A fiber dimension theorem for essential and canonical dimension

Lötscher, Roland

doi:10.1112/s0010437x12000565

Cited by 15 publications

(15 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our Theorem 1.1 gives upper bounds on the essential dimension of and regardless of the characteristic of . Combining these with the results of [BRV10, Mer09, CM14, Lot13] quickly gives the following, see §6 for details.…”

Section: Introductionmentioning

confidence: 58%

See 1 more Smart Citation

Spinors and essential dimension

Garibaldi¹,

Guralnick²

2017

Compositio Math.

View full text Add to dashboard Cite

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan--Reichstein--Vistoli (Annals of Math., 2010) and Chernousov--Merkurjev (Algebra & Number Theory, 2014) to fields of characteristic different from 2.Comment: v2 adds an appendix by Alexander Premet on generic stabilizer in HSpin_16 in characteristic

show abstract

Section: Introductionmentioning

confidence: 58%

“…Although Corollary 1.3 is stated and proved for split groups, it quickly implies analogous results for nonsplit forms of these groups, see [Lot13, §4] for details.…”

Section: Introductionmentioning

confidence: 91%

Spinors and essential dimension

Garibaldi¹,

Guralnick²

2017

Compositio Math.

View full text Add to dashboard Cite

show abstract

“…The essential dimension of a category fibered in groupoids was defined by Lötscher in [28] as follows. Let X be a CF G over F , let x be an object in the fiber X (K) of X over Spec(K), and let K ⊂ K be a subfield over F .…”

Section: The Generic Fibermentioning

confidence: 99%

“…In this section, we give a more general definition of the canonical dimension of a functor (see [23, §2] and [29, §1.6]). A more general definition of the canonical dimension of a category fibered in groupoids was given in [28].…”

Section: Canonical Dimensionmentioning

confidence: 99%

Essential dimension

Merkurjev

2016

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

In this paper we survey research on the essential dimension that was introduced by J. Buhler and Z. Reichstein. Informally speaking, the essential dimension of a class of algebraic objects is the minimal number of algebraically independent parameters one needs to define any object in the class. The notion of essential dimension, which is defined in elementary terms, has surprising connections to many areas of algebra, such as algebraic geometry, algebraic K-theory, Galois cohomology, representation theory of algebraic groups, theory of fibered categories and valuation theory. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups.

show abstract

“…These gerbes have been used extensively in the computation of the essential dimension of algebraic groups G like finite p-groups, algebraic tori, Spin groups etc. when A is diagonalizable (see [Lö12] for a survey). A key ingredient is the inequality from [Me09, Theorem 4.8], which bounds ed(G) from below in terms of the essential dimension of the gerbe [X/G].…”

Section: Introductionmentioning

confidence: 99%