Scheme theory begins by refusing to throw away algebraic information merely because ordinary point sets cannot see it.
The first four R24 guides built classical algebraic geometry from polynomial zero sets, ideals, coordinate rings, projective closure, morphisms, tangent spaces and singularities. They also exposed a recurring limitation: different rings can have the same classical point set. The ideals (x) and (x²) cut out the same reduced zero set in affine space, yet the rings k[x]/(x) and k[x]/(x²) are not the same algebraic object.
Schemes keep that difference. They replace “a variety is a set of points satisfying equations” with a richer object built from prime ideals and a sheaf of local rings. Nilpotents, generic points, residue fields and localisations become part of the geometry rather than discarded bookkeeping.
This guide continues directly from Ideals and Coordinate Rings and Morphisms, Singularities and Dimension. The goal is not to compress an entire graduate course into one page. It is to make the scheme-theoretic change of viewpoint usable through explicit examples.
Level: advanced undergraduate to beginning graduate enrichment. Prerequisites are commutative rings, ideals, prime and maximal ideals, localisation, affine varieties and basic topology.
Reading route: Spec and prime points → Zariski topology → structure sheaf → stalks and local rings → nilpotents → scheme morphisms → gluing → fibres and base change → scheme-theoretic intersections → practice → solutions.
1. Spec A turns every prime ideal into a point
Let A be a commutative ring with identity. Its prime spectrum is
Spec A={p⊂A : p is a prime ideal}.
This is the first major departure from classical affine geometry. Over an algebraically closed field k, maximal ideals of k[x₁,…,xₙ] correspond to ordinary k-points. Spec includes those maximal ideals but also every nonmaximal prime ideal.
Those extra points are not decorative. They encode irreducible subspaces and how points specialise. In k[x,y], the prime ideal (x) represents the whole irreducible y-axis as a generic algebraic location, while the maximal ideal (x,y−a) represents one closed point on that axis.
The Stacks Project defines affine schemes by equipping this spectrum with a sheaf of rings; its current scheme chapter gives the complete formal route from locally ringed spaces to affine schemes and gluing: Stacks Project, Schemes.
2. Prime points remember irreducible algebraic directions
A prime ideal p has the property that A/p is an integral domain. Geometrically, this matches irreducibility: a product fg can vanish generically on the corresponding irreducible piece only when one factor vanishes generically there.
For A=k[x,y]/(xy), the spectrum has two minimal prime ideals, represented by (x) and (y) in the quotient. They correspond to the two coordinate-axis components.
The maximal ideal (x,y) lies on both components. Scheme topology therefore stores containment data among irreducible pieces and closed points in one space.
For a domain A, the zero ideal (0) is prime. Its point in Spec A is the generic point of the irreducible affine scheme: its closure is the whole space.
3. Closed sets are V(I), now inside the prime spectrum
For an ideal I⊂A, define
V(I)={p∈Spec A : I⊆p}.
These sets are the Zariski-closed subsets of Spec A.
The formulas mirror classical affine geometry: V(0)=Spec A, V(A)=∅, V(I)∪V(J)=V(IJ)=V(I∩J), and intersections correspond to sums of ideals through ∩V(Iα)=V(ΣIα).
Only the radical matters topologically: V(I)=V(√I). Thus Spec A and Spec(A/√0) have the same underlying topological space even though their structure sheaves can be different.
This separation is crucial: the topological space sees which primes exist and how they specialise; the sheaf remembers additional ring structure, including nilpotents.
4. Standard opens D(f) are the basic local windows
For f∈A, define the principal or standard open
D(f)=Spec A\V(f)={p : f∉p}.
The D(f) form a basis for the Zariski topology. Their intersections satisfy D(f)∩D(g)=D(fg).
The algebraic meaning is localisation: on D(f), the function f is nowhere contained in the relevant prime ideals and is therefore made invertible.
The resulting affine open is naturally
D(f)≈Spec A_f.
This relation between standard opens and localised rings is one of the most-used interfaces in scheme calculations. See the Stacks Project’s Affine schemes section.
5. Worked spectrum: Spec k[x] has a generic point as well as closed points
Let k be algebraically closed and A=k[x]. The prime ideals are (0) and the maximal ideals (x−a) for a∈k.
Each (x−a) is a closed point because V(x−a) contains only that maximal ideal.
The point (0) is not closed. Its closure is V(0)=Spec k[x]. It is the generic point.
A standard open D(x) contains the generic point and every closed point except (x), corresponding to the affine line with the origin removed.
Thus the scheme-theoretic affine line is not merely the set k. It has one additional generic point whose specialisations are all the closed points.
6. Worked spectrum: Spec Z turns primes into geometric points
The prime ideals of Z are (0) and (p) for each ordinary prime number p.
Therefore Spec Z consists of a generic point (0) and one closed point for each prime integer.
The closed set V(n) consists of prime ideals containing n. If n≠0, these are exactly the points (p) where p divides n.
For example, V(12)={(2),(3)}. Its complement D(12) contains the generic point and every prime except 2 and 3.
This picture is one reason arithmetic geometry can treat number rings as geometric spaces. Prime numbers behave like closed points on an arithmetic curve, while the generic point records characteristic-zero information.
7. Specialisation reverses ideal containment
For p,q∈Spec A, the point q lies in the closure of {p} exactly when p⊆q.
Thus the smaller prime ideal is more generic and the larger prime ideal is more specialised.
In Spec k[x,y], (0) specialises to (x), and (x) specialises further to (x,y−a). This corresponds to the chain
whole affine plane → y-axis → a point on the y-axis.
Krull dimension measures the maximum length of such prime chains. The topological specialisation order and the algebraic inclusion order are therefore two views of the same structure.
8. A topological spectrum alone is not yet an affine scheme
An affine scheme is not merely Spec A as a set with topology. It is the pair
(Spec A, O_{Spec A}),
where O is the structure sheaf.
The structure sheaf assigns a ring of regular functions to each open set, together with restriction maps between nested opens. It is designed so that standard opens recover localisations:
O(D(f))≈A_f.
This local ring-valued data is what distinguishes schemes having the same underlying topological space but different algebraic thickness.
The formal construction and its local properties are developed in the Stacks Project’s Affine schemes section.
9. Why localisation is the correct ring on D(f)
Suppose a function f does not vanish on the open region D(f). Then powers of f should be permitted as denominators there.
The localisation A_f contains fractions a/fⁿ. Two such fractions are identified when they agree after multiplying by a sufficiently high power of f.
Example: on D(x)⊂Spec k[x], the function 1/x is regular. Algebraically the section ring is k[x]_x=k[x,x⁻¹].
On all of Spec k[x], 1/x is not regular because x is not invertible at the prime (x).
Regularity is therefore local and denominator-sensitive. The sheaf formalism records exactly which fractions are allowed on which open sets.
10. Sheaf gluing means compatible local functions determine one function
A sheaf satisfies two key principles. First, if two sections agree on an open cover, then they are the same section. Second, compatible sections on an open cover glue uniquely to a section on the union.
This is the algebraic version of a familiar geometric idea: a global function is determined by local formulas that agree wherever their domains overlap.
For schemes, the sheaf is not optional packaging. It allows a space to be built from affine pieces whose rings differ from chart to chart while preserving a coherent notion of local regular function.
R24.06 will develop sheaves as mathematical objects in their own right and explain why their cohomology measures failures of global gluing.
11. The stalk O_{X,p} records germs near one point
Given a scheme X and point p, the stalk O_{X,p} is formed from sections defined on neighbourhoods of p, identifying two sections when they agree on some smaller neighbourhood.
For an affine scheme X=Spec A and prime p⊂A,
O_{X,p}≈A_p.
The localisation A_p inverts every element outside p. It keeps precisely the algebraic information that survives arbitrarily close to the point p.
The ring A_p is local: it has one maximal ideal pA_p. This is why schemes are locally ringed spaces rather than merely ringed spaces.
12. Residue fields say what scalars live at a point
For p∈Spec A, define the residue field
κ(p)=A_p/pA_p.
This is a field because pA_p is the unique maximal ideal of A_p.
It is naturally isomorphic to the fraction field of A/p.
At a classical k-rational point a of affine space over an algebraically closed field, κ(a)=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.
13. Local dimension is the dimension of the local ring
For a Noetherian scheme X and point p, the local dimension is dim O_{X,p}, the Krull dimension of the stalk.
Prime chains in A_p correspond to prime chains of A contained in p. Localisation filters out components and closed sets that do not pass through p.
For A=k[x,y] and p=(x,y), the local ring A_p has dimension 2. For p=(x), the local ring has dimension 1 because only prime chains ending inside (x) remain.
This local perspective was already used in the preceding singularity guide. Scheme theory makes it part of the basic language rather than an auxiliary construction.
14. Regular local rings identify smooth algebraic behaviour
For a Noetherian local ring (A,m) with residue field κ, the embedding dimension is dim_κ(m/m²).
The ring is regular when
dim A=dim_κ(m/m²).
For finite-type varieties over a perfect field, regularity is closely tied to geometric smoothness, though scheme-theoretic smoothness is a relative notion and needs the base morphism to be specified.
The distinction matters especially in positive characteristic, where a scheme can be regular while a morphism to the base field fails to be smooth after field extension.
This guide uses regularity as a local-ring property and leaves the full relative smoothness theory to a later route.
15. Nilpotents are invisible to points but visible to schemes
An element a∈A is nilpotent if aⁿ=0 for some n>0. The nilradical √(0) is the ideal of all nilpotent elements.
Every prime ideal contains every nilpotent. Therefore Spec A and Spec(A/√0) have exactly the same prime ideals as a topological space.
But their structure sheaves differ. The original scheme can contain infinitesimal thickness that disappears after quotienting by the nilradical.
The reduced scheme X_red has the same underlying topological space as X but removes all nilpotents from the structure sheaf.
Classical algebraic varieties are usually reduced. Scheme theory deliberately allows nonreduced objects because multiplicity and infinitesimal deformation information can be geometrically meaningful.
16. Worked example: the dual-number point is thicker than an ordinary point
Let A=k[ε]/(ε²).
The only prime ideal is (ε), because A/(ε)≈k. Thus Spec A has one topological point.
Yet ε is nonzero in A and satisfies ε²=0. Therefore A is not isomorphic to k.
As a scheme, Spec k[ε]/(ε²) is a first-order infinitesimal thickening of a point.
This object is called the spectrum of the dual numbers. It is used to probe tangent vectors and first-order deformations.
The example demonstrates exactly why point sets are insufficient: one point can carry nontrivial infinitesimal algebra.
17. Tangent vectors can be read as maps from the dual numbers
Let X be a scheme over k and x a k-rational point. A tangent vector at x can be described by a k-morphism
Spec k[ε]/(ε²)→X
whose restriction to the reduced point Spec k is x.
For X=A¹=Spec k[t], such a morphism corresponds contravariantly to a k-algebra map k[t]→k[ε]/(ε²).
If the underlying point is a∈k, then t must map to a+vε for some v∈k. The coefficient v is the tangent vector.
This reproduces the one-dimensional tangent space without referring to a Euclidean limit. Infinitesimal test schemes replace “move by an infinitely small number” with exact algebra.
18. Ring homomorphisms reverse into affine-scheme morphisms
A ring homomorphism φ:A→B induces a morphism
Spec B→Spec A.
On points, a prime q⊂B maps to its inverse image φ⁻¹(q), which is prime in A.
On structure sheaves, localised ring maps provide the compatible pullback of regular functions.
Thus the category of affine schemes is opposite to the category of commutative rings. This is the scheme-theoretic continuation of the coordinate-ring contravariance from classical affine varieties.
The Stacks Project’s Schemes chapter and Points of schemes develop this correspondence through locally ringed spaces and local homomorphisms.
19. Worked morphism: squaring A¹→A¹
Let k be a field and define the ring map
k[t]→k[s], t↦s².
Contravariantly this gives a scheme morphism f:A¹_s→A¹_t.
On ordinary k-points it is s↦s².
But the scheme map contains more information than that point function. Its fibres are computed by tensor products, and the fibre over t=0 can carry nilpotent multiplicity.
This will be our running example for base change.
20. Closed subschemes come from quotient rings
Given an ideal I⊂A, the quotient map A→A/I induces a closed immersion
Spec(A/I)→Spec A.
Its underlying image is V(I).
If I and √I differ, the closed subscheme carries more information than the closed subset. For example, Spec k[x]/(x²) and Spec k[x]/(x) have the same underlying point but different structure sheaves.
Thus a closed subscheme remembers the actual ideal, while an ordinary Zariski-closed subset remembers only its radical.
21. Open subschemes come from localisation
The standard open D(f)⊂Spec A is itself an affine scheme:
D(f)=Spec A_f.
The map A→A_f induces the open immersion in the opposite direction.
Closed subschemes are therefore governed by quotients, while principal open subschemes are governed by localisations.
These two operations—quotient and localisation—form much of the basic local algebraic toolkit.
22. Schemes are spaces locally modelled on affine schemes
A scheme is a locally ringed space in which every point has an open neighbourhood isomorphic to an affine scheme.
This is the exact analogue of defining a manifold by requiring local Euclidean charts, except the local models are spectra of rings rather than open subsets of Rⁿ.
The analogy should not be pushed too far. Scheme topology can be highly non-Hausdorff, points can carry different residue fields, and nilpotent structure has no ordinary manifold analogue.
The formal definition appears in the Stacks Project: a scheme is a locally ringed space with an affine open neighbourhood around every point. See Stacks Project, Schemes §26.9.
23. Affine pieces can be glued along isomorphic open subschemes
Suppose X₁ and X₂ are schemes with opens U₁⊂X₁ and U₂⊂X₂ and an isomorphism U₁≈U₂ satisfying the required compatibility.
We can identify corresponding points and glue the structure sheaves to build a larger scheme X.
Projective space P¹ is the standard example. Glue two copies of A¹ with coordinates t and s along D(t) and D(s) by s=1/t.
Algebraically, the overlap rings are k[t,t⁻¹] and k[s,s⁻¹], and the isomorphism sends s to t⁻¹.
The resulting scheme is not affine, but it is covered by two affine opens. This is the scheme-theoretic meaning of projective charts.
24. A scheme need not come from one global ring
Every affine scheme is Spec A for one ring A, but general schemes are assembled from affine schemes.
P¹ cannot be represented as Spec of its global regular-function ring, because over an algebraically closed field its global regular functions are only constants while P¹ has many points and nontrivial geometry.
This is why the affine coordinate-ring dictionary cannot simply be applied globally without modification.
Sheaves repair the problem: instead of demanding one global coordinate ring, they keep compatible rings on every open region.
R24.06 will make this local-to-global machinery explicit.
25. Fibre products are tensor products in affine coordinates
Suppose ring maps R→A and R→B are given. Then
Spec A ×_{Spec R} Spec B ≈ Spec(A⊗_R B).
This universal construction combines two spaces over the same base.
The tensor product is not merely an algebraic trick. It is the coordinate ring forced by the universal property of the geometric fibre product.
The Stacks Project records this affine fibre-product formula explicitly: Affine scheme fibre products.
26. A fibre is a base change to one point
Let f:X→S be a scheme morphism and let s∈S. The fibre over s is
X_s=X×_S Spec κ(s).
This definition uses the residue field of the point, not merely its name as an element of a coordinate set.
If S=Spec R and X=Spec A, with s corresponding to prime p⊂R, then
X_s≈Spec(A⊗_R κ(p)).
The fibre can therefore change structure as the base point changes. It may split, merge, become singular or acquire nilpotents.
27. Worked fibre: squaring produces a nonreduced special fibre
Return to k[t]→k[s], t↦s², with k algebraically closed and characteristic not 2.
The fibre over t=a is obtained by imposing t=a:
k[s]⊗_{k[t]} k ≈ k[s]/(s²−a).
If a≠0, the polynomial s²−a has two distinct roots ±√a, so the fibre consists of two reduced points.
If a=0, the fibre is Spec k[s]/(s²). Topologically it has one point, but scheme-theoretically it is doubled by a nilpotent.
The special fibre remembers ramification multiplicity that the ordinary point set would compress into “one preimage”.
28. Base change asks the same family over a new base
Given X→S and a morphism S′→S, the base-changed scheme is
X_{S′}=X×_S S′.
Base change can represent extending scalars, restricting to a fibre, passing to a field extension or studying a family after changing parameters.
If X=Spec A over S=Spec R and S′=Spec R′, then X_{S′}=Spec(A⊗_R R′).
A property that survives every base change is stronger than one that holds only before scalar extension. Modern definitions such as smooth, proper and flat morphisms are designed with base change in mind.
29. Flatness is the algebraic condition behind controlled family variation
An R-module M is flat when tensoring with M preserves exact sequences. A morphism Spec A→Spec R is flat when A is flat as an R-module; general flatness is checked on local affine charts.
Flatness prevents certain algebraic collapses under base change and is central to well-behaved families.
It should not be reduced to “all fibres have the same dimension”. Under suitable finite-presentation hypotheses flat families often have controlled Hilbert polynomials and dimension behaviour, but flatness is an exactness condition, not a simple fibre-count slogan.
Example: k[t]→k[s] with t↦s² makes k[s] a free k[t]-module of rank 2 with basis {1,s}, so the morphism is finite and flat. The fibre at zero is nonreduced but still has scheme-theoretic length 2.
30. Scheme-theoretic intersections retain multiplicity
Let X and Y be closed subschemes of an affine scheme S=Spec A, defined by ideals I and J.
Their scheme-theoretic intersection is the fibre product X×_S Y, whose affine coordinate ring is
A/(I+J).
The underlying topological intersection depends only on radicals, but the quotient A/(I+J) can retain multiplicity and nilpotent structure.
This turns intersection multiplicity from an external counting correction into geometry stored inside the intersection scheme.
31. Worked tangent intersection: a line meets a parabola with multiplicity two
In A²=Spec k[x,y], let L be the x-axis y=0 and let C be the parabola y=x².
The ideals are I=(y) and J=(y−x²).
The scheme-theoretic intersection has coordinate ring
k[x,y]/(y,y−x²)≈k[x]/(x²).
Topologically the intersection is only the origin. Scheme-theoretically it is a doubled point.
This records the fact that the line is tangent to the parabola at the origin. The intersection multiplicity two is visible in the nilpotent class of x.
By contrast, intersecting y=0 with y=x gives k[x,y]/(y,y−x)≈k, a reduced single point corresponding to transverse intersection multiplicity one.
32. Scheme structure separates support from multiplicity
The support of a scheme is its underlying topological space. A nonreduced scheme can have the same support as a reduced one while carrying extra infinitesimal layers.
For Spec k[x]/(xⁿ), the support is the origin for every n≥1.
But the rings have increasing nilpotent thickness. As finite-dimensional k-vector spaces they have dimensions n, with basis 1,x,…,x^{n−1}.
In zero-dimensional settings this vector-space dimension is the scheme length. Length becomes a precise way to count multiplicity that survives deformation and intersection calculations better than merely counting geometric points.
33. Generic and closed fibres connect geometry to arithmetic
Consider a scheme X over Spec Z.
The fibre over the generic point (0) uses κ((0))=Q and describes the characteristic-zero geometry.
The fibre over a closed point (p) uses κ((p))=F_p and describes reduction modulo p.
This makes one scheme capable of organising an entire family of geometric objects across all prime characteristics.
Arithmetic geometry repeatedly studies what changes between the generic fibre over Q and special fibres over finite fields. R24.07 will make that bridge explicit.
34. Example: one equation over Z creates many fibres
Let X=Spec Z[x,y]/(y²−x³−x).
Over Q, the generic fibre is the affine cubic y²=x³+x.
Over F_p, the fibre is defined by the same polynomial reduced modulo p.
For primes where the discriminant is nonzero modulo p, the projective completion has good reduction as a smooth elliptic curve. At primes dividing the discriminant, the special fibre becomes singular.
The exact set of bad primes depends on the chosen integral model and discriminant. The structural point is that scheme language keeps all fibres in one arithmetic family instead of treating each modulus as an unrelated equation.
35. Local algebra tests properties point by point
Many scheme properties are local. Reducedness, regularity, dimension and normality can be checked through local rings or affine covers.
For X=Spec A, a point p carries A_p. If A_p has a nilpotent, the scheme is nonreduced near p. If A_p fails the regular-local-ring criterion, p is singular in the classical finite-type setting.
Normality asks whether local domains are integrally closed. Cohen–Macaulay and Gorenstein properties impose deeper homological conditions on local rings.
Scheme theory therefore turns geometry into a network of local algebra problems that can be compared and glued.
36. The functor-of-points viewpoint asks how every test scheme sees X
A scheme X determines a functor
h_X(T)=Hom(T,X).
Rather than asking only for ordinary points Hom(Spec k,X), we may ask for T-valued points for every test scheme T.
Dual-number points reveal tangent vectors. Points over finite extensions reveal field-of-definition information. Maps from infinitesimal schemes probe deformation structure.
The Yoneda viewpoint says that an object is determined, up to isomorphism, by how all other objects map into it.
The Stacks Project’s Points of schemes section develops this perspective from local rings and morphisms.
37. Proj is the projective partner of Spec
Affine schemes are built by Spec of rings. Projective schemes are built from graded rings using Proj.
For S=k[x₀,…,xₙ] with its standard grading, Proj S is projective n-space Pⁿ_k.
A homogeneous ideal I defines the closed projective subscheme Proj(S/I), subject to the standard irrelevant-ideal framework.
The affine opens D_+(x_i) are spectra of degree-zero localisations and recover the familiar projective coordinate charts.
This article keeps Proj as a bridge because the previous projective-variety guide already owns homogeneous coordinates. The deeper graded-module and sheaf machinery belongs naturally with R24.06.
38. A dependable scheme calculation workflow
First identify the ring and whether the problem concerns the underlying topological space, the structure sheaf or both.
List the relevant prime ideals or describe them structurally. Use V(I) for closed subsets and D(f) for standard opens.
At a point p, localise to A_p and compute the residue field κ(p) when fibre or tangent information is needed.
For a closed subscheme, keep the actual ideal I rather than replacing it automatically by √I. For an open restriction, localise.
For a morphism, reverse the corresponding ring map. For a fibre product, tensor over the base ring. For a fibre, tensor with the residue field of the chosen base point.
Finally, ask whether nilpotents, residue-field extensions or special fibres carry information that an ordinary point picture would miss.
39. Common misconceptions
Every scheme point is a classical geometric point. No. Nonmaximal prime ideals give generic and intermediate points.
Spec A is only a topological space. An affine scheme also carries the structure sheaf.
Nilpotents do not matter because they vanish at every prime. They are invisible topologically but remain visible in the local rings and can record multiplicity or infinitesimal structure.
A fibre is merely the set-theoretic inverse image. Scheme fibres are base changes and can be nonreduced or have residue-field structure invisible to the point set.
Every scheme is Spec of one ring. Only affine schemes are. General schemes are glued from affine opens.
Smooth, regular and reduced mean the same thing. They are different properties. Smoothness is relative to a base morphism; regularity is local-ring structure; reducedness only removes nilpotents.
40. Independent practice: twenty questions
- List the prime ideals of k[x] when k is algebraically closed.
- Which point of Spec k[x] is generic?
- Describe D(x) in Spec k[x].
- Describe V(30) in Spec Z.
- For primes p⊆q, which point is a specialisation of which?
- What is O(D(f)) on Spec A?
- What is the stalk at p∈Spec A?
- Compute the residue field at (x−a)∈Spec k[x].
- Compute the residue field at the generic point (0)∈Spec k[x].
- Why do Spec k[ε]/(ε²) and Spec k have the same underlying point set?
- What extra information distinguishes them as schemes?
- What affine-scheme morphism is induced by k[t]→k[s], t↦s²?
- Compute the fibre of that morphism over t=a.
- What happens to the fibre at a=0 in characteristic not 2?
- What closed subscheme of Spec A is defined by an ideal I?
- Compute the scheme-theoretic intersection of y=0 and y=x² in A².
- Why is that intersection not reduced?
- What ring describes Spec A×_{Spec R}Spec B?
- What is the fibre of X→S over s in intrinsic scheme notation?
- Explain why P¹ needs at least two affine charts even though its global regular functions over an algebraically closed field are constants.
41. Worked solutions and checks
1. The primes are (0) and (x−a) for a∈k. The latter are maximal because k[x]/(x−a)≈k.
2. (0). Its closure is V(0)=Spec k[x].
3. D(x) consists of all primes not containing x: the generic point and every closed point (x−a) with a≠0.
4. The prime ideals containing 30 are (2),(3),(5). Thus V(30) is those three closed points.
5. q is a specialisation of p because q lies in the closure of {p}. Equivalently p is more generic.
6. O(D(f))≈A_f.
7. O_{Spec A,p}≈A_p.
8. The local ring quotient gives κ((x−a))≈k.
9. κ((0)) is the fraction field k(x).
10. The dual-number ring has only one prime, (ε), because every prime contains the nilpotent ε and the quotient by (ε) is k.
11. The structure sheaf/local ring contains a nonzero nilpotent ε with ε²=0. Spec k does not.
12. It induces A¹_s→A¹_t, which on k-points sends s to s².
13. The fibre ring is k[s]/(s²−a), so the fibre is Spec k[s]/(s²−a).
14. At a=0 the fibre is Spec k[s]/(s²), a single nonreduced doubled point.
15. Spec(A/I) with its closed immersion into Spec A and underlying support V(I).
16. Add the ideals: (y)+(y−x²)=(y,x²). The intersection ring is k[x,y]/(y,x²)≈k[x]/(x²).
17. The class of x is nonzero but squares to zero, so the ring contains a nilpotent.
18. A⊗_R B.
19. X_s=X×_S Spec κ(s).
20. P¹ is not affine and cannot be reconstructed as Spec of its global function ring k. Instead it is glued from two affine lines whose overlap is the multiplicative group Spec k[t,t⁻¹].
42. What R24.05 adds to the Atlas
The classical sequence now has a scheme-theoretic upgrade: zero sets become spectra of primes; polynomial functions become structure sheaves; neighbourhoods become localisations; points carry local and residue fields; nilpotents retain infinitesimal information; intersections are fibre products; and families are organised by base change.
The next R24 cell, Sheaves and Cohomological Methods, will explain how local data are glued, why global sections can fail to capture the whole space, how quasi-coherent sheaves translate modules into geometry, and why cohomology measures obstructions to global solving.
Sources and further study
The primary reference for this guide is the Stacks Project chapter on Schemes, including Affine schemes and standard opens, the definition and basic properties of schemes, points of schemes and local rings, and affine fibre products. For the classical bridge, return to the earlier BTT guides on coordinate rings and local algebraic geometry.