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The Chow Ring of the Classifying Space of GO(2N) Saurav Bhaumik

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The Chow Ring of the Classifying Space of GO(2N) Saurav Bhaumik Math. Res. Lett. Volume 22, Number 4, 989–1003, 2015 The Chow ring of the classifying space of GO(2n) Saurav Bhaumik Let GO(2n) be the general orthogonal group scheme (the group of orthogonal similitudes). In the topological category, Y. Holla and ∗ N. Nitsure determined the singular cohomology ring Hsing(BGO(2n, C), F2) of the classifying space BGO(2n, C) of the corresponding complex Lie group GO(2n, C) in terms of explicit generators and relations. The author of the present note showed that over any alge- braically closed field of characteristic not equal to 2, the smooth- ∗ F ´etale cohomology ring Hsm-´et(BGO(2n), 2) of the classifying alge- braic stack BGO(2n) has the same description in terms of genera- ∗ tors and relations as the singular cohomology ring Hsing(BGO(2n, C), F2). Totaro defined for any reductive group G over a field, the ∗ Chow ring AG, which is canonically identified with the ring of characteristic classes in the sense of intersection theory, for prin- cipal G-bundles, locally trivial in ´etale topology. In this paper, we ∗ calculate the Chow group AGO(2n) over any field of characteristic different from 2 in terms of generators and relations. 1. Introduction The Chow ring of the classifying space of a reductive group was introduced by Totaro in [10], where he calculated the Chow rings of the classifying spaces of several finite groups and algebraic groups including O(n), Sp(2n), etc. Edidin and Graham in [3] introduced the equivariant Chow ring. Molina Rojas and Vistoli in [7], using the techniques of equivariant Chow groups, ∗ calculated the Chow ring ASO(n) in case n is even (the odd case was already addressed in Pandharipande [9] and Totaro [10]). Holla and Nitsure in [5] considered the general orthogonal group GO(n, C), which is also called the group of orthogonal similitudes, and calculated ∗ C F the singular cohomology ring of its classifying space Hsing(BGO(n, ); 2). In [1], the author of the present paper considered the algebraic version of the above Lie group, namely the general orthogonal group scheme GO(n)over an algebraically closed field of characteristic different from 2, and showed 989 990 Saurav Bhaumik ∗ F that the smooth-´etale cohomology ring Hsm-et(BGO(n); 2) of the algebraic stack BGO(n) has the same description in terms of generators and relations over F2 as the singular cohomology ring computed by Holla and Nitsure in [5]. In this present note, we calculate the Chow ring of the classifying space of GO(n) over a field of characteristic different from 2 in the sense of Totaro [10], using the methods of equivariant Chow groups. By the results of Totaro [10], this ring can be canonically identified with the ring of characteristic classes for principal GO(n)-bundles on smooth, quasi-projective schemes. In other words, the Chow ring of the classifying space of GO(n) is the ring of all intersection theoretical invariants for families of line bundle valued nondegenerate quadratic forms. Henceforth, all schemes and morphisms are over a fixed field k (not nec- essarily algebraically closed) with characteristic different from 2. We recall the definition of the algebraic group GO(n)overk.LetV = kn,andlet q : V → k be the quadratic form, defined by q(x1,...,x2m)=x1xm+1 + ···+ xmx2m, for the even case n =2m,andby ··· 2 q(x1,...,x2m+1)=x1xm+1 + + xmx2m + x2m+1, for the odd case n =2m +1.LetGO(n) be the affine algebraic group scheme of invertible linear automorphisms of V that preserve the quadratic form q up to a scalar. In terms of matrices, let J denote the nonsingular symmetric matrix of the bilinear form corresponding to q. Then as a functor of points, GO(n) attaches to each k-algebra S the group × t GO(n)(S)={A ∈ GLn(S): ∃ a ∈ S , AJA = aJ}. The algebraic group GO(n) is reductive, since its defining representation on n k is irreducible. Note that if k /k is a field extension such that the quadratic n 2 form q extended to V ⊗k k = k is equivalent to the quadratic form i xi , given by the identity matrix In,thenoverk , the algebraic group GO(n) defined above is isomorphic to the algebraic group GO(n) defined in [1]. The scalar a in the definition determines the character σ : GO(n) → Gm that satisfies tAJA = σ(A)J. Given a scheme X,andaprincipalGO(n)- bundle P on X (locally trivial in the ´etale topology), consider the rank n vector bundle E associated to the defining representation GO(n) ⊂ GLn, and the line bundle L determined by the character σ. The nondegenerate The Chow ring of the classifying space of GO(2n) 991 symmetric bilinear form corresponding to q induces a nondegenerate sym- ⊗ → metric bilinear form b : E OX E L. Conversely, given a nondegenerate quadratic triple of rank n (E,L,b), which is a triple consisting of a vector bundle E of rank n, a line bundle L and a nondegenerate symmetric bilin- ⊗ → ear b : E OX E L, we can reduce the structure group of E to GOn by applying Gram-Schmidt orthonormalization ´etale locally on X. ∗ Let AG denote the Chow ring of the classifying space of a reductive group G in the sense of Totaro [10]. Note that for any n ≥ 1, there is a canonical isomorphism ∼ SO (2n +1)× Gm → GO(2n +1). m Note that BGm is approximated by the projective spaces Pk in the sense of Totaro [10], and we have for any smooth scheme X the following natural isomorphisms. ∗ ∗ m ∼ ∗ m A (X) ⊗ A (Pk )→ A (X × Pk ) Since BSO(2n + 1) is approximated by smooth schemes, there is the K¨unneth isomorphism ∗ ⊗ ∗ →∼ ∗ ASO(2n+1) AGm AGO(2n+1). ∗ ∼ Z This determines the Chow ring for GO(2n + 1), because AGm = [λ], and the Chow ring for SO(2n + 1) is given by [9] and [10]. Therefore we are left with the task of calculating the Chow ring only in the even case GO(2n). ∗ The rest of this note is devoted to the calculation of AGO(2n). In Section 2, we recall Totaro’s definition of the Chow ring of a classifying space from [10], and the basic notions of equivariant Chow groups from ∗ Molina Rojas and Vistoli [7]. In Section 3, we calculate AGO(2n) in terms of explicit generators and relations over Z. We show, in terms of quadratic ∗ triples (E,L,b), AGO(2n) is generated by the Chern classes ci(E)andthe Chern class λ of L. The nondegenerate symmetric bilinear form determines ∼ an isomorphism of vector bundles E → E∨ ⊗ L. This isomorphism gives the following relations among Chern classes of E and the class λ. p i 2n − i p−i cp = (−1) ciλ ,p=1,...,2n p − i i=0 The main theorem of this note is Theorem 3.1, which asserts that the ideal ∗ of relations in AGO(2n) is generated by the above relations. ∗ TheinvariantsinAGO(2n) transform under the cycle map to the corre- ∗ sponding classes in Hsm-´et(BGO(2n), F2) (see Remark 3.2). 992 Saurav Bhaumik To prove the main theorem (Theorem 3.1), we first observe that λ and ∗ the even Chern classes are algebraically independent in AGO(2n). Eventually ∗ we focus on the torsion subgroup of AGO(2n) (which is an ideal) in order to carry out the rest of the proof. The inclusion O(2n) ⊂ GO(2n), where O(2n) 2n is looked upon as the algebraic group of linear automorphisms of k that preserve the quadratic form i xixi+n, gives rise to a ring homomorphism ∗ → ∗ AGO(2n) AO(2n). A biproduct of the proof of the main theorem is that this homomorphism determines an isomorphism of the corresponding torsion subgroups. Acknowledgements The author would like to thank Yogish Holla and Nitin Nitsure for useful discussions and suggestions. Thanks are also due to the referee, who informed the author that the cycle map in ´etale cohomology with torsion coefficients factors through rational equivalence. 2. Basic notions recalled 2.1. Chow ring of the classifying space Totaro [10] defined the Chow groups of the classifying space of a linear algebraic group G as follows. Let N>0 be an integer. Then there is a finite dimensional linear representation V of G,withaG-equivariant closed subset S of codimension ≥ N such that the action of G on (V − S) is free, such that the quotient (V − S)/G exists in the category of schemes and (V − S) → (V − S)/G is a principal bundle, locally trivial in ´etale topology. Foranysuchotherpair(V ,S), there are canonical isomorphisms for i<N i (V − S) ∼ i (V − S ) A → A . G G i The i-th Chow group AG of the classifying space of G is defined to be i V −S A G ,where(V,S) satisfies the above condition. The isomorphism above i shows that AG is independent of the choice of the particular pair (V,S). ∗ The graded group AG naturally has the structure of a graded ring under intersection product. Note that in the above approach, one defines the Chow ring of the clas- sifying space without referring to a possible classifying space BG whether in The Chow ring of the classifying space of GO(2n) 993 the category of algebraic stacks or simplicial spaces. But Morel and Voevod- sky defined the classifying space of a linear algebraic group as an object in the A1-homotopy category, and their construction gives the same Chow ring (Chapter 4, Proposition 2.6, [8]).
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