Field of definition
Encyclopedia
In mathematics
, the field of definition of an algebraic variety
V is essentially the smallest field
to which the coefficients of the polynomial
s defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.
The issue of field of definition is of concern in diophantine geometry
.
of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field
containing p elements. Affine n-space
over a field F is denoted by An(F).
of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous
.
A k-algebraic set
is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.
One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections
X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed.
A k-variety is called a variety if it is absolutely irreducible
, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, …, xn] that vanishes on V is the linear combination
(over kalg) of polynomials in k[x1, …, xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.
Equivalently, a k-variety V is a variety defined over k if and only if the function field
k(V) of V is a regular extension of k, in the sense of Weil
. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint
.
André Weil
proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
In mathematics
, the field of definition of an algebraic variety
V is essentially the smallest field
to which the coefficients of the polynomial
s defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.
The issue of field of definition is of concern in diophantine geometry
.
of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field
containing p elements. Affine n-space
over a field F is denoted by An(F).
of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous
.
A k-algebraic set
is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.
One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections
X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed.
A k-variety is called a variety if it is absolutely irreducible
, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, …, xn] that vanishes on V is the linear combination
(over kalg) of polynomials in k[x1, …, xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.
Equivalently, a k-variety V is a variety defined over k if and only if the function field
k(V) of V is a regular extension of k, in the sense of Weil
. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint
.
André Weil
proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
In mathematics
, the field of definition of an algebraic variety
V is essentially the smallest field
to which the coefficients of the polynomial
s defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.
The issue of field of definition is of concern in diophantine geometry
.
of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field
containing p elements. Affine n-space
over a field F is denoted by An(F).
of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous
.
A k-algebraic set
is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.
One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections
X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed.
A k-variety is called a variety if it is absolutely irreducible
, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, …, xn] that vanishes on V is the linear combination
(over kalg) of polynomials in k[x1, …, xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.
Equivalently, a k-variety V is a variety defined over k if and only if the function field
k(V) of V is a regular extension of k, in the sense of Weil
. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint
.
André Weil
proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the field of definition of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V is essentially the smallest field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
to which the coefficients of the polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.
The issue of field of definition is of concern in diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
.
Notation
Throughout this article, k denotes a field. The algebraic closureAlgebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
containing p elements. Affine n-space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
over a field F is denoted by An(F).
Definitions for affine and projective varieties
Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing An(kalg) with projective spaceProjective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
.
A k-algebraic set
Algebraic set
In mathematics, an algebraic set over an algebraically closed field K is the set of solutions in Kn of a set of simultaneous equationsand so on up to...
is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.
One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed.
A k-variety is called a variety if it is absolutely irreducible
Absolutely irreducible
In mathematics, absolutely irreducible is a term applied to linear representations or algebraic varieties over a field. It means that the object in question remains irreducible, even after any finite extension of the field of coefficients...
, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, …, xn] that vanishes on V is the linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
(over kalg) of polynomials in k[x1, …, xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.
Equivalently, a k-variety V is a variety defined over k if and only if the function field
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
k(V) of V is a regular extension of k, in the sense of Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint
Linearly disjoint
In mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met:...
.
André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
Examples
- The zero-locus of x12+ x22 is both a Q-variety and a Qalg-algebraic set but neither a variety nor a Qalg-variety, since it is the union of the Qalg-varieties defined by the polynomials x1 + ix2 and x1 - ix2.
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the field of definition of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V is essentially the smallest field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
to which the coefficients of the polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.
The issue of field of definition is of concern in diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
.
Notation
Throughout this article, k denotes a field. The algebraic closureAlgebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
containing p elements. Affine n-space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
over a field F is denoted by An(F).
Definitions for affine and projective varieties
Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing An(kalg) with projective spaceProjective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
.
A k-algebraic set
Algebraic set
In mathematics, an algebraic set over an algebraically closed field K is the set of solutions in Kn of a set of simultaneous equationsand so on up to...
is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.
One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed.
A k-variety is called a variety if it is absolutely irreducible
Absolutely irreducible
In mathematics, absolutely irreducible is a term applied to linear representations or algebraic varieties over a field. It means that the object in question remains irreducible, even after any finite extension of the field of coefficients...
, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, …, xn] that vanishes on V is the linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
(over kalg) of polynomials in k[x1, …, xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.
Equivalently, a k-variety V is a variety defined over k if and only if the function field
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
k(V) of V is a regular extension of k, in the sense of Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint
Linearly disjoint
In mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met:...
.
André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
Examples
- The zero-locus of x12+ x22 is both a Q-variety and a Qalg-algebraic set but neither a variety nor a Qalg-variety, since it is the union of the Qalg-varieties defined by the polynomials x1 + ix2 and x1 - ix2.
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the field of definition of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V is essentially the smallest field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
to which the coefficients of the polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.
The issue of field of definition is of concern in diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
.
Notation
Throughout this article, k denotes a field. The algebraic closureAlgebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
containing p elements. Affine n-space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
over a field F is denoted by An(F).
Definitions for affine and projective varieties
Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing An(kalg) with projective spaceProjective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
.
A k-algebraic set
Algebraic set
In mathematics, an algebraic set over an algebraically closed field K is the set of solutions in Kn of a set of simultaneous equationsand so on up to...
is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.
One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed.
A k-variety is called a variety if it is absolutely irreducible
Absolutely irreducible
In mathematics, absolutely irreducible is a term applied to linear representations or algebraic varieties over a field. It means that the object in question remains irreducible, even after any finite extension of the field of coefficients...
, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, …, xn] that vanishes on V is the linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
(over kalg) of polynomials in k[x1, …, xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.
Equivalently, a k-variety V is a variety defined over k if and only if the function field
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
k(V) of V is a regular extension of k, in the sense of Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint
Linearly disjoint
In mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met:...
.
André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
Examples
- The zero-locus of x12+ x22 is both a Q-variety and a Qalg-algebraic set but neither a variety nor a Qalg-variety, since it is the union of the Qalg-varieties defined by the polynomials x1 + ix2 and x1 - ix2.
- With Fp(t) a transcendental extension of Fp, the polynomial x1p- t equals (x1 - t1/p) p in the polynomial ring (Fp(t))alg[x1]. The Fp(t)-algebraic set V defined by x1p- t is a variety; it is absolutely irreducible because it consists of a single point. But V is not defined over Fp(t), since V is also the zero-locus of x1 - t1/p.
- The complex projective line is a projective R-variety. (In fact, it is a variety with Q as its minimal field of definition.) Viewing the real projective line as being the equator on the Riemann sphere, the coordinate-wise action of complex conjugation on the complex projective line swaps points with the same longitude but opposite latitudes.
- The projective R-variety W defined by the homogeneous polynomial x12+ x22+ x32 is also a variety with minimal field of definition Q. The following map defines a C-isomorphism from the complex projective line to W: (a,b) → (2ab, a2-b2, -i(a2+b2)). Identifying W with the Riemann sphere using this map, the coordinate-wise action of complex conjugation on W interchanges opposite points of the sphere. The complex projective line cannot be R-isomorphic to W because the former has real points, points fixed by complex conjugation, while the latter does not.
Scheme-theoretic definitions
One advantage of defining varieties over arbitrary fields through the theory of schemesScheme (mathematics)In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
is that such definitions are intrinsic and free of embeddings into ambient affine n-space.
A k-algebraic set is a separated and reduced scheme of finite type over Spec(k)Spectrum of a ringIn abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
. A k-variety is an irreducible k-algebraic set. A k-morphism is a morphism between k-algebraic sets regarded as schemes over Spec(k).
To every algebraic extension L of k, the L-algebraic set associated to a given k-algebraic set V is the fiber product V ×Spec(k) Spec(L). A k-variety is absolutely irreducible if the associated kalg-algebraic set is an irreducible scheme; in this case, the k-variety is called a variety. An absolutely irreducible k-variety is defined over k if the associated kalg-algebraic set is a reduced scheme. A field of definition of a variety V is a subfield L of kalg such that there exists a k∩L-variety W such that W ×Spec(k∩L) Spec(k) is isomorphic to V and the final object in the category of reduced schemes over W ×Spec(k∩L) Spec(L) is an L-variety defined over L.
Analogously to the definitions for affine and projective varieties, a k-variety is a variety defined over k if the stalk of the structure sheafRinged spaceIn mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
at the generic pointGeneric pointIn mathematics, in the fields general topology and particularly of algebraic geometry, a generic point P of a topological space X is an algebraic way of capturing the notion of a generic property: a generic property is a property of the generic point.- Definition and motivation :A generic point of...
is a regular extension of k; furthermore, every variety has a minimal field of definition.
One disadvantage of the scheme-theoretic definition is that a scheme over k cannot have an L-valued point if L is not an extension of k. For example, the rational point (1,1,1) is a solution to the equation x1 + ix2 - (1+i)x3 but the corresponding Q[i]-variety V has no Spec(Q)-valued point. The two definitions of field of definition are also discrepant, e.g. the (scheme-theoretic) minimal field of definition of V is Q, while in the first definition it would have been Q[i]. The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set up to change of basis. In this example, one way to avoid these problems is to use the Q-variety Spec(Q[x1,x2,x3]/(x12+ x22+ 2x32- 2x1x3 - 2x2x3)),
whose associated Q[i]-algebraic set is the union of the Q[i]-variety Spec(Q[i][x1,x2,x3]/(x1 + ix2 - (1+i)x3)) and its complex conjugate.
Action of the absolute Galois group
The absolute Galois groupAbsolute Galois groupIn mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
Gal(kalg/k) of k naturally actsGroup actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. In general, if V is a scheme over k (e.g. a k-algebraic set), Gal(kalg/k) naturally acts on V ×Spec(k) Spec(kalg) via its action on Spec(kalg).
When V is a variety defined over a perfect fieldPerfect fieldIn algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
k, the scheme V can be recovered from the scheme V ×Spec(k) Spec(kalg) together with the action of Gal(kalg/k) on the latter scheme: the sections of the structure sheaf of V on an open subset U are exactly the sections of the structure sheaf of V ×Spec(k) Spec(kalg) on U ×Spec(k) Spec(kalg) whose residueResidue (complex analysis)In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
s are constant on each Gal(kalg/k)-orbit in U ×Spec(k) Spec(kalg). In the affine case, this means the action of the absolute Galois group on the zero-locus is sufficient to recover the subset of k[x1, …, xn] consisting of vanishing polynomials.
In general, this information is not sufficient to recover V. In the example of the zero-locus of x1p- t in (Fp(t))alg, the variety consists of a single point and so the action of the absolute Galois group cannot distinguish whether the ideal of vanishing polynomials was generated by x1 - t1/p, by x1p- t, or, indeed, by x1 - t1/p raised to some other power of p.
For any subfield L of kalg and any L-variety V, an automorphism σ of kalg will map V isomorphically onto a σ(L)-variety.
Further reading
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- The terminology in this article matches the terminology in the text of Fried and Jarden, who adopt Weil's nomenclature for varieties. The second edition reference here also contains a subsection providing a dictionary between this nomenclature and the more modern one of schemes.
- Kunz deals strictly with affine and projective varieties and schemes but to some extent covers the relationship between Weil's definitions for varieties and GrothendieckAlexander GrothendieckAlexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
's definitions for schemes. - Mumford only spends one section of the book on arithmetic concerns like the field of definition, but in it covers in full generality many scheme-theoretic results stated in this article.
