Essential Dimension

WHAT IS...Essential Dimension?

“…These developments are beyond the scope of the present note; please see the surveys [2,3] for an overview and further references. I will, however, mention one unexpected result that has come up in my joint work with P. Brosnan and A. Vistoli: it turns out that the essential dimension of the spinor group Spin n increases exponentially with n. This has led to surprising consequences in the theory of quadratic forms.…”

“…The inequality ed(PGL n ) n 2 proved by Procesi has since been strengthened (see [3]), but the new upper bounds are still quadratic in n. In the 1990s B. Kahn asked if ed(PGL n ) grows sublinearly in n, i.e., if there exists a C > 0 such that ed(PGL n ) Cn for every n. By the primary decomposition theorem we lose little if we assume that n is a prime power. Until recently, the best known lower bound was ed(PGL p r ) 2r .…”

Essential dimension and canonical dimension of gerbes banded by groups of multiplicative type

“…In almost all situations, when the (absolute) essential dimension of an algebraic group or algebraic stack is known, it is equal to its essential p-dimension for some prime p. This is a limitation of most methods we have in Galois cohomology and related algebraic areas. For a broad discussion of this phenomenon, see [Re10,§5]. As an example it is well known that ed p (PGL p ) = 2 for all primes p, but the question if ed(PGL p ) = 2 for all primes p is widely open and a negative answer to this question would disprove the cyclicity conjecture of Albert for central simple algebras of degree p. When the absolute essential dimension differs from essential p-dimension for all primes p (or is not known to coincide with essential p-dimension for some prime p) it often becomes significantly more difficult to compute it.…”

Essential Dimension of Generic Symbols in Characteristic

“…In this paper we consider the essential dimension of objects and functors relating to central simple algebras and higher symbols in Milne-Kato cohomology, focusing on the bad characteristic case. Since its introduction in [7], most of the upper and lower bounds on the essential dimension of central simple algebras have required that the degree of the algebra be relatively prime to the characteristic of the base field k. Two excellent surveys on essential dimension, [18] and [20], contain many of these results, algebraic and functorial definitions of essential dimension, pessential dimension and much more.…”

“…H 1 (K , PGL n ), the set of isomorphism classes of PGL n -torsors over Spec(K ), has a bijective correspondence with Alg n (K ), the set of isomorphism classes of central simple algebras of degree n over K . In particular, using the standard notation in [20], ed k (PGL n ) = ed k (Alg n ).…”