Scheme theory begins when algebraic geometry stops treating polynomial zero sets as the whole object and instead lets the entire commutative ring—prime ideals, nilpotents, localisations and all—determine the geometry.
The classical variety route in Affine Varieties and Algebraic Sets, Ideals and Coordinate Rings, Projective Varieties and Morphisms, Singularities and Dimension already showed why local rings and nilpotents matter. Schemes make that local algebra the basic language rather than an afterthought.
A scheme can encode an ordinary smooth curve, a singular surface, a finite set of arithmetic points, an infinitesimal thickening, the prime numbers inside Spec Z, or a family whose fibres change as a parameter changes. The same formalism works across geometry and arithmetic.
Level: advanced undergraduate / beginning graduate enrichment. The main prerequisite is commutative algebra: ideals, prime and maximal ideals, localisation, quotient rings and tensor products.
Reading route: Spec → Zariski topology → distinguished opens → structure sheaf → stalks/local rings → residue fields → nilpotents → morphisms → gluing → fibres and base change → practice → solutions.
1. The spectrum of a ring turns prime ideals into points
For a commutative ring A with identity, define
Spec A={p⊂A : p is a prime ideal}.
This is already a conceptual enlargement. In a classical affine variety over an algebraically closed field, closed points correspond to maximal ideals. Spec A keeps every prime ideal, including nonmaximal primes that behave like generic points of irreducible subspaces.
For A=k[x], the prime ideals include (0) and the irreducible principal ideals. Over algebraically closed k, the maximal ideals are (x−a), one for each a∈k. The point (0) is not an ordinary affine k-point; it is the generic point of the whole affine line.
2. Closed sets in Spec A come from ideals
For an ideal I⊂A define
V(I)={p∈Spec A : I⊆p}.
These V(I) are the closed sets of the Zariski topology on Spec A.
The familiar identities survive: V(I)∪V(J)=V(IJ), arbitrary intersections correspond to sums of ideals, and V(I)=V(√I).
The order reverses: I⊆J implies V(J)⊆V(I).
The topology therefore records containment among primes and the algebraic specialisation structure of the ring.
3. Generic points make irreducible closed sets visible as actual points
If p is a prime ideal, the closure of the point {p} in Spec A is V(p).
Thus p is a generic point of the irreducible closed subset V(p): its closure is the whole subset.
In Spec k[x,y], the prime (x) represents the generic point of the y-axis. The maximal ideals (x,y−a) are closed points lying on that axis.
This is one of scheme theory’s most useful changes of viewpoint. Irreducible subvarieties no longer need to be represented only as collections of points; each has its own generic point.
4. Distinguished opens are the basic computational neighbourhoods
For f∈A define the distinguished open
D(f)=Spec A\V((f))={p : f∉p}.
The D(f) form a basis for the Zariski topology.
Algebraically, D(f) corresponds to the localisation A_f in which powers of f become invertible:
D(f)≈Spec A_f.
This makes localisation geometric. “Move to the region where f is nonzero” and “invert f in the ring” are the same operation viewed from opposite sides.
5. Worked example: D(x) inside the affine line
Let A=k[x]. Then D(x) consists of primes that do not contain x.
Over algebraically closed k, the closed points are (x−a). The point (x) corresponding to a=0 is excluded, while (x−a) for a≠0 remain.
The coordinate ring is k[x]_x=k[x,x⁻¹]. Thus D(x) is the multiplicative line G_m: the affine line with zero removed.
The reciprocal function 1/x is regular on this open because x is invertible there.
6. A scheme needs a structure sheaf, not only a topological space
The prime spectrum by itself records where algebraic phenomena live, but geometry also needs functions on every open set.
The affine scheme associated to A is the pair
(Spec A,O_{Spec A})
where O is the structure sheaf.
On a distinguished open D(f), the structure sheaf satisfies
O(D(f))≈A_f.
Thus the same ring A controls the global space and all its local function rings through localisation.
7. Stalks recover the local ring at a prime
For p∈Spec A, the stalk of the structure sheaf at p is
O_{Spec A,p}≈A_p.
Here A_p localises A by inverting every element outside p.
A_p is a local ring whose unique maximal ideal is pA_p.
The stalk contains germs of functions near p: two functions become equal if they agree on some sufficiently small neighbourhood of p.
This recovers and generalises the local rings used to define singularities in the preceding guide.
8. Residue fields tell us what scalars live at a point
For p∈Spec A define its residue field
κ(p)=Frac(A/p)
more precisely as the fraction field of A/p, equivalently A_p/pA_p.
At a closed k-rational point of an affine k-variety, κ(p)=k.
At the generic point (0) of Spec k[x], the residue field is k(x), the rational function field.
Thus different scheme points can carry different fields. Arithmetic geometry relies heavily on this flexibility.
9. Nilpotents survive in schemes
Consider A=k[ε]/(ε²).
The ring has one prime ideal (ε), so Spec A has only one underlying point.
But A is not isomorphic to k because ε is nonzero and nilpotent.
The corresponding scheme is an infinitesimal thickening of a point, often called the dual-number point.
Classical reduced geometry would replace A by A/√(0)=k and lose ε. Scheme theory keeps the first-order infinitesimal direction.
This is why schemes can encode tangent vectors, multiplicities, infinitesimal deformations and nontransverse intersections more faithfully than reduced point sets.
10. Reduced schemes remove nilpotents
A scheme X is reduced if every local ring O_{X,p} has no nonzero nilpotents.
For an affine scheme Spec A, this is equivalent to A being reduced.
The reduction X_red has the same underlying topological space but replaces the structure sheaf by its quotient modulo nilpotents.
For Spec k[ε]/(ε²), the reduction is Spec k.
Reduction therefore preserves locations but deletes infinitesimal thickness.
11. Spec Z turns arithmetic into geometry
The prime ideals of Z are (0) and (p) for prime numbers p.
Thus Spec Z has one generic point (0) and one closed point for each prime number.
The residue field at (p) is F_p. The residue field at (0) is Q.
Geometrically, the “generic fibre” lives over Q while each prime p provides a special fibre over F_p.
This analogy is not merely poetic. Arithmetic schemes allow equations over Z to be studied simultaneously over Q and modulo every prime.
12. Closed immersions are quotient rings
For an ideal I⊂A, the quotient map A→A/I induces a morphism
Spec(A/I)→Spec A.
Its image is V(I), and it is a closed immersion.
If I is radical, the resulting closed subscheme is reduced. If I is not radical, the same underlying closed set carries additional nilpotent structure.
Example: Spec k[x]/(x²) and Spec k[x]/(x) have the same underlying point inside A¹, but the former is a doubled point.
13. Morphisms of affine schemes reverse ring homomorphisms
A ring homomorphism φ:A→B induces a scheme morphism
Spec B→Spec A
by sending a prime q⊂B to its inverse image φ⁻¹(q), which is prime in A.
This is the scheme-theoretic continuation of the coordinate-ring contravariance of affine varieties.
The map on structure sheaves is also part of the data. A scheme morphism is not just a continuous map of prime spectra.
14. Worked morphism: projection from an affine plane
The inclusion k[t]→k[t,x] induces
Spec k[t,x]→Spec k[t].
On classical k-points, this is the projection (t,x)↦t.
But the scheme morphism also knows how nonclosed primes and residue fields map.
The fibre over a point p∈Spec k[t] is constructed by tensor product with κ(p), not merely by informal substitution.
15. Fibres are base changes to residue fields
For f:X→Y and y∈Y, the scheme-theoretic fibre is
X_y=X×_Y Spec κ(y).
For affine schemes Spec B→Spec A and p∈Spec A,
X_p≈Spec(B⊗_A κ(p)).
This formula automatically handles residue-field extensions and nilpotent structure.
It is a central reason tensor products appear everywhere in algebraic geometry: they compute geometric base change.
16. Worked arithmetic family: x²−t over the parameter line
Let B=k[t,x]/(x²−t), viewed as a k[t]-algebra. This gives a morphism X=Spec B→A¹_t.
Over t=a, the fibre is Spec k[x]/(x²−a).
For generic a over an algebraically closed field of characteristic not two, the fibre has two reduced points when a≠0.
At a=0, the fibre is Spec k[x]/(x²), one point with nilpotent thickness.
Scheme fibres record that the two points collide into a nonreduced double point rather than simply “becoming one point”.
17. Fibre products are the universal way to impose compatibility
Given X→S and Y→S, the fibre product X×_S Y represents pairs of points or maps that agree over S.
Affinely, if X=Spec B, Y=Spec C and S=Spec A, then
X×_S Y≈Spec(B⊗_A C).
This single construction produces intersections, fibres, extension of scalars and many parameter changes.
For example, base-extending an R-variety to C corresponds to tensoring its coordinate ring with C over R.
18. Schemes are glued from affine schemes
A scheme is a locally ringed space covered by open subsets each isomorphic to Spec A_i for some commutative ring A_i.
This is analogous to a manifold being covered by coordinate charts, but the local models are affine schemes rather than Euclidean open sets.
The overlaps carry compatible ringed-space isomorphisms satisfying cocycle conditions.
Projective space is the standard example. Pⁿ is covered by n+1 affine opens U_i≈Aⁿ whose coordinate rings are glued by localisation relations.
Gluing is why local algebra can build genuinely global spaces.
19. Worked gluing: projective line from two affine lines
Take U=Spec k[t] and V=Spec k[s].
Remove t=0 from U and s=0 from V, obtaining D(t)=Spec k[t,t⁻¹] and D(s)=Spec k[s,s⁻¹].
Identify the overlaps by s=t⁻¹.
The resulting glued scheme is P¹_k.
The point missing from the t-chart is the projective point at infinity, visible in the s-chart as s=0.
20. Local properties are checked on affine neighbourhoods
Many scheme properties are local in the Zariski topology: reduced, locally Noetherian, regular, normal and dimension questions can be checked on suitable affine opens or local rings.
This is operationally important. A global scheme may be complicated, but near a point p one can choose an affine neighbourhood Spec A and work with A_p.
The local-to-global strategy is not merely pedagogical. It is built into the definitions.
21. Noetherian schemes impose finite algebraic control
A scheme is locally Noetherian if it can be covered by spectra of Noetherian rings. It is Noetherian if it is quasi-compact and locally Noetherian.
Finite-type schemes over a field are Noetherian because polynomial rings over fields are Noetherian.
Noetherianity ensures ascending chains of ideals stabilise and descending chains of closed subsets stabilise.
Many dimension, decomposition and coherence theorems rely on this finiteness condition.
22. Regular, normal and reduced remain distinct
A reduced scheme has no nonzero nilpotents.
A normal integral scheme has integrally closed local domains.
A regular scheme has regular local rings.
Regular implies normal under standard Noetherian hypotheses, and normal implies reduced for integral schemes, but converses fail.
Keeping these properties separate prevents a common collapse of different local algebraic defects into one vague notion of “niceness”.
23. Dimension is prime-chain depth
The Krull dimension of Spec A is the supremum of lengths of chains
p₀⊊p₁⊊⋯⊊p_n
of prime ideals.
For Spec k[x,y], the chain (0)⊊(x)⊊(x,y) has length 2.
For Spec Z, (0)⊊(p) has length 1, so Spec Z has dimension 1.
This is why arithmetic geometers sometimes speak of Spec Z as behaving like a one-dimensional arithmetic curve.
24. Scheme-theoretic intersections keep multiplicity
In A², intersect the x-axis y=0 with the parabola y=x².
The combined ideal is (y,y−x²)=(y,x²).
The intersection scheme is Spec k[x,y]/(y,x²)≈Spec k[x]/(x²).
Its underlying set is only the origin, but the nilpotent x records intersection multiplicity two.
This is a concrete payoff of nonreduced scheme structure: tangency becomes algebraically visible.
25. Base change changes the field without rebuilding the theory
Suppose X is a scheme over Spec k and K/k is a field extension.
The base change
X_K=X×_{Spec k}Spec K
is the same geometric construction viewed over K.
Affinely, A becomes A⊗_k K.
Irreducibility, point counts and factorisation may change after base extension. This distinguishes properties over the base field from geometric properties after passing to an algebraic closure.
26. A dependable scheme workflow
Start with the ring A. List relevant prime ideals or the ideal relations that define the region of interest.
Use V(I) for closed subsets and D(f) for basic opens. Translate D(f) into A_f.
At a point p, localise to A_p and compute the residue field κ(p).
Do not discard nilpotents unless reduction is explicitly intended.
For a ring map A→B, reverse direction to Spec B→Spec A. Compute fibres using B⊗_Aκ(p).
For a global scheme, reduce questions to affine opens when the property is local, then check compatibility on overlaps.
27. Common misconceptions
Spec A is just the set of maximal ideals. No. Every prime ideal is a scheme point.
A scheme is just a topological space of primes. No. The structure sheaf is essential.
Nilpotents are algebraic noise. They can encode multiplicity and infinitesimal geometry.
Fibres are found only by substituting coordinate values. Scheme-theoretic fibres are base changes to residue fields and can retain nonreduced structure.
All points have the base field as residue field. Generic and arithmetic points often carry larger or different fields.
28. Independent practice: twenty questions
- List the prime ideals of Z.
- What is the closure of p∈Spec A?
- Describe D(f).
- What ring corresponds to D(f)?
- What is the local ring at p?
- What is the residue field at (0) in Spec k[x]?
- What is the residue field at (x−a)?
- How many underlying points does Spec k[ε]/(ε²) have?
- Why is that scheme not reduced?
- Describe Spec(A/I)→Spec A.
- What scheme map comes from A→B?
- For Spec k[t,x]→Spec k[t], what is the fibre over t=a?
- For k[t,x]/(x²−t), what happens to the fibre at t=0?
- State the affine fibre-product formula.
- How is P¹ glued from two affine lines?
- Give a prime chain showing dim Spec k[x,y]≥2.
- Why is dim Spec Z=1?
- Compute the scheme-theoretic intersection of y=0 and y=x².
- What does base extension k→K do to an affine coordinate ring?
- Distinguish reduced, normal and regular.
29. Worked solutions and checks
1. (0) and (p) for every prime integer p.
2. V(p).
3. D(f)={p:f∉p}=Spec A\V(f).
4. A_f, so D(f)≈Spec A_f.
5. A_p.
6. k(x), the fraction field of k[x].
7. k, when a∈k.
8. One underlying point.
9. ε is nonzero but ε²=0.
10. It is the closed immersion whose image is V(I), retaining any nilpotent structure from A/I.
11. Spec B→Spec A, q↦φ⁻¹(q).
12. Spec(k[t,x]⊗_{k[t]}k)≈Spec k[x] after evaluating t at a through the residue field.
13. The fibre is Spec k[x]/(x²), a nonreduced double point.
14. Spec B×_{Spec A}Spec C≈Spec(B⊗_A C).
15. Glue Spec k[t] and Spec k[s] along D(t) and D(s) via s=t⁻¹.
16. (0)⊊(x)⊊(x,y).
17. The longest prime chain has the form (0)⊊(p), of length one.
18. The combined ideal is (y,x²), giving Spec k[x]/(x²), a doubled point.
19. A becomes A⊗_k K.
20. Reduced means no nonzero nilpotents; normal concerns integral closure of local domains; regular means local rings have minimal embedding dimension equal to Krull dimension.
30. Where this guide hands off
The next R24 cell develops sheaves and cohomology: how compatible local algebraic data glue globally, why some local solutions fail to globalise, and how derived invariants measure those failures.
After that, arithmetic geometry uses schemes over Z and number fields to organise rational points, reduction modulo primes and elliptic curves, while the final cell turns toward Gröbner bases, elimination and enumerative computation.
Sources and further study
The Stacks Project is the principal open reference for schemes, spectra, localisation, fibre products, morphisms and sheaves. See also MIT 18.726 Algebraic Geometry for graduate-level scheme foundations and the earlier BTT R24 guides for the classical variety route.