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 Joe Buhler and Zinovy Reichstein^[1] and in its full generality defined by Alexander Merkurjev.^[2]

Basically, essential dimension measures the complexity of algebraic structures via their fields of definition. For example, if $K$ is a field and $V$ a $K$ -vector space, a quadratic form $q:V\to K$ is said to be defined over a subfield $L$ of $K$ if there exists a $K$ -basis $e_{1},\dots ,e_{n}$ of $V$ such that $q$ can be expressed in the form

q{\bigg (}\sum _{i=1}^{n}x_{i}e_{i}{\bigg )}=\sum a_{ij}x_{i}x_{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 $F:$ $Fields$ $/k\to$ $Set$ . 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 $F(L/k)\to F(K/k)$ induced by the inclusion of $L$ in $K$ .

The essential dimension of $a$ , denoted by $\operatorname {ed} (a)$ , is the least transcendence degree (over $k$ ) of a field of definition for $a$ . The essential dimension of the functor F, denoted by $\operatorname {ed} (F)$ , is the supremum of $\operatorname {ed} (a)$ 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 $Q_{n}:$ $Fields$ $/k\to$ $Set$ 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 $L/k\to K/k$ (given by the inclusion of $L$ in $K$ ) to the map sending the isomorphism class of a quadratic form $q:V\to L$ 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 $H^{1}(-,G):$ $Fields$ $/k\to$ $Set$ 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 $\operatorname {ed} (G)$ .

Essential dimension of a fibered category: let ${\mathcal {F}}$ be a category fibered over the category $\operatorname {Aff} /k$ of affine $k$ -schemes, given by a functor $p:{\mathcal {F}}\to \operatorname {Aff} /k$ . 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 $A\in \operatorname {Aff} /k$ the isomorphism classes of objects in the fiber $p^{-1}(A)$ form a set. Then we get a functor $F_{p}:$ $Fields$ $/k\to$ $Set$ taking a field extension $K/k$ to the set of isomorphism classes in the fiber $p^{-1}(\operatorname {Spec} (K))$ . The essential dimension of the fibered category ${\mathcal {F}}$ is defined as the essential dimension of the corresponding functor $F_{p}$ . In the 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 $G$ .

For $G$ a Spin group over an algebraically closed field $k$ , the essential dimension is listed in OEIS: A280191.

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 an algebraic group over $k$ ) is known for $n\leq 5$ (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 $\operatorname {Gal} (L/k)$ -lattices $P\to X(T)$ with cokernel finite and of order coprime to $p$ , where $P$ is a permutation lattice.