Affine algebraic geometry becomes powerful when a geometric zero set is replaced by the ring of polynomial functions that live on it.
The preceding guide, Affine Varieties and Algebraic Sets, began with polynomial equations and their zero loci. This guide reverses direction: starting from a geometric set X, we ask which polynomial functions are genuinely different on X, what algebraic information records irreducibility and points, and why geometric maps naturally reverse into ring homomorphisms.
The central object is the coordinate ring. If X⊂Aⁿ is affine algebraic, then functions that differ by a polynomial vanishing on X should represent the same function on X. Quotienting by I(X) performs exactly that identification.
Level: advanced undergraduate enrichment. We use commutative rings with identity and usually work over an algebraically closed field k when invoking the classical Nullstellensatz.
Reading route: ideals → radicals → prime/maximal ideals → coordinate rings → Nullstellensatz → points and functions → regular maps → localisation → practice → solutions.
1. Ideals record polynomial consequences
Let R=k[x₁,…,xₙ]. An ideal I⊂R is closed under addition and under multiplication by arbitrary ring elements.
If I=(f₁,…,f_r), then every element of I has the form h₁f₁+⋯+h_rf_r. Any point where all fᵢ vanish also makes every polynomial in I vanish.
Thus a finite equation list is only a generating interface. The ideal contains all algebraic consequences obtained by polynomial combination.
For example, (x,y) contains x², xy, y³ and x+y, but not the constant polynomial 1. Geometrically V(x,y) is the origin in A².
2. The unit ideal corresponds to the empty zero set
If I=R, then 1∈I. Since 1 vanishes nowhere, V(I)=∅.
Conversely, over an algebraically closed field, the weak Nullstellensatz implies that a proper ideal in k[x₁,…,xₙ] has a common zero. Therefore V(I)=∅ implies I=R.
This equivalence can fail over a non-algebraically-closed field if interpreted using only rational points. For instance, (x²+1) is proper in R[x] but has no real zero.
The field hypothesis is therefore not decoration. It determines whether algebraic consistency guarantees a point in the chosen affine space.
3. Radical ideals forget multiplicity at the level of point sets
The radical of an ideal I is
√I={f∈R : fᵐ∈I for some m≥1}.
An ideal is radical when √I=I.
Geometrically, V(I)=V(√I). If fᵐ vanishes at a point, then f vanishes there because fields have no nonzero nilpotents.
For example, V(x²)=V(x). The ideals (x²) and (x) are different, but their point sets are identical. The radical operation removes the multiplicity information invisible to classical reduced point-set geometry.
This is precisely one reason schemes later keep nonradical ideals rather than replacing everything by its zero set.
4. Prime ideals correspond to irreducible geometry
An ideal P is prime when P≠R and fg∈P implies f∈P or g∈P.
Equivalently, R/P is an integral domain.
Over an algebraically closed field, an affine algebraic set X is irreducible exactly when I(X) is prime.
The example X=V(y−x²) has coordinate ring k[x,y]/(y−x²)≈k[x], an integral domain, so the parabola is irreducible.
By contrast, V(xy) has coordinate ring k[x,y]/(xy), which has zero divisors: the classes of x and y are nonzero but their product is zero. This algebraically detects the union of two components.
5. Maximal ideals correspond to points in classical affine geometry
An ideal m is maximal when m≠R and no proper ideal lies strictly between m and R. Equivalently R/m is a field.
For a point a=(a₁,…,aₙ)∈Aⁿ(k), the ideal
m_a=(x₁−a₁,…,xₙ−aₙ)
is maximal because evaluation at a gives a surjective homomorphism R→k with kernel m_a.
The weak Nullstellensatz says that when k is algebraically closed, every maximal ideal of k[x₁,…,xₙ] has this form.
Thus ordinary affine points become maximal ideals. Modern scheme theory enlarges the space by including all prime ideals, not just maximal ones.
6. The coordinate ring identifies polynomials that agree on X
For an affine algebraic set X⊂Aⁿ, define its coordinate ring
k[X]=k[x₁,…,xₙ]/I(X).
Two polynomials f and g define the same polynomial function on X exactly when f−g∈I(X). Their classes in the quotient are then equal.
For the parabola X=V(y−x²), the class of y equals the class of x². Every polynomial function can therefore be reduced to a polynomial in x alone, producing k[X]≈k[x].
For the hyperbola X=V(xy−1), the coordinate relation xy=1 makes y the inverse of x. Hence k[X]≈k[x,x⁻¹], the Laurent polynomial ring.
Harvard’s undergraduate algebraic-geometry notes state the affine coordinate ring as k[z₁,…,zₙ]/I(X) and emphasise that it determines an affine variety up to isomorphism in the classical setting: Harvard Math 137 notes.
7. Coordinate rings are intrinsic even when embeddings differ
The affine line A¹ and the parabola V(y−x²) sit in different ambient affine spaces, but their coordinate rings are isomorphic.
This ring isomorphism is not a coincidence. Affine varieties are isomorphic exactly when their coordinate rings are isomorphic as k-algebras.
The direction reverses: an isomorphism φ:X→Y produces a pullback φ*:k[Y]→k[X].
Thus the coordinate ring captures intrinsic affine algebraic structure rather than merely the equations of one embedding.
Different embeddings can still matter for degree, projective closure, tangent geometry and elimination problems. Intrinsic isomorphism does not erase all useful ambient information.
8. Hilbert’s basis theorem makes polynomial geometry Noetherian
A ring is Noetherian if every ideal is finitely generated. Hilbert’s basis theorem says that if R is Noetherian, then R[x] is Noetherian.
Since a field k is Noetherian, k[x₁,…,xₙ] is Noetherian.
Consequently every ideal of the polynomial ring has a finite generating set. Therefore every affine algebraic set can be cut out by finitely many equations, even if originally defined by an infinite family.
Noetherianity also implies descending chains of algebraic closed sets stabilise, because ideal inclusions reverse closed-set inclusions.
This finiteness is structural, not computational efficiency. A finite generating set can still be extremely large or difficult to find.
9. The strong Nullstellensatz closes the V–I dictionary
For an ideal I⊂k[x₁,…,xₙ] with k algebraically closed, Hilbert’s Nullstellensatz states
I(V(I))=√I.
Thus if we start from equations, take their common zero set, then collect every polynomial vanishing on that zero set, we recover exactly the radical of the original ideal.
In particular, radical ideals correspond bijectively to affine algebraic sets through I↔V.
This is the classical algebra–geometry dictionary: geometric information is encoded not by arbitrary ideals but by radical ideals when only reduced point sets are being considered.
MIT’s algebraic-geometry notes place the Nullstellensatz at the centre of affine algebraic geometry before spectra, localisation and morphisms: MIT 18.721 notes.
10. Worked Nullstellensatz example: (x²,y) and the origin
Let I=(x²,y)⊂k[x,y], with k algebraically closed.
The common zero set satisfies y=0 and x²=0, hence x=0. Therefore V(I)={(0,0)}.
The vanishing ideal of the origin is (x,y). Thus
I(V(I))=(x,y).
Also √(x²,y)=(x,y), because x²∈I implies x∈√I and y∈I.
The Nullstellensatz matches the direct calculation. The original ideal remembers a doubled x-direction that the reduced point set does not.
11. Radical, prime and maximal form a hierarchy
Every maximal ideal is prime. Every prime ideal is radical. The converses fail in general.
Example: (xy) in k[x,y] is radical when x and y are distinct irreducible factors, but it is not prime because xy∈(xy) while neither x nor y belongs to (xy).
Geometrically V(xy) is reduced but reducible.
The ideal (x) is prime but not maximal in k[x,y], because k[x,y]/(x)≈k[y], which is an integral domain but not a field.
Geometrically V(x) is the whole y-axis: irreducible but not a single point.
12. Functions on X are quotient classes, not raw formulas
On X=V(y−x²), the expressions y, x² and y+(y−x²)(x+1) all define the same function.
The coordinate ring handles this automatically. Each expression represents the same coset modulo I(X).
This prevents a common mistake: comparing formulas syntactically rather than comparing their restrictions to the variety.
A coordinate ring is therefore a compression device. It forgets differences that are invisible on X while preserving every polynomial function that can be observed on X.
13. Regular functions can include fractions locally
On an affine variety, a regular function is locally expressible as g/h where g and h are polynomial functions and h does not vanish on the neighbourhood under consideration.
A central theorem says every globally regular function on an affine variety is represented by an element of its coordinate ring.
Thus local rational expressions can glue to a global polynomial-class function even when no one local expression is convenient everywhere.
On the hyperbola xy=1, the function 1/x is globally regular because x never vanishes and 1/x=y in the coordinate ring.
By contrast, 1/x is not a globally regular function on A¹ because x vanishes at the origin.
14. Principal open sets correspond to localisation
For f∈k[X], the principal open set is
D(f)={p∈X : f(p)≠0}.
On D(f), powers of f are allowed in denominators. Algebraically this is the localisation k[X]_f.
For X=A¹ and f=x, D(x)=A¹\{0}. Its regular functions include x⁻¹, x⁻² and polynomial combinations of positive and negative powers: k[x,x⁻¹].
Localisation is the algebraic mechanism for focusing on a region where selected functions are invertible.
This operation becomes foundational in scheme theory, where local rings are obtained by localising at prime ideals.
15. Regular maps reverse to k-algebra homomorphisms
Let φ:X→Y be a regular map of affine varieties. Every regular function f on Y can be pulled back to f∘φ on X.
This defines
φ*:k[Y]→k[X], f↦f∘φ.
The direction reverses: geometry X→Y becomes algebra k[Y]→k[X].
Composition reverses accordingly: if X→Y→Z, then (ψ∘φ)*=φ*∘ψ*.
This contravariance is not a technical annoyance; it is the organising bridge of affine algebraic geometry.
16. Worked map example: t↦(t,t²)
Let X=V(y−x²) and φ:A¹→X be φ(t)=(t,t²).
The pullback sends the coordinate classes x↦t and y↦t².
Since y−x²=0 in k[X], the assignment respects the relation. Therefore it defines a k-algebra homomorphism
k[x,y]/(y−x²)→k[t].
This homomorphism is an isomorphism, mirroring the geometric isomorphism between the parabola and affine line.
Checking a candidate regular map algebraically often means checking that the defining ideal of the target maps to zero.
17. Affine geometry and finitely generated reduced k-algebras form opposite categories
Classically, affine algebraic sets over an algebraically closed field correspond to finitely generated reduced k-algebras.
Irreducible affine varieties correspond to finitely generated integral-domain k-algebras.
The word “opposite” appears because maps reverse. A geometric morphism X→Y corresponds to a k-algebra homomorphism k[Y]→k[X].
This equivalence allows many geometric questions to be solved algebraically. Products become tensor products, closed subvarieties become quotient rings, open localisation becomes ring localisation, and points become maximal ideals.
Modern algebraic geometry extends this correspondence from reduced finitely generated algebras to arbitrary commutative rings through the spectrum construction.
18. Closed subvarieties correspond to quotient rings
Let X be affine with coordinate ring A=k[X], and let Y⊂X be a closed algebraic subset.
The functions on X that vanish on Y form an ideal I_X(Y)⊂A. The coordinate ring of Y is
k[Y]≈A/I_X(Y).
Thus imposing additional polynomial equations geometrically corresponds to quotienting the coordinate ring algebraically.
For the y-axis Y=V(x) inside A², the quotient k[x,y]/(x)≈k[y].
The quotient removes functions that vanish identically on the subspace and simplifies every remaining function using the new relation x=0.
19. Products correspond to tensor products
For affine varieties X and Y over k,
k[X×Y]≈k[X]⊗_k k[Y].
For X=A¹ and Y=A¹, k[x]⊗_k k[y]≈k[x,y], matching A¹×A¹=A².
If X is the hyperbola xy=1 with coordinate ring k[t,t⁻¹] and Y=A¹ with coordinate ring k[s], then the product has ring k[t,t⁻¹,s].
Again the algebraic construction mirrors the geometric one, but in reversed categorical orientation for maps.
20. Dimension becomes Krull dimension
The Krull dimension of a commutative ring A is the supremum of lengths of strict chains of prime ideals
p₀⊊p₁⊊⋯⊊p_d.
For an affine variety X, geometric dimension equals the Krull dimension of k[X].
In k[x,y], the chain (0)⊊(x)⊊(x,y) has length 2, reflecting dim A²=2.
Prime ideals therefore represent irreducible closed subspaces at different scales: the zero ideal corresponds to all A², (x) to the y-axis, and (x,y) to the origin.
This is one conceptual reason scheme theory treats every prime ideal as a point: the prime spectrum records all irreducible algebraic locations, not only classical closed points.
21. Nilpotents carry infinitesimal information invisible to reduced varieties
Consider A=k[ε]/(ε²). The element ε is nonzero but ε²=0.
If one takes only classical k-points, this algebra looks like a single point because every k-algebra map A→k must send ε to 0.
Yet A is not the same ring as k. The nilpotent element stores an infinitesimal thickening direction.
Classical varieties deliberately discard this information by requiring reduced coordinate rings. Schemes retain it.
This is the bridge to R24.05: scheme theory does not merely add abstraction. It preserves algebraic information that ordinary zero sets erase.
22. A dependable algebra–geometry workflow
When given equations, form the ideal they generate and ask whether it is radical or prime. Do not infer geometry from a generator list alone.
When given a variety X, compute or describe I(X) and then k[X]. Use quotient relations aggressively to simplify functions.
When given a point, identify its maximal ideal. When given an irreducible closed subset, identify the corresponding prime ideal.
When given a regular map, pull back coordinate functions and verify the target relations vanish after substitution.
When restricting to D(f), localise by f. When imposing new closed equations, quotient by the corresponding ideal.
Always state whether the base field is algebraically closed before invoking the classical point-maximal-ideal form of the Nullstellensatz.
23. Common misconceptions
Ideal equals zero set. No. Different ideals can define the same point set; radicalisation is the relevant reduced correction.
Prime means one point. No. Prime ideals correspond to irreducible closed sets; maximal ideals correspond to classical points over an algebraically closed field.
Coordinate rings depend on the ambient embedding. Their presentations do, but the ring up to k-algebra isomorphism is intrinsic to the affine variety.
Every fraction is a regular function. Only where its denominator is invertible. Localisation records the allowed denominators.
Zero nilpotents are harmless notation. Nonzero nilpotents carry infinitesimal scheme structure that reduced varieties intentionally forget.
24. Independent practice: eighteen questions
- Compute √(x²) in k[x].
- Compute V(x²,y) in A².
- Find I(V(x²,y)) over an algebraically closed field.
- Why is (x) prime in k[x,y]?
- Why is (x) not maximal in k[x,y]?
- Why is (x,y) maximal?
- Show k[x,y]/(y−x²)≈k[x].
- Show k[x,y]/(xy−1)≈k[x,x⁻¹].
- Is (xy) prime? Is it radical?
- State the strong Nullstellensatz.
- Why does the weak Nullstellensatz fail over R if one asks only for real zeros?
- What ring represents regular functions on D(x)⊂A¹?
- For φ(t)=(t,t²), what are φ*(x) and φ*(y)?
- What geometric operation corresponds to quotienting k[X] by an ideal?
- What geometric operation corresponds to localising at f?
- Give a prime-ideal chain of length two in k[x,y].
- Why does k[ε]/(ε²) have more structure than a reduced point?
- Explain why maps reverse between affine varieties and coordinate rings.
25. Worked solutions and checks
1. √(x²)=(x), since fᵐ∈(x²) exactly forces sufficient divisibility by x, and x itself has square in the ideal.
2. x²=0 and y=0 imply x=y=0, so the zero set is the origin.
3. By direct evaluation or Nullstellensatz, the vanishing ideal of the origin is (x,y).
4. The quotient k[x,y]/(x)≈k[y] is an integral domain, so (x) is prime.
5. The quotient k[y] is not a field, so (x) is not maximal.
6. k[x,y]/(x,y)≈k, a field.
7. Send the class of x to x and y to x². The relation y−x² becomes zero. Every class reduces uniquely to a polynomial in x.
8. The relation xy=1 makes y=x⁻¹. The quotient is generated by x and its inverse, giving the Laurent polynomial ring.
9. It is not prime because xy belongs to the ideal but neither x nor y does. It is radical because xy is square-free and (xy)=(x)∩(y) in k[x,y].
10. Over an algebraically closed field, I(V(I))=√I for every ideal I in k[x₁,…,xₙ].
11. The proper ideal (x²+1)⊂R[x] has no real zero, so proper ideal does not guarantee a point in A¹(R).
12. k[x]_x=k[x,x⁻¹].
13. φ*(x)=t and φ*(y)=t².
14. Passing to a closed algebraic subset cut out by those additional equations.
15. Restricting to the principal open set D(f), where f does not vanish.
16. (0)⊊(x)⊊(x,y).
17. Its element ε is nonzero but nilpotent. Classical points cannot detect that infinitesimal direction, while the ring can.
18. A function on the target composes with a geometric map X→Y to produce a function on X, so pullback sends k[Y] to k[X].
26. Where this guide hands off
R24.03 introduces projective space, homogeneous coordinates and homogeneous ideals. Projective geometry lets algebraic curves and higher-dimensional varieties include their points at infinity in one global algebraic space.
R24.04 then studies morphisms, tangent spaces, singular points and dimension. R24.05 will retain nilpotent and local-ring information through schemes.
Sources and further study
See Harvard Math 137 undergraduate algebraic geometry notes for coordinate rings, regular functions and affine maps, and MIT 18.721 Algebraic Geometry notes for rings, Zariski topology, Nullstellensatz, spectrum, localisation and affine morphisms.