Essential dimension

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 ^[1] and in its most generality defined by A. Merkurjev. ^[2]

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,...,e_n of V such that q can be expressed in the form $q(\sum x_i e_i) = \sum a_{ij} x_ix_j$ with all coefficients a_ij 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) 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.

• 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 S_n (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 prime to p, where P is a permutation lattice.

References

1. Buhler, J.; Reichstein, Z. (1997). "On the essential dimension of a finite group". Compositio Mathematica 106 (2): 159–179. doi:10.1023/A:1000144403695.
2. Berhuy, G.; Favi, G. (2003). "Essential Dimension: a Functorial Point of View (after A. Merkurjev)". Documenta Mathematica 8: 279–330 (electronic).