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 : VK 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 (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

edit

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/kSet. 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

edit
  • Essential dimension of quadratic forms: For a natural number n consider the functor Qn : Fields/kSet 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/kK/k (given by the inclusion of L in K) to the map sending the isomorphism class of a quadratic form q : VL 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/kSet 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/kSet 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

edit

References

edit
  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).