Question

Let \(K=k\left(x_{1}, \ldots, x_{n}\right)\) be a function field in \(r\) variables over \(k\). (a) Show that there is an affine variety \(V \subset \mathbb{A}^{n}\) with \(k(V)=K\). (b) Show that we may find \(V \subset \mathbb{A}^{r+1}\) with \(k(V)=K, r=\operatorname{dim} V .\) (Assume char \((k)=0\) if you wish.)

Short answer

Answer: An affine variety V is a subset of an affine space \(\mathbb{A}^n\) defined by the vanishing set of a collection of polynomials {\(f_1, \dots, f_m\)}. The function field of an affine variety k(V) is the field of equivalence classes of rational functions that are defined and regular on the variety. Equivalently, if V is the zero locus of an ideal I, then k(V) is the field of fractions of the quotient algebra \(A=k[x_1, \dots, x_n]/I\).

Step-by-step solution

Definition: Affine Variety and Function Field

An affine variety V is a subset of an affine space \(\mathbb{A}^n\) defined by the vanishing set of a collection of polynomials {\(f_1, \dots, f_m\)}. The function field of an affine variety k(V) is the field of equivalence classes of rational functions that are defined and regular on the variety. Equivalently, if V is the zero locus of an ideal I, then k(V) is the field of fractions of the quotient algebra \(A=k[x_1, \dots, x_n]/I\).

Finding an affine variety V with k(V) = K

To find an affine variety V with its function field equal to K, we can start with the algebra of all regular functions K. Since K is a function field in r variables over k, it is given by the field of fractions of a finitely generated k-algebra. Let this be \(A=k[x_1, \dots, x_n]/I\), where I is the ideal corresponding to the relations between the variables. Let V be the zero set of the polynomials in I in \(\mathbb{A}^n\). Then, by definition, the function field of V is equal to K, which is what we need to prove.

Using Noether normalization lemma

To show that we can find V in \(\mathbb{A}^{r+1}\) with k(V) = K and r = dim(V), we can use the Noether normalization lemma. The lemma states that if A is a finitely generated k-algebra and \(K(\underline{x}) \supseteq A \supseteq k\), then there exists an algebraically independent subset {\(y_1, \dots, y_r\)} of A over k, such that A is an integral extension of the polynomial ring \(k[y_1, \dots, y_r]\). In other words, A is a finitely generated module over k[y_1, \dots, y_r]. Let {\(y_1, \dots, y_r\)} be the algebraically independent elements in A, and consider the affine variety V' defined by the ideal I' which is the vanishing set of the polynomial equations {\(y_i - f_i(x_1, \dots, x_n) = 0\)} in \(\mathbb{A}^{r+1}\). The coordinate ring of V' is given by the quotient \(k[x_0, \dots, x_r]/I'\), where {\(x_0, \dots, x_r\)} are the coordinates of \(\mathbb{A}^{r+1}\). Since each of the polynomial equations has degree 1, I' is a prime ideal, and thus V' is irreducible. Hence, there is a surjective homomorphism from the coordinate ring of V' to the coordinate ring of V, which induces an isomorphism between the function field of V' and the function field of V, which we know is equal to K. Therefore, k(V') = K and r = dim(V'), as required. In summary, we have proven that there exists an affine variety V in \(\mathbb{A}^n\) with k(V) = K and another affine variety V' in \(\mathbb{A}^{r+1}\) with k(V') = K, r = dim(V').

Key concepts

Function Field

In algebraic geometry, the concept of a function field is integral for understanding the properties and behavior of varieties. A function field over a field \(k\) is generally the collection of equivalence classes of rational functions that are well-defined on an array of point sets, known as varieties. To think of it simply, if you have a polynomial ring \(k[x_1, \, \ldots, \, x_n]\) and an ideal \(I\) that describes relationships between them, the function field of a variety \(V\) becomes the field of fractions of the quotient algebra \(k[x_1, \, \dots, \, x_n]/I\). This basically means we're considering fractions of polynomials that respect the zero situations imposed by ideal \(I\).

Function fields are significant because they bring numerically rich information from polynomial expressions to help depict the affine or projective spaces, which further facilitate understanding of higher-dimensional spaces and algebraic structures in geometry. This relationship and transformation are central to unraveling the complexity of algebraic structures, bridging raw algebra with geometric reality.

Noether Normalization Lemma

Noether's normalization lemma is a powerful tool in algebraic geometry and commutative algebra. It offers a systematic method to reduce complex algebraic structures to more manageable "normal" forms. In essence, the lemma says that for any finitely generated algebra \(A\) over a field \(k\), there exists a subalgebra that is a polynomial ring, such that \(A\) is an integral extension of this polynomial ring.

From a simplification perspective, if \(A\) represents a complicated k-algebra, there must be a subset \(\{y_1, \, \dots, \, y_r\}\) that is algebraically independent over \(k\). Furthermore, \(A\) is a finitely generated module over this polynomial ring \(k[y_1, \, \dots, \, y_r]\). This allows the reduction of the problem to studying the behaviors of polynomial equations with fewer complexities, simplifying the dimension-theoretic considerations of the varieties. This theorem is especially pertinent for showing dimensions of polarities of varieties, as it not only establishes isomorphic relations between different function fields but also assists in embedding varieties into affine spaces of certain dimensions.

Algebraic Geometry

Algebraic geometry is a branch of mathematics that merges the skills of algebra with the intuitive visuals of geometry. It studies the properties of spaces that arise from solutions to polynomial equations. An affine variety, a subset of affine space \(\mathbb{A}^{n}\), is one such fundamental entity, defined by the common zeros of a particular set of polynomials. The elegance of algebraic geometry lies in capturing geometric entities through algebraic expressions, offering insights into dimensions and intersections of these rich structures.

Helming the interplay between different branches of mathematics, algebraic geometry can unravel the nature of solutions and spaces, framed within a solid algebraic context. Through concepts like affine varieties and function fields, it allows the interpretation and manipulation of geometric spaces as algebraic objects, leveraging polynomial equations to express and solve complex spatial problems.

Algebraic Independence

Algebraic independence is an important concept when dealing with multivariable function fields or polynomial rings. A set \(\{f_1, \, f_2, \, \ldots, \, f_m\}\) of elements in a given ring is said to be algebraically independent over a field \(k\) if there is no non-zero polynomial \(P\) with coefficents in \(k\) such that \(P(f_1, \, f_2, \, \ldots, \, f_m) = 0\).

This concept serves as a measure of "freedom" within the function field or algebra, providing a criterion for simplifying the structure. For instance, in the context of Noether's normalization lemma, finding an algebraically independent set of elements is key for demonstrating that a complex algebraic stratum can be represented by simpler, lower-dimensional polynomial rings. It is akin to saying certain variables in polynomial form do not impose algebraic dependence on one another, thereby allowing the synthesis of polynomial equations and formulating new substitutions or embeddings in variety studies.