On the notion of canonical dimension for algebraic groups

Berhuy, Grégory; Reichstein, Zinovy

doi:10.1016/j.aim.2004.12.004

Cited by 30 publications

(57 citation statements)

References 30 publications

(19 reference statements)

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“…has a dense image. The notion of incompressibility appears to be very important in the study of the splitting properties of Gtorsors, where G is a linear algebraic group, in computations of the essential and the canonical dimension of G (see [1], [2] and [4]). For instance, using the Rost degree formula (1) Merkurjev provided a uniform and shortend proof of the incompressibility of certain Severi-Brauer varieties, involution varieties and quadrics (see [9, §5 and §7]).…”

Section: 14])mentioning

confidence: 99%

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Degree formula for connective K-theory

Zainoulline

2009

Invent. math.

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Section: 14])mentioning

confidence: 99%

“…Let X be a smooth projective 3-fold. In this case we have three partitions (1, 1, 1), (1,2) and (3), where the first and the last one belong to Λ 2 . We have χ(O X ) = − …”

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confidence: 99%

Degree formula for connective K-theory

Zainoulline

2009

Invent. math.

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“…where ∂ is the connecting map for the exact sequence (3). Note that as µ p ⊂ F × , the intersection of Ker(χ * ) over all characters χ ∈ C * is trivial.…”

Section: Main Theoremmentioning

confidence: 99%

“…(cf. [3], [11]) Let F be a field and C a class of field extensions of F . A field E ∈ C is called generic if for any L ∈ C there is an F -place E L. The canonical dimension cdim(C) of the class C is the minimum of the tr.…”

Section: Example 122mentioning

confidence: 99%

Essential dimension of finite p-groups

Karpenko

Merkurjev

2008

Invent. math.

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Abstract. We prove that the essential dimension and p-dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F .The notion of the essential dimension ed(G) of a finite group G over a field F was introduced in [5]. The integer ed(G) is equal to the smallest number of algebraically independent parameters required to define a Galois G-algebras over any field extension of F . If V is a faithful linear representation of G over Prop. 4.15]). The essential dimension of G can be smaller than dim(V ) for every faithful representation V of G over F . For example, we have ed(Z/3Z) = 1 over Q or any field F of characteristic 3 (cf. [2, Cor. 7.5]) and ed(S 3 ) = 1 over C (cf. [5, Th. 6.5]).In this paper we prove that if G is a p-group and F is a field of characteristic different from p containing p-th roots of unity, then ed(G) coincides with the least dimension of a faithful representation of G over F (cf. Theorem 4.1).We also compute the essential p-dimension of a p-group G introduced in [15]. We show that ed p (G) = ed(G) over a field F containing p-th roots of unity.In the paper the word "scheme" means a separated scheme of finite type over a field and "variety" an integral scheme. Acknowledgment:We are grateful to Zinovy Reichstein for useful conversations and comments.

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“…The concept of canonical dimension was introduced by G. Berhuy and Z. Reichstein [BR05]. See also [Me09].…”

Section: Introductionmentioning

confidence: 99%

Essential dimension of involutions and subalgebras

Lötscher

2012

Isr. J. Math.

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Abstract. We use theorems of N. Karpenko about the incompressibility of Severi Brauer varieties and quadratic Weil transfers of Severi Brauer varieties to compute the minimal number of parameters needed to define conjugacy classes of involutions of given type on division algebras of 2-primary index. Similarly we compute the minimal number of parameters needed to define conjugacy classes ofétale subalgebras of given type and central simple subalgebras of given degree in division algebras of p-primary index.

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On the notion of canonical dimension for algebraic groups

Abstract: We define and study a numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P n−1 .

Cited by 30 publications

References 30 publications

Degree formula for connective K-theory

Degree formula for connective K-theory

Essential dimension of finite p-groups

Essential dimension of involutions and subalgebras

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