Examples
- Essential dimension of quadratic forms: For a natural number n 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.
Read more about this topic: Essential Dimension
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