04.02.01 · algebraic-geometry / schemes

Scheme

shipped3 tiersLean: partial

Anchor (Master): Hartshorne §II; Vakil; Grothendieck-Dieudonné EGA

Intuition [Beginner]

A scheme is a geometric object built from commutative rings. The slogan: a commutative ring is the ring of functions on some "geometric space," and that space is the scheme associated to the ring.

The basic example: the ring $C [x, y]$ of polynomials in two variables. Geometrically, this ring is the "ring of functions on the plane $C^{2}$ ." The corresponding scheme $Spec (C [x, y])$ has points corresponding to ideals of the polynomial ring — including the maximal ideals (corresponding to actual points of the plane) and the prime ideals (corresponding to subvarieties like curves).

Schemes generalize the algebraic varieties of classical algebraic geometry by allowing arbitrary commutative rings — including rings with nilpotents (e.g., $k [ϵ] / (ϵ^{2})$ for "infinitesimal" calculations) and rings of integers (for arithmetic geometry). They are the universal language of modern algebraic geometry, enabling Grothendieck's revolutionary unification of algebra and geometry.

Visual [Beginner]

A commutative ring on the left, an associated geometric "scheme" on the right. The scheme's points are prime ideals of the ring; functions on the scheme are elements of the ring.

Worked example [Beginner]

The simplest scheme: $Spec (Z)$ , the spectrum of the integers. Its points are the prime ideals of $Z$ :

The maximal ideals $(p) = p Z$ for each prime number $p$ — these correspond to "ordinary" closed points.
The zero ideal $(0)$ — the generic point, geometrically the "whole space."

So $Spec (Z)$ has one point per prime number, plus a "generic point." The closed points are like a discrete set indexed by primes; the generic point is dense (its closure is the whole space).

The structure sheaf $O$ assigns to each open set the appropriate ring of "rational" functions — for example, $O (Spec (Z)) = Z$ , the ring of integers itself.

This is the prototype of all schemes. More generally, $Spec (Z [x, y, \dots])$ adds more polynomial dimensions; $Spec (Z [x] / (f (x)))$ corresponds to the "scheme of $f = 0$ ." Every commutative ring has its own scheme.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $R$ be a commutative ring (with unit) 03.01.02. The spectrum of $R$ , denoted $Spec (R)$ , is the topological space whose points are the prime ideals of $R$ , with the Zariski topology: closed sets are $V (I) := {p \in Spec (R) : I \subseteq p}$ for ideals $I \subseteq R$ .

The structure sheaf $O_{Spec (R)}$ on $Spec (R)$ assigns to a basic open $D (f) := {p : f \in / p}$ (for $f \in R$ ) the localisation $R_{f}$ . For a general open $U$ , $O (U)$ is determined by the sheaf condition. The pair $(Spec (R), O_{Spec (R)})$ is an affine scheme.

A scheme is a locally ringed space $(X, O_{X})$ — a topological space with a sheaf of commutative rings whose stalks $O_{X, x}$ are local rings — such that $X$ is locally isomorphic to an affine scheme. Concretely: every $x \in X$ has an open neighbourhood $U$ with $(U, O_{X} ∣_{U}) ≅ (Spec (R), O_{Spec (R)})$ for some commutative ring $R$ .

A morphism of schemes $f : X \to Y$ is a morphism of locally ringed spaces — a continuous map of underlying spaces together with a compatible map of sheaves of rings $O_{Y} \to f_{*} O_{X}$ such that the induced map on stalks is a local ring homomorphism (preserves the maximal ideal).

Key constructions:

Affine scheme: $Spec (R)$ for a ring $R$ .
Projective scheme: $Proj (R)$ for a graded ring $R$ . Examples: $P_{k}^{n} = Proj (k [x_{0}, \dots, x_{n}])$ .
Affine line over a ring $R$ : $A_{R}^{1} := Spec (R [t])$ .
Fibre product: schemes admit pullbacks $X \times_{Z} Y$ for any morphisms $X \to Z \leftarrow Y$ .

A scheme over a field $k$ (or more generally a base scheme $S$ ) is a scheme $X$ together with a structural morphism $X \to Spec (k)$ . For algebraically closed $k$ , classical algebraic varieties are reduced, irreducible, separated, finite-type schemes over $k$ — a substantial restriction of the general scheme notion.

Key theorem with proof [Intermediate+]

Theorem (Anti-equivalence of affine schemes and commutative rings). The functor $Spec : CRing^{op} \to AffSch$ from commutative rings (opposite category) to affine schemes is an equivalence of categories. Its inverse is the global-sections functor $Γ : AffSch \to CRing^{op}$ , which sends $(Spec (R), O) \mapsto O (Spec (R)) = R$ .

Proof sketch. Two directions:

Affine schemes are determined by their global sections. For an affine scheme $Spec (R)$ , $Γ (Spec (R), O_{Spec (R)}) = R$ by construction (the structure sheaf is glued from localisations, and the global sections are the original ring). So $Γ \circ Spec = id_{CRing^{op}}$ .

Morphisms of affine schemes correspond to ring homomorphisms. A morphism $Spec (R) \to Spec (S)$ as locally ringed spaces gives a ring map $S \to R$ (apply $Γ$ ). Conversely, a ring map $φ : S \to R$ induces a continuous map $Spec (R) \to Spec (S)$ by $p \mapsto φ^{- 1} (p)$ , with the structure-sheaf map coming from $φ$ . Verifying that these are inverse uses the local-ring-map condition (which lifts to the maximal-ideal-preserving condition on the corresponding prime ideal).

The full equivalence follows by combining objects and morphisms; the locally-ringed-space framework ensures the structure-sheaf data is preserved correctly. $□$

This anti-equivalence is the foundation of Grothendieck's algebraic geometry: every commutative ring becomes a geometric object, and every algebraic operation has a geometric counterpart. It is what makes "schemes" the right generalisation of varieties.

Bridge. The construction here builds toward 04.03.01 (sheaf cohomology), where the same data is upgraded, and the symmetry side is taken up in 04.04.01 (riemann-roch theorem for curves). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 4 (medium, conceptual).

What is the difference between a reduced and non-reduced scheme?

Hint

Reduced means the structure sheaf has no nilpotent local sections.

Answer

A scheme $X$ is reduced if the structure sheaf $O_{X}$ has no nilpotent local sections — equivalently, every stalk $O_{X, x}$ is a reduced ring (no $r \neq = 0$ with $r^{n} = 0$ ).

Non-reduced schemes have "infinitesimal thickenings": $Spec (k [ϵ] / (ϵ^{2}))$ is the dual numbers scheme, a single point with a 1-dimensional tangent direction realised algebraically. It is not reduced because $ϵ \neq = 0$ but $ϵ^{2} = 0$ .

Non-reduced schemes are essential for moduli theory and deformation theory: tangent vectors at a point of a moduli space correspond to morphisms from $Spec (k [ϵ] / ϵ^{2})$ . In classical algebraic geometry (varieties over algebraically closed fields), schemes are usually assumed reduced.

Exercise 5 (medium, conceptual).

State the functor of points perspective: every scheme is determined by what it represents.

Hint

For a scheme $X$ and a ring $R$ , $X (R) := Hom (Spec (R), X)$ .

Answer

Each scheme $X$ defines a functor $X (-) : CRing \to Set$ by

X (R) := Hom_{Sch} (Spec (R), X) .

This functor encodes the scheme up to isomorphism (Yoneda lemma applied to the category of schemes). For example, the affine line $A_{S}^{1}$ has $A_{S}^{1} (R) = Hom_{S} (Spec (R), A^{1}) = R$ — the "points" of $A^{1}$ valued in $R$ are just elements of $R$ .

The functor-of-points perspective recasts schemes as representable functors on commutative rings, making algebraic geometry into a flavour of functorial commutative algebra. It is the foundation of moduli problems (representable by moduli schemes) and of stacks (functors that aren't quite representable but represented by groupoids).

Exercise 7 (hard, conceptual).

State Grothendieck's "scheme = locally ringed space locally isomorphic to $Spec (R)$ " definition and explain why the local-ringed-space condition is essential.

Hint

Without local rings, morphisms wouldn't preserve the relevant geometric structure.

Answer

A locally ringed space is a topological space $X$ with a sheaf of rings $O_{X}$ such that every stalk $O_{X, x}$ is a local ring (has a unique maximal ideal). A morphism of locally ringed spaces $f : (X, O_{X}) \to (Y, O_{Y})$ is a continuous map plus a sheaf morphism $O_{Y} \to f_{*} O_{X}$ such that the induced stalk map at every point sends maximal ideals to maximal ideals.

A scheme is a locally ringed space locally isomorphic to an affine scheme.

The local-ringed-space condition is essential because:

Stalks of structure sheaves at points are local rings — the unique maximal ideal corresponds to "functions vanishing at that point."
Morphisms must preserve this structure: a function vanishing at $f (x)$ in $Y$ should pull back to a function vanishing at $x$ in $X$ .

Without the locality condition, morphisms would forget which functions vanish where — the geometric content of "evaluating a function at a point" would be lost.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has AlgebraicGeometry.Scheme, AlgebraicGeometry.Spec, with the affine-scheme/ring duality formalised.

[object Promise]

The companion module re-exports Mathlib's scheme API and Codex conventions.

Advanced results [Master]

Functor of points. Detailed in Exercise 5. Schemes form a full subcategory of functors $CRing \to Set$ via the representability $X \mapsto X (-)$ . Yoneda's lemma identifies scheme morphisms with natural transformations of functors.

Properties of schemes / morphisms. Hartshorne and Vakil classify properties: separated (diagonal closed), proper (universally closed), flat, smooth, étale, finite type, finite presentation, etc. These properties characterise the morphisms one cares about; they are the algebraic-geometric analogues of "compact," "surjective," "differentiable," etc., from differential topology.

Quasi-coherent sheaves on schemes. A sheaf $F$ on a scheme $X$ is quasi-coherent if locally it is given by an $O_{X} (U)$ -module via the localisation pattern. Quasi-coherent sheaves on $Spec (R)$ correspond to $R$ -modules (the Serre-Swan theorem for affine schemes). They are the natural "vector-bundle-like" objects.

Coherent sheaves. Quasi-coherent + finite-type. On Noetherian schemes, coherent sheaves are well-behaved (kernels, cokernels, images are coherent) and have computable cohomology.

Cohomological properties. Affine schemes have vanishing higher cohomology of quasi-coherent sheaves (Serre's vanishing theorem). Projective schemes have computable coherent cohomology via the Serre formula. These properties drive Riemann-Roch and Hodge-theoretic computations 04.04.01.

Connection to other geometries. Smooth schemes over $C$ are complex manifolds; schemes over $R$ extend to the real-algebraic setting. Arithmetic schemes (over $Z$ ) connect algebraic geometry with number theory. Stacks generalise schemes to allow "geometric objects with automorphisms."

Synthesis. This construction generalises the pattern fixed in 04.01.01 (sheaf), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

$Spec (R) = CRing^{op}$ . Proved in §"Key theorem".

Zariski topology axioms. Proved in Exercise 2.

Krull dimension and chains. Sketched in Exercise 3.

Projective space affine cover. Proved in Exercise 6.

Local ring structure of stalks. For an affine scheme $Spec (R)$ at point $p$ , the stalk $O_{Spec (R), p}$ is the localisation $R_{p}$ — a local ring with maximal ideal $p R_{p}$ . The "local ring at $p$ " captures the local algebraic information at that point.

Quasi-coherent ↔ module. On $Spec (R)$ , the category of quasi-coherent sheaves is equivalent to the category of $R$ -modules: $F \mapsto F (Spec (R))$ and conversely $M \mapsto M$ via the localisation construction.

Connections [Master]

Sheaf 04.01.01 — the structure sheaf of a scheme is a sheaf of rings.
Associative algebra 03.01.02, Ideal 03.01.03 — the ring-theoretic input.
Topological space 02.01.01 — the underlying point-set with Zariski topology.
Sheaf cohomology 04.03.01 — the central computational tool on schemes.
Riemann-Roch theorem 04.04.01 — applies to coherent sheaves on schemes (curves and surfaces).

Historical & philosophical context [Master]

The notion of a scheme was developed by Alexander Grothendieck and Jean Dieudonné in the Éléments de géométrie algébrique (EGA, 1960–1967). Grothendieck's revolutionary idea: replace the classical varieties (subsets of $A^{n}$ defined by polynomial equations) with the more flexible notion of locally ringed spaces locally isomorphic to spectra of rings. This generalisation includes nilpotents, arithmetic rings, and arbitrary base schemes, dramatically expanding the scope of algebraic geometry.

Earlier algebraic geometers (Weil, Zariski, Serre) had developed the theory of varieties in a functorial language but stopped short of the full ring-theoretic generalisation. Grothendieck's EGA, together with the Séminaire de géométrie algébrique (SGA) volumes, set the modern technology in place: schemes, morphisms, properties, sheaves, cohomology, étale cohomology.

The scheme-theoretic perspective transformed not just algebraic geometry but adjacent fields: arithmetic geometry (Wiles' proof of Fermat's Last Theorem; the Langlands programme), derived algebraic geometry (Lurie, Toën-Vezzosi), and category theory (sheaves on a Grothendieck site). Schemes are the universal language for "spaces studied via their rings of functions."

Bibliography [Master]

Hartshorne, R., Algebraic Geometry, Springer GTM 52, 1977. Chapters II–III.
Vakil, R., The Rising Sea: Foundations of Algebraic Geometry, draft monograph.
Eisenbud, D. & Harris, J., The Geometry of Schemes, Springer GTM 197, 2000.
Grothendieck, A. & Dieudonné, J., Éléments de géométrie algébrique (EGA), Publications Mathématiques de l'IHÉS, 1960–1967.
Mumford, D., The Red Book of Varieties and Schemes, Springer Lecture Notes, 1988.

v0.5 Strand A unit #2. Scheme — the geometric realisation of a commutative ring; Grothendieck's universal generalisation of algebraic varieties to arbitrary commutative rings, including nilpotents and arithmetic rings.