In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, essential dimension is an
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
defined for certain
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s such as
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
s and
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s. It was introduced by
J. Buhler and
Z. Reichstein
and in its most generality defined by
A. Merkurjev.
Basically, essential dimension measures the complexity of algebraic structures via their
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
of definition. For example, a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
''q'' : ''V'' → ''K'' over a field ''K'', where ''V'' is a ''K''-
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, is said to be defined over a
subfield ''L'' of ''K'' if there exists a ''K''-
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
''e''
1,...,''e''
''n'' of ''V'' such that ''q'' can be expressed in the form
with all coefficients ''a''
''ij'' belonging to ''L''. If ''K'' has
characteristic different from 2, every quadratic form is
diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
. 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
In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
over ''k'' of a subfield ''L'' of ''K'' over which ''q'' is defined.
Formal definition
Fix an arbitrary field ''k'' and let /''k'' denote the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of finitely generated
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s of ''k'' with inclusions as
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s. Consider a (covariant)
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
''F'' : /''k'' → .
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
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of ''ed''(''a'') taken over all elements ''a'' of ''F''(''K''/''k'') and objects ''K''/''k'' of /''k''.
Examples
* Essential dimension of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s: For a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
''n'' consider the functor ''Q''
''n'' : /''k'' → taking a field extension ''K''/''k'' to the set of
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the stru ...
es 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 group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
s: For an algebraic group ''G'' over ''k'' denote by H
1(−,''G'') : /''k'' → the functor taking a field extension ''K''/''k'' to the set of isomorphism classes of ''G''-
torsor
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
s 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
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of ...
: Let
be a category fibered over the category
of affine ''k''-schemes, given by a functor
For example,
may be the
moduli stack
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such space ...
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 ''F''
''p'' : /''k'' → 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 ''F''
''p''. In case of the classifying stack
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
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...
''G'' is always finite and bounded by the minimal dimension of a generically free
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
minus the dimension of ''G''.
* For ''G'' a
Spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
''k'', the essential dimension is listed in .
* The essential dimension of a finite algebraic
''p''-group over ''k'' equals the minimal dimension of a
faithful representation In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group on a vector space is a linear representation in which different elements of are represented by distinct linear mapp ...
, provided that the base field ''k'' contains a primitive ''p''-th
root of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
.
* The essential dimension of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
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 In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher ...
admitting a
Galois splitting field ''L''/''k'' of degree a power of a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p''. Then the essential dimension of ''T'' equals the least rank of the kernel of a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
of Gal(''L''/''k'')-
lattices ''P'' → ''X''(''T'') with
cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of .
Cokernels are dual to the kernels of category theory, hence the name: ...
finite and of order
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to ''p'', where ''P'' is a permutation lattice.
References
{{reflist
Algebraic structures