In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichstein and in its most generality defined by A. Merkurjev. Basically, essential dimension measures the complexity of algebraic structures via their fields of definition. For example, a quadratic form q : V → K over a field K, where V is a K-vector space, is said to be defined over a subfield L of K if there exists a K-basis e1,...,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 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 functor F : Fields/k → Set. For a field extension K/k and an element a of F 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 → F induced by the inclusion of L in K. The essential dimension of a, denoted by ed, is the least transcendence degree of a field of definition for a. The essential dimension of the functor F, denoted by ed, is the supremum of ed taken over all elements a of F 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 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 : Fields/k → Set the functor taking a field extension K/k to the set of isomorphism classes of G-torsors over K. The essential dimension of this functor is called the essential dimension of the algebraic group G, denoted by ed.
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 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 group G is always finite and bounded by the minimal dimension of a generically freerepresentation minus the dimension of G.
