Ideals in K[x1, x2,..., xn] and Algebraic Varieties

 

Definition 1. A subset I of the polynomial ring K[x1, x2,..., xn] is an ideal if it satisfies:
  1. 0 is an element of I,
  2. If f and g are any two elements in I, then f+g is still an element of I,
  3. If f is an element of I, then for any h in K[x1, x2,..., xn] hf is still an element of I.
The first example of ideal in K[x1, x2,..., xn] is the ideal generated by a finite number of polynomials:
Lemma 2. Let F = {f1,f2,...,fs} be a finite subset of K[x1, x2,..., xn]. Then the set

{ Ssi=1 hifi : h1,h2,...,hs are in K[x1, x2,..., xn]}

is an ideal.
The proof is really straightforward.
Definition 3. The set { Ssi=1 hifi : h1,h2,...,hs are in K[x1, x2,..., xn]} is called the ideal generated by F and it is denoted by <f1,f2,...,fs>. The polynomials f1, f2,..., fs are called generators.
Definition 4. If an ideal I has finitely many generators it is said to be finitely generated and the set {f1,f2,...,fs} is called a basis of I.
Actually, the Hilbert Basis Theorem states that:
Theorem 5 (Hilbert Basis Theorem). Every ideal in K[x1, x2,..., xn] is finitely generated.
A very important consequence of this theorem is that any ascending chain of ideals I1 C I2 C I3 C ... in K[x1, x2,..., xn] stabilizes with In for some n. This is called ACC (Ascending Chain Condition) and it is used to prove that the Buchberger's algorithm terminates in a finite number of steps.

Note that a given ideal may have many different bases. A particular kind of bases is given by Groebner bases which have very useful properties.

There is a nice analogy between ideals and subspaces in linear algebra. Both have to be closed under addition and multiplication (except that for a subspace we multiply by scalars, whereas for an ideal by polynomials). But the analogy ends here.
For example, in linear algebra a basis must span and be linear independent over K, while a basis for an ideal is concerned only with spanning, not with independence. In fact, if we allow multiplication by polynomials, no independence is possible. To see this, consider the ideal <x,y> in K[x,y]. Then 0 can be written as a linear combination of x and y with nonzero polynomials in this way: 0 = (y)x - (x)y.
A consequence of the lack of independence is that when we write an element f of <f1,f2,...,fs> as Ssi=1 hifi then the coefficients hi are not unique. As an example, consider f = x2+xy+y2 as an element of <x,y>. Then f can be written as a combination of x and y in these two different ways: f = (x+y)x+(y)y and f = (x)x+(x+y)y.
Another difference with linear algebra will be explained after the following definition:
Definition 6. A basis {f1,f2,...,fs} of an ideal I is said to be minimal if there exist no proper subset of {f1,f2,...,fs} which is a basis of I.
Example.
  1. {x,x2} is not a minimal basis, since x generates the same ideal.

Unlike subspaces in linear algebra, ideals may have bases with a different number of generators, even though these bases are minimal. For example, consider {x} and {x+x2, x2}. They generate the same ideal in K[x] and they both are minimal. To prove the first part of the statement we must show that <x2,x+x2> is a subset of <x,y> and viceversa that <x,y> is a subset of <x2,x+x2>.
An element f of <x2,x+x2> can be written as f  = h1x2 + h2(x+x2) where h1 and h2 are polynomials of K[x]. Sorting it out we get f  = (h1+ h2)x2 +h2x = [(h1+ h2)x+h2]x which shows that f is an element of <x>. Conversely, an element g of <x> can be written as g  = h(x+x2) - hx2, where h is a polynomial of K[x].  So g is also an element of <x2,x+x2>. Hence <x>=<x2,x+x2>.

An ideal generated by only one element is called principal. By the division algorithm in K[x], the one taught in the high school, it is possible to prove that every ideal in K[x] is principal.

Now we will define what we mean by affine (algebraic) variety and explain the relationship with ideals.
Definition 7. Let S = {f1, f2,..., fs} be a subset of K[x1,x2,...,xn]. Then we set

V(S) = {(a1,a2,...,an) of Kn : fi(a1,a2,...,an) = 0 for all 1<=i<=s}

We call V(S) the affine (algebraic) variety defined by S.

So the affine variety V(S) is the set of all solutions of the system of equations f1(x1,x2,...,xn) = f2(x1,x2,...,xn) = ... = fs(x1,x2,...,xn) = 0.
Observation 8. If I is the ideal defined by { Ssi=1 hifi : h1,h2,...,hs are in K[x1, x2,..., xn]}, which is the smallest ideal containing S, then V(S)=V(I).

V(S) C V(I). In fact, if P(a1,a2,...,an) is in V(S), then fi(P) = 0 for all 1<=i<=s; hence Ssi=1 hifi(P) = 0 and P is in V(I).
V(I) C V(S). In fact, if P(a1,a2,...,an) is in V(I), then f(P) = 0 for all f in I; but SCI, so f(P) = 0 for all f in S. Hence P is in V(S).

Thus, we can now on consider V(I) instead of V(S).
If {f1,f2,..., fs} and {g1, g2,..., gt} are two different bases of I then V(f1, f2,..., fs) = V(g1, g2,..., gt).

Let's see some basic properties of affine varieties:
Lemma 9. Let V, W be affine varieties of Kn. Then V I W and V U W are affine varieties. In particular, if V = V(I) = V(f1, f2,..., fs) and W = V(J) = V(g1, g2 ,..., gt) we have:
  1. V I W = V(f1, f2,..., fs, g1, g2 ,..., gt) = V(I U J)
  2. V U W = V(figj: 1<=i<=s, 1<=j<=t) = V(I I J)
Examples.
  1. V = V(x-z), W = V(y-z, z-1). Then V I W = V(x-z,y-z,z-1) = {(1,1,1)}.
  2. V(x) U V(x-z) = V(x2-zx).
Definition 10. Let V be an affine variety of Kn. Then we set

I(V) = {f of K[x1,x2,...,xn]: f(a1,a2,...,an) = 0 for all (a1,a2,...,an) of V}
It's really easy to prove that I(V) is an ideal. We call it the ideal of V.

Examples.

  1. In K2, I({(0,0)}) = <x,y>. In fact, any polynomial of the form A(x,y)x+B(x,y)y vanishes at the origin and, conversely, any polynomial which vanishes at the origin is of the form A(x,y)x+B(x,y)y (suppose f=Si,j aijxiyj, then a00=f(0,0)=0, so f= 0+(Si>0,j aijxi-1yj)x+(Si,j>0 aijxiyj-1)y).
  2. I(Kn) = {0} if K is infinite ("0" stands for the zero polynomial, i.e. the polynomial with all zero coefficients).
  3. If K = Zp with p prime then xp-x vanishes in all points of Zp since ap = a for all a of Zp. Hence xp-x is in I(Zp), which does not result to be {0}.

What about I(V(I))? It is true that I C I(V(I)), but equality not always occurs. For example I(V(<x2,y2>)) = <x,y> which is bigger than <x2,y2> since neither x nor y is in <x2,y2>. If K is a closed field then I(V(I)) is rad(I) (Nullstellensatz), where rad(I) is defined as follows:
Definition 11. Given an ideal I of K[x1,x2,...,xn], we call radical of I, the set

rad(I) = {f of K[ x1,x2,...,xn]: fn is on I for a positive integer n}.
Example.
  1. rad(<x2,y2>) = <x,y>.
Definition 12. An ideal I is said to be a radical ideal if I = rad(I).
Example.
  1. <x,y> is a radical ideal. Hence I(V(<x,y>)) = <x,y>.

We conclude this section with some properties of affine varities and their ideals.
Lemma 13. Let V, W be affine varieties of Kn. Then:
  1. V C W if and only if I(W) C I(V).
  2. V = W if and only if I(V) = I(W).

 

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