Basically, essential dimension measures the complexity of algebraic structures via their
fields of definition. For example, a
quadratic formq : V → K over a field K, where V is a K-
vector space, is said to be defined over a
subfieldL of K if there exists a K-
basise1,...,en of V such that q can be expressed in the form with all coefficients aij belonging to L. If K has
characteristic different from 2, every quadratic form is
diagonalizable. Therefore, q has a field of definition generated by n elements. Technically, one always works over a (fixed) base field k and the fields K and L in consideration are supposed to contain k. The essential dimension of q is then defined as the least
transcendence degree over k of a subfield L of K over which q is defined.
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)
functorF : 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 fieldK/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 numbern consider the functor Qn : 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 .
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 be a category fibered over the category of affine k-schemes, given by a functor For example, may be the
moduli stack of genus g curves or the classifying stack of an algebraic group. Assume that for each the isomorphism classes of objects in the fiber p−1(A) form a set. Then we get a functor Fp : Fields/k → Set taking a field extension K/k to the set of isomorphism classes in the fiber . The essential dimension of the fibered category is defined as the essential dimension of the corresponding functor Fp. In case of the classifying stack 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 groupG is always finite and bounded by the minimal dimension of a generically free
representation minus the dimension of G.
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).