} is degenerate, with radical equal to L_{m i}; hence onQ_ithere lives a vector bundleE_i^defˆ L^?_{m i}=L_{m i}of rank 2iwith a non-degenerate quadratic form. For each iˆ1;. . .;m 1 wehavea projection pi:Qi 1!Qi, obtained by dropping the last totally isotropic subbundlein thechain; and Q_{i 1} is canonically isomorphic, as a scheme overQ_i, to thesmooth quadric bundleinP(E_i) defined by the quadratic form on E_i. This me ans that Q_{i 1} is a family of quadrics of dimension 2(i 1) overQ_i. Le t us de note byh_i2A¹(Q_{i 1}) therestriction toQ_{i 1} of theclass c1 OP(Ei)(1)

2A¹ P(Ei) .

Each bundleE_ihas a canonical SO_{n 2i}-structure. Callp_i:L^?_{m i}!E_ithe projection. From each totally isotropic vector subbundle LE_i of rank m i, we get a totally isotropic vector subbundle p_iLL^?_{m i}f_iE of rankm; thenLis admissibleif and only ifp_iLis admissible.

Theuniversal flagL1L2. . .Lm 1f₁EonQ1can becompleted in a uniqueway to a maximal totally isotropic flag L1. . .Lm 1 L_mf₁Ein such a way thatL_mis admissible. Then Edidin and Graham define

ym(E)ˆf1 scm(Lm)

2A^m(X) where we have set

sˆh²₂h⁴₃. . .h^2m_m ²2A(Q₁):

REMARK 5.2. In this formula each of the classes h_ishould bepulled back to Q₁. Here, and in what follows, we use the following convention:

whenf:Y!X is a morphism of smooth varieties, andj2A(X), wewill also writejforfj2A(Y). Similarly, ifE!Xis a vector bundle, we will also write EforfE. This has theadvantageof considerably simplifying notation, and should not lead to confusion. With this notation, when f is proper the projection formula reads: ifj2A(X) andh2A(Y), then

f(jh)ˆjfh: