Английская Википедия: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,...,en of V such that q can be expressed in the form <math>q\left(\sum x_i e_i\right) = \sum a_{ij} x_ix_j</math> 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 Шаблон:Math/k denote the category of finitely generated field extensions of k with inclusions as morphisms. Consider a (covariant) functor F : Шаблон:Math/k → Шаблон:Math. 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 Шаблон:Math/k.
Examples
- Essential dimension of quadratic forms: For a natural number n consider the functor Qn : Шаблон:Math/k → Шаблон:Math 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 <math>q_K : V \otimes_L K \to K</math>.
- Essential dimension of algebraic groups: For an algebraic group G over k denote by H1(−,G) : Шаблон:Math/k → Шаблон:Math 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 <math>\mathcal{F}</math> be a category fibered over the category <math>Aff/k</math> of affine k-schemes, given by a functor <math>p : \mathcal{F} \to Aff/k.</math> For example, <math>\mathcal{F}</math> may be the moduli stack <math>\mathcal{M}_g</math> of genus g curves or the classifying stack <math>\mathcal{BG}</math> of an algebraic group. Assume that for each <math>A \in Aff/k</math> the isomorphism classes of objects in the fiber p−1(A) form a set. Then we get a functor Fp : Шаблон:Math/k → Шаблон:Math taking a field extension K/k to the set of isomorphism classes in the fiber <math>p^{-1}(Spec(K))</math>. The essential dimension of the fibered category <math>\mathcal{F}</math> is defined as the essential dimension of the corresponding functor Fp. In case of the classifying stack <math>\mathcal{F} = \mathcal{BG}</math> 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 Шаблон:OEIS2C.
- 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.
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