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The Picard group of the moduli of Gbundles on a curve
 Compositio Math. 112
, 1998
"... This paper is concerned with the moduli space of principal Gbundles on an algebraic curve, for G a complex semisimple group. While the case G = SLr, which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until rec ..."
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Cited by 34 (3 self)
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This paper is concerned with the moduli space of principal Gbundles on an algebraic curve, for G a complex semisimple group. While the case G = SLr, which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until recently, when it
The Arason invariant and mod 2 algebraic cycles
 J. A.M.S
, 1998
"... 2. The special Clifford group 5 3. Kcohomology of split reductive algebraic groups 7 ..."
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Cited by 33 (8 self)
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2. The special Clifford group 5 3. Kcohomology of split reductive algebraic groups 7
On The Chow Motive Of Some Moduli Spaces.
, 1998
"... . We study the motive of moduli spaces of stable vector bundles over a smooth projective curve. We prove this motive lies in the category generated by the motive of the curve and we compute its class in the Grothendieck ring of the category of motives. As applications we compute the Poincar'eH ..."
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Cited by 17 (1 self)
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. We study the motive of moduli spaces of stable vector bundles over a smooth projective curve. We prove this motive lies in the category generated by the motive of the curve and we compute its class in the Grothendieck ring of the category of motives. As applications we compute the Poincar'eHodge polynomials and the number of points over a finite field and we study some conjectures on algebraic cycles on these moduli spaces. Introduction The cohomology of the moduli spaces of stable vector bundles over a smooth projective curve has been thoroughly studied over the last years. This study has been accomplished by topological methods involving the NarasimhanSeshadri correspondence ([34]), by number theoretical methods ([22], [23]) and using differential geometry ([1]). The work we present is a generalisation of some of these results to the more general setup of motives. We use a geometric construction due to E. Bifet, M. Letizia and F. Ghione to compute the motivic Poincar'e polynomia...
On Grothendieck—Serre’s conjecture concerning principal Gbundles over reductive group schemes
 I, Preprint (2009), http://www.math.uiuc.edu/Ktheory/ Popescu, D. General
, 1986
"... Let R be a regular semilocal ring containing an infinite perfect subfield and let K be its field of fractions. Let G be a reductive Rgroup scheme satifying a mild ”isotropy condition”. Then each principal Gbundle P which becomes trivial over K is trivial itself. If R is of geometric type, then it ..."
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Cited by 15 (7 self)
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Let R be a regular semilocal ring containing an infinite perfect subfield and let K be its field of fractions. Let G be a reductive Rgroup scheme satifying a mild ”isotropy condition”. Then each principal Gbundle P which becomes trivial over K is trivial itself. If R is of geometric type, then it suffices to assume that R is of geometric type over an infinite field. Two main Theorems of Panin’s, Stavrova’s and Vavilov’s [PSV] state the same results for semisimple simply connected Rgroup schemes. Our proof is heavily based on those two Theorems of [PSV, Thm.1.1], on the main result of [CT/S] and on two purity Theorems proven in the present preprint. Those purity result look as follows. Given an Rtorus C and a smooth Rgroup scheme morphism µ: G → C one can form a functor from Ralgebras to abelian groups S ↦ → F(S): = C(S)/µ(G(S)). We prove that this functor satisfies a purity theorem for R. If R is of geometric type, then it suffices to assume that R is of geometric type over an infinite field. Examples to mentioned purity results are considered in the very end of the preprint. 1
Applications of AtiyahHirzebruch spectral sequence for motivic cobordism
 Department of Mathematics, Faculty of Education, Ibaraki University
"... Abstract. We study applications of AtiyahHirzebruch spectral sequences for motivic cobordisms found by Hopkins and Morel. 1. ..."
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Abstract. We study applications of AtiyahHirzebruch spectral sequences for motivic cobordisms found by Hopkins and Morel. 1.
ESSENTIAL DIMENSION
"... Abstract. Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions [BuR97]. Essential dimension has since b ..."
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Cited by 13 (4 self)
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Abstract. Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions [BuR97]. Essential dimension has since been investigated in several broader contexts, by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. The goal of this paper is to survey some of this research. I have tried to explain the underlying ideas informally through motivational remarks, examples and proof outlines (often in special cases, where the argument is more transparent), referring an interested reader to the literature for a more detailed treatment. The sections are arranged in rough chronological order, from the definition of essential dimension to open problems. 1. Definition of essential dimension
Splitting fields for E8torsors
 Department of Mathematics, University of Southern
"... Several of the fundamental problems of algebra can be unified into the problem of classifying Gtorsors over an arbitrary field k, for a linear algebraic group G. (A Gtorsor can be defined as a principal Gbundle over Spec k, or as an algebraic variety over k with a free transitive action of G.) Fo ..."
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Cited by 9 (2 self)
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Several of the fundamental problems of algebra can be unified into the problem of classifying Gtorsors over an arbitrary field k, for a linear algebraic group G. (A Gtorsor can be defined as a principal Gbundle over Spec k, or as an algebraic variety over k with a free transitive action of G.) For example, PGL(n)torsors are equivalent to central simple algebras, and torsors for the orthogonal group are equivalent to quadratic forms. See Serre [36] for a recent survey of the classification problem for Gtorsors over a field. The study of Gtorsors is still in its early stages. Indeed, it is not completely known how complicated Gtorsors can be, if we fix the type of the group but allow arbitrary base fields. Tits showed that there is a bound on how complicated they can be. For each split semisimple group G, there is an integer d(G) depending only on the type of G, not on the field, such that every Gtorsor over a field k becomes trivial over some finite extension E of k of degree dividing d(G) [40]. For example, it is easy to see that one can take d(G) to be the order of the Weyl group of G. But it is a fundamental problem to determine the best possible number d(G) for
ALGEBRAIC COBORDISM OF SIMPLY CONNECTED LIE GROUPS
"... Abstract. Let GC be the algebraic group over C corresponding a simply connected Lie group G. The algebraic cobordism Ω(GC) defined by Levine and Morel is showed isomorphic to MU ∗subalgebra of MU ∗ (G) with some modulous and is computed explicitely. 1. ..."
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Abstract. Let GC be the algebraic group over C corresponding a simply connected Lie group G. The algebraic cobordism Ω(GC) defined by Levine and Morel is showed isomorphic to MU ∗subalgebra of MU ∗ (G) with some modulous and is computed explicitely. 1.
Relative padic Hodge theory, I: Foundations
, 2011
"... We initiate a new approach to relative padic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry. In this paper, we give a thorough development of ϕmodules over a relative Robba ring associated to a perfect Banach ring of characteristic p, includi ..."
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We initiate a new approach to relative padic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry. In this paper, we give a thorough development of ϕmodules over a relative Robba ring associated to a perfect Banach ring of characteristic p, including the relationship between these objects and étale Zplocal systems and Qplocal systems on the algebraic and analytic spaces associated to the base ring. We also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite étale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic p and the finite étale algebras over a corresponding Qpalgebra. This recovers the homeomorphism between the absolute Galois groups of Fp((π)) and Qp(µp∞) given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, and Scholl. Applications to the description of étale local systems on nonarchimedean analytic spaces will be described in subsequent papers. Contents 0