Jan 23, 2012 - Let GO(2n) be the general orthogonal group (the group of similitudes) ... space BGO(2n) of the complex Lie group GO(2n) in terms of exp...

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´ Characteristic Classes for GO(2n) in Etale Cohomology Saurav Bhaumik Abstract Let GO(2n) be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic , 2. We determine the e´ tale cohomology ring with F2 coefficients of the algebraic stack BGO(2n). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring of the classifying space BGO(2n) of the complex Lie group GO(2n) in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are (i) an extension to e´ tale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and (ii) construction of a Gysin sequence for the Gm fibration BO(2n) → BGO(2n) of algebraic stacks.

Mathematics Subject Classification: 14F20; 14L30

1 Introduction Let k be an algebraically closed field of characteristic , 2. The general orthogonal group, also known as the group of similitudes, is the closed subgroup scheme GO(n) ⊂ GLn,k whose set of R-valued points, for any k-algebra R, is GO(n)(R) = {A ∈ GLn (R) : ∃ a ∈ R× , t AA = aIn }. This is a reductive group scheme, since it is reduced and its defining representation on kn is irreducible. In the topological category, Y. Holla and N. Nitsure [2] determined the characteristic classes for the complex Lie group GO(n) with coefficients in F2 , i.e. they explicitly determined the singular cohomology ring H ∗ (BGO(n); F2 ) in terms of generators and relations. The present note proves a similar result in the algebraic category for the e´ tale cohomology for the algebraic stack BGO(n). There are two basic components in our extension of the methods of [2] to the algebraic category. The first of them is a good functorial description of the cohomology of the stack BGO(n). This can be achieved by extending to e´ tale cohomology an idea of Totaro [7], which he originally used for treating Chow groups. The second component is the Gysin sequence for the Gm -torsor BO(n) → BGO(n) in e´ tale cohomology, which is actually a 1

by-product of the first component. Once these points are established, the determination of the ring of characteristic classes proceeds exactly as in [2]. As was remarked in section 1 of [2], the group scheme GO(2n + 1) is a product Gm × SO(2n + 1), so BGO(2n + 1) BGm × BSO(2n + 1), and Het∗ (BGO(2n + 1); F2 ) Het∗ (BGm ; F2 ) ⊗ Het∗ (BSO(2n + 1); F2 ) = F2 [λ] ⊗ F2 [w2 , . . . , w2n+1 ] = F2 [λ, w2 , . . . , w2n+1 ]. It is perhaps well known that Het∗ (BSO(2n + 1); F2 ) F2 [w2 , . . . , w2n+1 ]. Nonetheless, we sketch a proof of this fact at the end of Section 5 for the reader’s reference. Thus, the case of interest is the cohomology ring of BGO(2n).

2 Notations and Preliminaries Henceforth, all schemes considered in this note will be quasi-projective over an algebraically closed field k of characteristic , 2, and all morphisms will be over k. For such a scheme X, we will denote Het∗ (X; F2 ) simply by H ∗ (X) in what follows. A morphism X → Y of schemes will be called n-acyclic, if the induced map H i (Y) → H i (X) is a bijection for i ≤ n. A scheme X will be called n-acyclic if the structure morphism X → Spec k is n-acyclic. If a morphism is n-acyclic for all n, it will be called an acyclic morphism. A morphism from a scheme to an algebraic stack X → X over k will be called n-acyclic if the map induced in cohomology H i (X) → H i (X) is a bijection for i ≤ n. For any smooth group scheme G over k, BG will denote the algebraic stack of all principal bundles on the category of quasi-projective k-schemes, locally trivial in e´ tale topology. A characteristic class for G in e´ tale cohomology with coefficient F2 is a natural transformation |BG| → Het∗ ( ; F2 ) of functors from the category of quasi projective schemes over k to the category of sets, where |BG| is defined by taking |BG|(X) to be the set of isomorphism classes in BG(X) for any quasi projective scheme X. All such characteristic classes for G form a ring, whose addition and multiplication come from the addition and cup product structure on H ∗ . Definition 1. Let G be a group scheme over k, acting on a k-scheme X. A bundle quotient for this action is a morphism φ : X → Y of k-schemes, such that X together with the given G-action is a principal G-bundle on Y, locally trivial in the e´ tale topology. Remark 2.1. Clearly, whenever such a bundle quotient exists, it is unique up to a unique isomorphism. The existence of bundle quotient is the same as the representability of the quotient stack [X/G] by a scheme. In what follows, bundle quotients will naturally occur in two different ways: a) If a reductive group G acts on a finite type, affine scheme X, and U ⊂ X is an invariant open subscheme such that for each closed point x ∈ U, the orbit O(x) is closed in X, and the induced action of G on U is free, then U admits a bundle quotient by G. Indeed, since G is reductive, we have a uniform categorical quotient φ : X → X/G with X/G affine and φ universally submersive (Theorem 1.1, [6]). As U is invariant, its image φ(U) ⊂ X/G is open. It follows that φ : U → φ(U) is a geometric quotient. Again, since the action on U 2

is free, φ : U → φ(U) is a principal G-bundle (Proposition 0.9 of [6]). b) An affine algebraic group scheme G has a bundle quotient by any of its reduced closed subgroup schemes H, which acts on G by right translations. If G is reduced, hence smooth over k, then G/H, being a reduced homogeneous space, is smooth over k (k being algebraically closed). As H is reduced, hence smooth over k, G → G/H is e´ tale locally trivial. Note that in each of the above cases, the quotient space is a quasi-projective scheme over k.

3 Characteristic Classes for G and the Cohomology of the Stack BG There is a natural map ΘG from the cohomology of the classifying stack BG to the ring of characteristic classes, which takes a cohomology class ν ∈ H ∗ (BG) to the characteristic class whose value on a principal G-bundle P → X is f ∗ν, where f : X → BG is the classifying morphism of P. Our treatment of this map is inspired by the article Totaro [7], which introduced a similar map (actually the inverse of ΘG ) for chow groups, and proved that it was a bijection. Proposition 3.1. When G is a reductive group scheme over k, the map ΘG is a bijection. Before we write down the proof of this theorem, let us recall some facts and make a few observations. Let Z ֒→ X be a closed immersion of two smooth schemes over k, where X has relative dimension n over k, and all the irreducible components of Z have codimension at least (s + 1) in X. Then each point z ∈ Z has an open neighbourhood z ∈ U ⊂ X, and there is an e´ tale morphism π : U → Ank such that Z ∩ U = π−1 (An−t k ) with t ≥ s + 1. Then the theorem of cohomological purity (Theorem 5.1 of [5]) shows that H iZ (X) = 0 for i < 2s + 2. From the spectral sequence H i (Z, H Zj (X)) ⇒ HZi+ j (X), which shows that HZi (X) = 0 for i < 2s + 2, and the following long exact sequence · · · → HZi (X) → H i (X) → H i (X − Z) → HZi+1 (X) · · · we conclude that the open immersion (X − Z) ֒→ X is 2s-acyclic. However, we need a little stronger statement, where Z need not be smooth: Lemma 3.2. Let Z ֒→ X be a closed immersion of finite type schemes over k with X smooth over k. If all the irreducible components of Z have codimension ≥ s + 1 in X, then the open immersion (X − Z) ֒→ X is 2s-acyclic. Proof. The underlying reduced subscheme Zred has a finite filtration by reduced, closed subschemes Zred = Z0 ⊃ Z1 ⊃ · · · ⊃ Zℓ , 3

where for each i ≥ 1, Zi is the singular locus of Zi−1 and Zℓ is non-empty, smooth over k. Then codim(Zi , X) ≥ s + i + 1 for each i. As Zℓ is smooth over k, X − Zℓ ֒→ X is (2s + 2l)-acyclic by the paragraph preceding the Lemma. Note that Zi−1 − Zi is a smooth over k, closed subscheme of X − Zi of codimension ≥ s + i + 1, so that X − Zi−1 ֒→ X − Zi is (2s + 2i)-acyclic for each i ≥ 1, by the same argument. Therefore, X − Z ֒→ X is 2s-acyclic. Lemma 3.3. Let P → X be a principal G-bundle over k with G reduced and X quasiprojective over k. Let f be its classifying morphism X → BG. If P is n-acyclic over k, then the morphism f is also n-acyclic. Proof. We have to show that Ri f∗ F2,X = 0 where F2,X is the constant sheaf on X. Let π : Spec k → BG be the classifying morphism of the trivial bundle G → Spec k. As π is smooth and surjective, to show that Ri f∗ F2,X = 0, it is enough to show that π∗ Ri f∗ F2,X = 0. Note that we have a Cartesian square /X P f′

f

π

Spec k ∗ i

By smooth base-change, π R f∗ F2,X =

Ri f∗′ F2,P

/

BG

= H i (P), which is zero by hypothesis.

A morphism X ′ → X will be called an affine space bundle if X admits an e´ tale cover {Xi → X}i∈I such that each base change X ′ ×X Xi → Xi is an affine space over Xi . Lemma 3.4. Let X be a quasi-projective scheme over k, E a principal G-bundle on X, where G is reductive over k, and s > 0 an integer. Then there is an affine space bundle π : X ′ → X, an affine space V over k with a linear G-action on it, and a closed subset S ⊂ V such that G acts freely on (V −S ) and the pullback π∗ E is isomorphic as a G-bundle over X ′ to the pullback of the principal G-bundle (V − S ) → (V − S )/G by a morphism f : X ′ → (V − S )/G. Moreover, the quotient space (V − S )/G is a quasi-projective scheme over k, and the classifying morphism (V − S )/G → BG for the principal G-bundle (V − S ) → (V − S )/G is 2s-acyclic. Proof. First see that if V is an affine space with a linear G-action, and if S is a closed subset of V such that the orbit of each closed point in (V − S ) is closed in V, and G acts freely on (V − S ), then by Remark 2.1 a), we have a bundle quotient (V − S ) → (V − S )/G. Therefore, the first statement follows from Lemma 1.6 of [7] (and the proof of it). The quotient space (V − S )/G is quasi-projective by Remark 2.1. By Lemma 3.3, we only have to show that the inclusion (V − S ) ֒→ V is 2s-acyclic, as V is acyclic over k. The proof of Lemma 1.6 of [7] shows that V and S can be so chosen that S has codimension ≥ s + 1 in V. Lemma 3.5. Let G be any reduced algebraic group over k. Then, given any positive integer s, there exists a linear representation V of G, a closed subset S of V such that G acts freely on (V − S ), the bundle quotient (V − S ) → (V − S )/G exists, the total space

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(V − S ) is 2s-acyclic over k, and the quotient space is smooth, quasi-projective over k. In particular, the classifying morphism of this principal G-bundle is 2s-acyclic. Proof. This is precisely Remark 1.4 of [7]. To recall the same, let G be regarded as a closed subgroup scheme of GLn,k for some n. For any positive integer s, choose a positive o integer p such that (p−n) ≥ s. Let Mn,p be the scheme of n× p matrices of rank n, regarded n×p o as an open subscheme of Mn,p = A . Recall that for the left action of GLn,k on Mn,p , p p the bundle quotient is the Grassmannian Gr(k , n) of n-dimensional quotients of k , and o Mn,p → Gr(k p , n) is the tautological principal GLn -bundle on the Grassmannian. Now, the 0 total space of the associated GLn /G-bundle on Gr(k p , n) is just the bundle quotient of Mn,p by the action of G, which exists as a smooth, quasi-projective scheme over k because it is a quotient of GL p,k by a smooth, closed subgroup scheme (Remark 2.1 b)). Also observe o that the codimension of the complement of Mn,p in Mn,p is ≥ (n − p + 1) ≥ (s + 1). Then o by Lemma 3.2, we can take V = Mn,p , and (V − S ) = Mn,p . The last sentence follows from Lemma 3.3. Proof of Proposition 3.1. The injectivity is immediate: if ΘG (β) = 0 for a nonzero β ∈ H i (BG) then by Lemma 3.5 we find a principal G-bundle P → X, whose classifying morphism X → BG is i-acyclic, yielding a contradiction to the definition of ΘG . To prove the surjectivity, starting off from a characteristic class ν of homogeneous degree s, say, we choose any pair (V, S V ), where V is a linear representation of G, and S V is a closed subset of V of codimension ≥ s + 1, such that G acts freely on (V − S V ), so that the classifying morphism (V − S V )/G → BG is 2s-acyclic. The value of ν on this principal G-bundle is the image, under the induced map in cohomology, of a unique element, say α ∈ H s (BG). We claim that ν = ΘG (α). The claim will follow immediately from Lemma 3.4, once we show that α is independent of the pair (V, S V ). But this in turn is a consequence of Totaro’s argument involving “independence of V and S ”, which we recall from the proof of Theorem 1.1 of [7]: let (W, S W ) be another pair such as (V, S V ). Then the bundle quotients ((V − S V ) × W)/G and (V × (W − S W ))/G exist as vector bundles over (V − S V )/G and (W − S W )/G, respectively. Both the closed subsets V × S W and S V × W of V ⊕ W have codimensions at least (s + 1), and outside each of them, G acts freely, so that in each case the bundle quotient exists. Therefore, there is an invariant open subscheme of V ⊕ W which contains both of these open complements and which consists of G-stable points. Let us call this open subscheme V ⊕ W − S V⊕W . Observe that S V⊕W , being contained in each of the two closed subsets mentioned above, has codimension at least (s + 1) in V ⊕ W. Since H i ((V − S )/G) for i ≤ 2s does not depend on S as long as the codimension of S is ≥ s + 1, we see that all the arrows in the following commutative diagram are isomorphisms for i ≤ 2s, where the right vertical, the diagonal, and the lower horizontal arrows are induced by the classifying morphisms, and the upper horizontal and the left vertical arrows, by obvious inclusions: / H i ((V × (W − S W ))/G) = H i ((W − S W )/G) H i ((V ⊕ W − S V⊕W )/G) l❨ O

❨❨❨❨❨❨ ❨❨❨❨❨❨ ❨❨❨❨❨❨ ❨❨❨❨❨❨ ❨❨❨❨❨❨ ❨❨❨ o

H i ((V − S V )/G) = H i (((V − S V ) × W)/G) This completes the proof.

H i (BG)

5

Remark 3.6. The independence of V and S as above can also be applied in a little different setup: suppose P → X is a principal G-bundle, and there are two G-equivariant pairs (V, S V ) and (W, S W ) with maps from X to the bundle quotients (V −S V )/G and (W −S W )/G, with properties as required in Lemma 3.4. Then the diagonal map P → V ⊕ W factors through an open inclusion (V ⊕ W − S V⊕W ) ֒→ V ⊕ W as above.

4 The Gysin Sequence We begin by observing the following: Remark 4.1. Let L → X be a line bundle over k. A line bundle being locally trivial in the Zariski topology, the zero section has pure codimension in L. Therefore, whenever there is a principal Gm -bundle with the quotient space, hence the total space as well, smooth over k, we have the Gysin sequence for the smooth pair (L, X), where X is regarded as the zero section. Lemma 4.2. Let 1 → N → G → G′ → 1 be an exact sequence of reduced, affine algebraic groups over k, and let E → X be a principal G-bundle. Then the morphism E → X admits a factorization E → E ′ → X, where E → E ′ is the bundle quotient of E by N, and E ′ → X is the G′ -bundle associated to E → X. Proof. The associated G′ -bundle E ′ → X exists by e´ tale descent of affine morphisms. It is seen locally that the morphism E → E ′ is the required bundle quotient. In the above, let G′ = Gm , the multiplicative group scheme. Let P → Q be a principal G-bundle, where Q is a smooth, quasi-projective scheme over k. Then the bundle quotient of P by the action of N exists as the Gm -bundle Q′ → Q associated to P → Q, by Lemma 4.2. We can recognise Q′ → Q as the complement of the zero section of the induced line bundle. The Gysin long exact sequence · · · → H i (Q) → H i (Q′ ) → H i−1 (Q) → H i+1 (Q) → · · · exists by Remark 4.1, and is functorial for maps of the quotient space. On the other hand, we have the universal principal G-bundle Spec k → BG, whose associated Gm -torsor is the 1-morphism BN → BG. Assuming that G is reductive, we will construct a long exact sequence for this Gm -torsor: · · · → H i (BG) → H i (BN) → H i−1 (BG) → H i+1 (BG) → · · · In order to do this, we observe that by Lemma 3.5, given any integer s > 0, there is a principal G-bundle P → Q with H i (P) = 0 for i ≤ 2s + 2, so that the classifying morphism Q → BG is (2s + 2)-acyclic. Similarly, if Q′ → Q is the associated Gm -bundle, then P → Q′ is a principal N-bundle, whose classifying morphism Q′ → BN is (2s + 2)acyclic. We define the Gysin sequence for BN → BG by requiring each of the squares in the following diagram to commute, where the vertical isomorphisms are induced by the classifying morphisms for i ≤ 2s: 6

···

/

H i (BG)

H i (BN) /

/

H i−1 (BG)

/

H i+1 (BG)

···

/

/ H i (Q) / H i (Q′ ) / H i−1 (Q) / H i+1 (Q) / ··· ··· That the Gysin sequence does not depend on the particular principal bundle chosen, but any principal G-bundle P → Q of the type mentioned in the statement of Lemma 3.5 defines the same Gysin sequence for BN → BG, follows from the “independence of (V, S ) argument”, which was used in the proof of Proposition 3.1. More generally, by the independence of (V, S ) argument and Lemma 3.4, the Gysin sequence for BN → BG is compatible with the Gysin sequence for principal G-bundles over schemes (even when the vertical morphisms are not isomorphisms) by the same independence of (V, S ) argument. Therefore we have proved the following proposition.

Proposition 4.3. If G is reductive, there is a Gysin long exact sequence for the Gm torsor BN → BG, which is compatible with Gysin sequence for principal G-bundles over schemes in the sense that if P → Q is a principal G-bundle with Q a smooth, quasiprojective scheme over k, and Q′ → Q the associated Gm -bundle, then all the squares in the following diagram are commutative, where the vertical maps are induced by the classifying morphisms. / H i (BG) / H i (BN) / H i−1 (BG) / H i+1 (BG) / ··· ···

···

/

i

H (Q)

/

i

′

/

H (Q )

H

i−1

(Q)

/

H

i+1

(Q)

/

···

5 Cohomology Calculation In the paper of Holla and Nitsure [2], the main ingredient of the determination of the ∗ singular cohomology ring H sing (BGO(2n); F2 ) was the Gysin sequence. Other conceptual points that were used their argument are the following : ∗ (1) The fact that H sing (BO(n); F2 ) F2 [w1 , . . . , wn ], where wi is the i-th Stiefel-Whitney class; (2) the splitting principle for O(n)-bundles; (3) the fact that under the map in cohomology induced by the inclusion BO(n) ⊂ BGLn , ci 7→ w2i , and ∗ ∗ ∗ (4) the K¨unneth formula, i.e. that the natural map H sing (C× ; F2 )⊗H sing (X; F2 ) → H sing (C× × X; F2 ) is an isomorphism. The Gysin sequence for BO(2n) → BG(2n) in e´ tale cohomology is already established in Proposition 4.3. Among the other conceptual points, an e´ tale cohomology version of (1) is available in [4]. (2) The splitting principle in e´ tale cohomology is the ‘Claim’ at page 180 of [1], while an e´ tale cohomology version of (3) appears in the same paper. We also have the K¨unneth formula in e´ tale cohomology for the following special case: The natural map Het∗ (X; F2 ) ⊗ Het∗ (Gm ; F2 ) → Het∗ (X × Gm ; F2 ) is an isomorphism for a finite type, smooth X over k. The proof is much the same as that of Lemma 10.2 in [5]; the only observation we need here is that there is a quasi-isomorphism f ∗ P• ≃ Rp∗ (q∗ F2 ), where 7

0

P• is the complex F2 → F2 → 0, and p and q are projections from X × Gm on the first and second components, respectively. This is immediate from the Gysin sequence of the trivial Gm -bundle on X. ∗ With this preparation, we observe that the cohomology classes in H sing (BGO(2n); F2 ) introduced in section 4.3 of Holla and Nitsure [2] make sense in the e´ tale setup, and the proof of Theorem 4.3.5 of [2] carries over word by word. Therefore the description in terms of explicit generators and relations of the e´ tale cohomology ring Het∗ (BGO(2n); F2 ) of the classifying stack of GO(2n) over any algebraically closed field is exactly the same as the reduced mod-2 singular cohomology ring of the topological space BGO(2n), where GO(2n) is considered as a Lie group over complex numbers. Also, the map induced in e´ tale cohomology by the inclusion BGO(2n) ⊂ BGL2n has exactly the same description as in Proposition 3.2 of [3]. Characteristic Classes for SO(2n + 1) It is perhaps well known that H ∗ (BSO(2n + 1)) F2 [w2 , . . . , w2n+1 ]. Nonetheless, for the lack of a suitable reference in the cadre of algebraic stacks, we sketch a proof of this. Note that O(2n+1) µ2 ×SO(2n+1), for n ≥ 1. Hence, BO(2n+1) Bµ2 ×BSO(2n+1) for n ≥ 1. We will show that in this specific case K¨unneth formula holds for the cohomology of algebraic stacks: H ∗ (Bµ2 × BSO(2n + 1)) H ∗ (Bµ2 ) ⊗ H ∗ (BSO(2n + 1)). This will imply that F2 [w1 , . . . , w2n+1 ] F2 [w1 ] ⊗ H ∗ (BSO(2n + 1)), from which it can be seen that H ∗ (BS O(2n + 1)) F2 [w2 , . . . , w2n+1 ]. Claim: If X is any finite type, smooth scheme over k, and if Br stands for the bundle quotient of Ar+1 −{0} by µ2 ⊂ Gm , then the natural map H ∗ (Br )⊗H ∗ (X) → H ∗ (Br ×X) is an isomorphism. To prove it, we begin by observing the following: Let E → B be a principal Gm -bundle, and let B′ = E/µ2 , so that we have a natural projection B′ → B, which is again a principal Gm -bundle (because Gm /µ2 Gm ). Then the class of the principal Gm -bundle B′ → B in H 1 (B, Gm ) is the square of that of E → B, and therefore, its image under the connecting morphism coming from the Kummer sequence is zero in H 2 (B, µ2 ). This means that the Gysin sequence for B′ → B splits into short exact sequences. Now look at the following commutative diagram, whose left column is a part of the Gysin sequence for Br → Pr tensored with H ∗ (X), and the right column is the corresponding part of the Gysin sequence for Br × X → Pr × X. 0 0

⊕i+ j=m H i (Pr ) ⊗ H j (X)

∼

/

⊕i+ j=m H i (Br ) ⊗ H j (X)

⊕i+ j=m H i−1 (Pr ) ⊗ H j (X)

/

∼

/

H m (Pr × X)

H m (Br × X)

H m−1 (Pr × X)

0

0 8

The first and the third horizontal maps are isomorphisms, by Lemma 10.2 of [5]. Therefore the middle row is an isomorphism H ∗ (Br ) ⊗ H ∗ (X) H ∗ (Br × X), as claimed. The K¨unneth formula for Bµ2 × BSO(2n + 1) can be deduced from this claim, as follows. As we have described in Section 3, we can use acyclic covers to approximate the cohomology of classifying stacks. In the formula H ∗ (Br ) ⊗ H ∗ (X) H ∗ (Br × X), we substitute for X an acyclic cover for BSO(2n + 1), and note that Br is an r-acyclic cover for Bµ2 . This completes the proof of the K¨unneth formula for Bµ2 × BSO(2n + 1). Acknowledgment: This result will form a part of the my PhD thesis. I thank my supervisor, Nitin Nitsure, for his guidance.

References [1] Helen Esnault et al, Coverings with odd ramification and Stiefel-Whitney classes, J. reine angew. Math. 441 (1993), 145-188. [2] Yogish I. Holla and Nitin Nitsure, Characteristic Classes for GO(2n, C), Asian Journal of Mathematics, Volume 5, 2001, p.169-182. [3] Holla, Yogish I.; Nitsure, Nitin Topology of quadric bundles. Internat. J. Math. 12 (2001), no. 9, p.1005-1047. [4] J. F. Jardine, Universal Hasse-Witt Classes, Contemporary Mathematics, Volume 83 (1989). ´ [5] J. Milne, Etale Cohomology [6] David Mumford, Geometric Invariant Theory, Third enlarged edition. [7] Burt Totaro, Chow Ring of a Classifying Space, Proceedings of Symposia in Pure Mathematics, Volume 67, 1999. Address: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India. e-mail: [email protected] 18-I-2012

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