Polynomial rings and ideals are the building blocks of algebraic geometry. They provide a powerful framework for studying geometric objects through algebraic equations, bridging the gap between algebra and geometry.
In this context, we'll explore how polynomial rings encode geometric information and how ideals represent sets of solutions. We'll see how these concepts form the foundation for understanding algebraic varieties and their properties.
Polynomial rings and their properties
Definition and notation
A , denoted as R[x], is a ring formed by polynomials with coefficients from a ring R
The elements of a polynomial ring are polynomials, and the operations of addition and multiplication are performed on these polynomials
For example, if R is the ring of integers, then R[x] consists of polynomials with integer coefficients, such as 3x2+2x−1
Properties inherited from the coefficient ring
Polynomial rings inherit many properties from the coefficient ring R, such as commutativity and the existence of identity elements for addition and multiplication
If R is commutative, then R[x] is also commutative, meaning that for any polynomials f(x) and g(x) in R[x], f(x)g(x)=g(x)f(x)
The zero polynomial and the constant polynomial 1 serve as the additive and multiplicative identity elements in R[x], respectively
If R is a field, then R[x] is a domain (PID), meaning that every ideal in R[x] is generated by a single polynomial
For instance, if R is the field of real numbers, then every ideal in R[x] is of the form (f(x)), where f(x) is a polynomial in R[x]
Degree of a polynomial
The is the highest power of the variable in the polynomial, and it plays a crucial role in determining the properties of the polynomial ring
For example, the polynomial 3x2+2x−1 has degree 2, as the highest power of x is 2
Polynomials of the same degree can be compared and ordered based on their leading coefficients, allowing for the division algorithm and the notion of greatest common divisors (GCDs) in R[x]
Ideals in algebraic geometry
Definition and properties
An ideal I in a ring R is a subset of R that is closed under addition and multiplication by elements of R
For any a,b∈I and r∈R, a+b∈I and ra∈I
Ideals generalize the concept of multiples in the ring of integers and allow for the study of congruences and quotient rings
In the ring of integers Z, the ideal (n) consists of all multiples of n, and the Z/(n) represents the congruence classes modulo n
Correspondence with algebraic sets
In algebraic geometry, ideals in polynomial rings are used to define algebraic sets, which are the solution sets of systems of polynomial equations
For example, the ideal I=(x2+y2−1,y−x2) in R[x,y] defines the V(I), which is the unit circle intersected with the parabola y=x2
The correspondence between ideals and algebraic sets is a fundamental concept in algebraic geometry, known as the
Closed sets in the Zariski topology are precisely the algebraic sets, and the topology is defined by taking finite unions and arbitrary intersections of algebraic sets
Prime and maximal ideals
Prime ideals, which are ideals P such that for any a,b∈R, if ab∈P, then either a∈P or b∈P, play a crucial role in understanding the geometry of algebraic sets
Prime ideals correspond to irreducible algebraic sets, which cannot be written as the union of two proper subsets
Maximal ideals, which are ideals that are maximal with respect to inclusion, correspond to points in the defined by the polynomial ring
For example, in C[x,y], the (x−1,y−2) corresponds to the point (1,2) in the complex affine plane
Operations on ideals
Sum, product, and intersection
The sum of two ideals I and J, denoted as I+J, is the smallest ideal containing both I and J
I+J={a+b:a∈I,b∈J}
The product of two ideals I and J, denoted as IJ, is the ideal generated by all products of elements from I and J
IJ={∑i=1naibi:ai∈I,bi∈J,n∈N}
The intersection of two ideals I and J, denoted as I∩J, is the largest ideal contained in both I and J
I∩J={a:a∈I and a∈J}
Quotient and radical
The quotient of two ideals I and J, denoted as (I:J), is the ideal consisting of all elements r in R such that rJ is a subset of I
(I:J)={r∈R:rJ⊆I}
The I, denoted as I, is the ideal consisting of all elements r in R such that some power of r belongs to I
I={r∈R:rn∈I for some n∈N}
Operations on ideals allow for the manipulation and simplification of systems of polynomial equations in algebraic geometry
For instance, the radical of an ideal corresponds to the vanishing set of the ideal, which is the set of all points where all polynomials in the ideal evaluate to zero
Geometric meaning of ideals
Algebraic sets and vanishing ideals
The algebraic set defined by an ideal I, denoted as V(I), is the set of all points in the affine space that satisfy all the polynomial equations in I
V(I)={(a1,…,an)∈kn:f(a1,…,an)=0 for all f∈I}, where k is the underlying field
The ideal of an algebraic set S, denoted as I(S), is the set of all polynomials that vanish on every point of S
I(S)={f∈k[x1,…,xn]:f(a1,…,an)=0 for all (a1,…,an)∈S}
Zariski topology and irreducible sets
The Zariski topology on the affine space is defined by taking algebraic sets as the closed sets, establishing a correspondence between ideals and closed sets
The closure of a set in the Zariski topology is the smallest algebraic set containing it, and it corresponds to the radical of the ideal of the set
Irreducible algebraic sets, which cannot be written as the union of two proper subsets, correspond to prime ideals in the polynomial ring
An algebraic set is irreducible if and only if its ideal is a
Dimension and local properties
The dimension of an algebraic set is related to the height of its corresponding prime ideal, providing a way to study the geometric properties of the set
The dimension of an irreducible algebraic set is the transcendence degree of its function field over the base field
The local ring at a point in an algebraic set is obtained by localizing the polynomial ring at the maximal ideal corresponding to that point, allowing for the study of local properties of the set
The localization of a ring R at a prime ideal P, denoted as RP, consists of elements of the form ba, where a∈R and b∈R∖P, and it captures the local behavior of the algebraic set near the point corresponding to P