Essential dimension

Formal definition

Fix an arbitrary field k and let Fields/k denote the category of finitely generated field extensions of k with inclusions as morphisms. Consider a (covariant) functor F : Fields/k → Set. For a field extension K/k and an element a of F(K/k) a field of definition of a is an intermediate field K/L/k such that a is contained in the image of the map F(L/k) → F(K/k) induced by the inclusion of L in K.

The essential dimension of a, denoted by ed(a), is the least transcendence degree (over k) of a field of definition for a. The '*essential dimension of the functor F''', denoted by ed(F), is the supremum of ed(a) taken over all elements a of F(K/k) and objects K/k of Fields/*k''.

Examples

• Essential dimension of quadratic forms: For a natural number n consider the functor Q**n : Fields/k → Set taking a field extension K/k to the set of isomorphism classes of non-degenerate n-dimensional quadratic forms over K and taking a morphism L/k → K/k (given by the inclusion of L in K) to the map sending the isomorphism class of a quadratic form q : V → L to the isomorphism class of the quadratic form q_K : V \otimes_L K \to K.
• Essential dimension of algebraic groups: For an algebraic group G over k denote by H1(−,G) : Fields/k → Set the functor taking a field extension K/k to the set of isomorphism classes of G-torsors over K (in the fppf-topology). The essential dimension of this functor is called the essential dimension of the algebraic group G, denoted by ed(G).
• Essential dimension of a fibered category: Let \mathcal{F} be a category fibered over the category Aff/k of affine k-schemes, given by a functor p : \mathcal{F} \to Aff/k. For example, \mathcal{F} may be the moduli stack \mathcal{M}_g of genus g curves or the classifying stack \mathcal{BG} of an algebraic group. Assume that for each A \in Aff/k the isomorphism classes of objects in the fiber p−1(A) form a set. Then we get a functor F**p : Fields/k → Set taking a field extension K/k to the set of isomorphism classes in the fiber p^{-1}(Spec(K)). The essential dimension of the fibered category \mathcal{F} is defined as the essential dimension of the corresponding functor F**p. In case of the classifying stack \mathcal{F} = \mathcal{BG} of an algebraic group G the value coincides with the previously defined essential dimension of G.

Known results

• The essential dimension of a linear algebraic group G is always finite and bounded by the minimal dimension of a generically free representation minus the dimension of G.
• For G a Spin group over an algebraically closed field k, the essential dimension is listed in .
• The essential dimension of a finite algebraic p-group over k equals the minimal dimension of a faithful representation, provided that the base field k contains a primitive p-th root of unity.
• The essential dimension of the symmetric group Sn (viewed as algebraic group over k) is known for n ≤ 5 (for every base field k), for n = 6 (for k of characteristic not 2) and for n = 7 (in characteristic 0).
• Let T be an algebraic torus admitting a Galois splitting field L/k of degree a power of a prime p. Then the essential dimension of T equals the least rank of the kernel of a homomorphism of Gal(L/k)-lattices P → X(T) with cokernel finite and of order coprime to p, where P is a permutation lattice.

References

References

1. Buhler, J.. (1997). "On the essential dimension of a finite group". Compositio Mathematica.
2. Berhuy, G.. (2003). "Essential Dimension: a Functorial Point of View (after A. Merkurjev)". Documenta Mathematica.