Schemes — Spec, Proj, and Morphisms

Schemes are locally ringed spaces locally isomorphic to the spectrum

Spec A

of a commutative ring. The affine spectrum

Spec

and the homogeneous spectrum

Proj

produce the local affine and projective models; morphisms of schemes are morphisms of locally ringed spaces, glued from ring homomorphisms. Applies to: algebraic geometry, arithmetic geometry, deformation theory, moduli problems. Who/when: Alexander Grothendieck and the Paris school, Éléments de géométrie algébrique (1960–1967), building on Serre's FAC (1955).

A scheme $(X, O_{X})$ is a topological space equipped with a sheaf of rings that is locally modeled on the spectrum of a commutative ring, the set of prime ideals topologized by the Zariski topology and endowed with its structure sheaf. The construction $Spec$ turns the category of commutative rings into the local building blocks of geometry; $Proj$ of a graded ring builds the projective objects by gluing affine charts on homogeneous localizations. The decisive innovation is that points include generic points (non-maximal primes) and that the structure sheaf may carry nilpotents, encoding infinitesimal and non-reduced data invisible to classical varieties. A surprising fact: $Spec Z$ is a one-dimensional "curve" whose closed points are the primes and whose generic point is $(0)$ , making number theory literally a geometry. The functor $A \mapsto Spec A$ is a contravariant equivalence between commutative rings and affine schemes, so all of commutative algebra becomes geometry.

The Core Insight

Geometry should be reconstructed from its ring of functions. For an affine variety the coordinate ring determines everything; Grothendieck's leap is to take any commutative ring $A$ , declare its prime ideals to be points, and recover functions by localization. The value of a "function" $f \in A$ at a point $p$ is its image in the residue field $κ (p)$ , so functions can vanish, be nilpotent, or have different residue fields at different points. Gluing $Spec$ 's along open sets gives schemes; a morphism is then completely determined by, and dual to, ring maps — geometry and algebra become two views of one category.

Formal Statement

The Prime Spectrum as a Space

Let $A$ be a commutative ring with $1$ . The prime spectrum $Spec A$ is the set of prime ideals $p ⊊ A$ . For an ideal $I \subseteq A$ define the vanishing locus

V (I) = {p \in Spec A : I \subseteq p} .

⟳a prime

p

"lies on"

V (I)

iff every element of

I

maps to

0

κ (p)

, i.e.

I \subseteq p

; check

V (I) \cup V (J) = V (I J)

and

⋂_{α} V (I_{α}) = V (\sum_{α} I_{α})

The sets $V (I)$ are the closed sets of the Zariski topology. A basis of opens is given by the distinguished opens $D (f) = Spec A ∖ V (f) = {p : f \in / p}$ for $f \in A$ .

The Structure Sheaf

The structure sheaf $O_{X}$ on $X = Spec A$ is the unique sheaf of rings with

O_{X} (D (f)) = A_{f} = A [f^{- 1}], O_{X, p} = A_{p} .

⟳define

O_{X} (U)

for general open

U

as compatible families of fractions

s : U \to ∐_{p} A_{p}

locally of the form

a / g

; verify the sheaf axioms and that on

D (f)

this recovers

A_{f}

via the localization–covering correspondence.

The pair $(X, O_{X})$ is a locally ringed space because each stalk $A_{p}$ is local with maximal ideal $p A_{p}$ . The residue field at $p$ is

κ (p) = A_{p} / p A_{p} = Frac (A / p) .

⟳

A_{p} / p A_{p}

is the localization of the domain

A / p

at its zero ideal, which is by definition its fraction field.

Affine Scheme and Scheme

An affine scheme is a locally ringed space isomorphic to $(Spec A, O_{Spec A})$ for some ring $A$ . A scheme is a locally ringed space $(X, O_{X})$ admitting an open cover ${U_{i}}$ with each $(U_{i}, O_{X} ∣_{U_{i}})$ an affine scheme. The global sections functor and $Spec$ form an adjunction; on affines they invert one another:

Hom_{Sch} (X, Spec A) ≅ Hom_{CRing} (A, Γ (X, O_{X})) .

⟳a scheme map to

Spec A

is determined by where the generators of

A

go as global functions; conversely a ring map

A \to Γ (X, O_{X})

assembles, on each affine chart, into local maps by the universal property of localization.

Spec is an Anti-equivalence

The functor

Spec : CRing^{op} ⟶ AffSch, A \mapsto Spec A,

is an equivalence of categories, with quasi-inverse the global sections functor

Γ

. A ring homomorphism

φ : A \to B

induces

Spec φ : Spec B \to Spec A

q \mapsto φ^{- 1} (q)

, continuous and a morphism of locally ringed spaces.

⟳the contraction

φ^{- 1} (q)

of a prime is prime; functoriality and the local-homomorphism condition follow from

φ^{- 1} (q) A_{φ^{- 1} (q)} \subseteq

preimage of

q B_{q}

Proj of a Graded Ring

Let $S = ⨁_{d \geq 0} S_{d}$ be a $Z_{\geq 0}$ -graded ring with irrelevant ideal $S_{+} = ⨁_{d > 0} S_{d}$ . The homogeneous spectrum

Proj S = {p \subset S homogeneous prime : S_{+} \neq \subseteq p} .

⟳discard

p \supseteq S_{+}

because these correspond to the "vertex" of the affine cone; a prime is homogeneous iff it is generated by its homogeneous components.

It is covered by affines $D_{+} (f) ≅ Spec ((S_{f})_{0})$ for homogeneous $f \in S_{+}$ , where $(S_{f})_{0}$ is the degree-zero part of the localization. For $S = A [x_{0}, \dots, x_{n}]$ this yields projective $n$ -space $P_{A}^{n} = Proj A [x_{0}, \dots, x_{n}]$ .

Why It Holds / Derivation

Spec is a Sheaf and a Locally Ringed Space

Step 1 — Topology. From [1],

V (0) = X

V (A) = \emptyset

V (I) \cup V (J) = V (I \cap J) = V (I J)

, and

⋂_{α} V (I_{α}) = V (\sum_{α} I_{α})

. Hence the

V (I)

are closed sets of a topology. The

D (f)

form a basis since

V (I) = ⋂_{f \in I} V (f)

, so its complement is

⋃_{f \in I} D (f)

. Step 2 — $D (f) \subseteq D (g)$ criterion.

D (f) \subseteq D (g)

iff

f \in (g)

, iff

f^{n} = g h

for some

n, h

, which is exactly when

g

becomes a unit in

A_{f}

. This gives canonical restriction maps

A_{g} \to A_{f}

and makes the assignment

D (f) \mapsto A_{f}

a presheaf on the basis. Step 3 — Sheaf on the basis. A cover

D (f) = ⋃_{i} D (g_{i})

corresponds to

f \in (g_{i})_{i}

, i.e. some power

f^{N} \in (g_{i})

. The sheaf gluing for

A_{f}

over

{A_{g_{i}}}

reduces to an exact sequence

0 \to A_{f} \to i \prod A_{g_{i}} \to i, j \prod A_{g_{i} g_{j}},

⟳this is the localized form of faithfully flat descent / a partition-of-unity argument: write

1 = \sum_{i} a_{i} g_{i}^{m}

A_{f}

to patch local fractions, and use injectivity of

A_{f} \to \prod A_{g_{i}}

to glue uniquely.

which is exact, so the presheaf on distinguished opens is a sheaf and extends uniquely to $O_{X}$ .

Step 4 — Stalks.

O_{X, p} = lim_{f \in / p} A_{f} = A_{p}

, a local ring; thus

(X, O_{X})

is a locally ringed space. Step 5 — Morphisms are local. A ring map

φ : A \to B

pulls back primes and, on stalks, gives

A_{φ^{- 1} (q)} \to B_{q}

sending the maximal ideal into the maximal ideal — a local homomorphism. This is the extra condition distinguishing scheme morphisms among arbitrary morphisms of ringed spaces.

■

Alexander Grothendieck | 1928–2014

Grothendieck recast algebraic geometry on the foundation of schemes between 1958 and 1970, mostly through the Éléments de géométrie algébrique (EGA), written with Jean Dieudonné, and the seminars SGA. His insight that every commutative ring is the ring of functions on a geometric object dissolved the boundary between geometry and arithmetic, enabling the relative viewpoint (everything is a morphism over a base), representable functors, and the machinery later used to prove the Weil conjectures. He emphasized working with the functor of points $T \mapsto Hom (T, X)$ and the systematic use of base change, sheaf cohomology, and nilpotent thickenings. His program reorganized the entire field around universal properties rather than coordinates.

Building Intuition

Think of $Spec A$ as the set of "ways to evaluate" elements of $A$ . Each prime $p$ gives a homomorphism $A \to κ (p)$ , and $f \in A$ becomes a "function" whose value at $p$ is its image $f (p) \in κ (p)$ — but the target field changes from point to point. Closed points behave like ordinary points; the generic point $(0)$ of an irreducible scheme lies in every nonempty open and represents "the whole component at once," so a property holding at the generic point holds on a dense open. Nilpotents are functions that are everywhere zero as values yet nonzero in the ring — fuzz that records multiplicity (a double point is $Spec k [ϵ] / (ϵ^{2})$ ). $Proj$ glues affine charts the way $P^{n}$ is built from $A^{n}$ charts $x_{i} \neq = 0$ , where the gluing data is exactly "rescale by ratios of coordinates," matched by degree-zero localizations.

Spec ℤ as an arithmetic curve: closed points are primes, the generic point (0) is dense

Worked Examples

Spec of a non-domain: a disconnected union

Given:

A = Z /6 Z

. Find

Spec A

and its topology. Steps:

By the Chinese Remainder Theorem $Z /6 ≅ Z /2 \times Z /3$ .
Primes of a product $R_{1} \times R_{2}$ are $p_{1} \times R_{2}$ and $R_{1} \times p_{2}$ . Here $Z /2, Z /3$ are fields, so each has the single prime $(0)$ .
Thus $Spec (Z /6) = {(2) / (6), (3) / (6)}$ — two points.
The idempotents $e = (1, 0), 1 - e = (0, 1)$ split the space; both points are open and closed.
Residue fields: $κ ((2)) = F_{2}$ , $κ ((3)) = F_{3}$ .

Answer:

Spec (Z /6) = {pt_{F_{2}}} ⊔ {pt_{F_{3}}}, dim = 0, # points = 2.

Proj of a polynomial ring: the projective line

Given:

S = k [x_{0}, x_{1}]

de g x_{i} = 1

k

a field. Describe

P_{k}^{1} = Proj S

. Steps:

Irrelevant ideal $S_{+} = (x_{0}, x_{1})$ . Standard cover by $D_{+} (x_{0}), D_{+} (x_{1})$ .
$D_{+} (x_{0}) = Spec ((S_{x_{0}})_{0})$ . Degree-zero elements of $k [x_{0}, x_{1}, x_{0}^{- 1}]$ are polynomials in $t = x_{1} / x_{0}$ , so $(S_{x_{0}})_{0} = k [t] = Spec A_{k}^{1}$ .
Likewise $D_{+} (x_{1}) = Spec k [u]$ , $u = x_{0} / x_{1}$ .
Overlap $D_{+} (x_{0} x_{1})$ : the gluing identifies $u = t^{- 1}$ on the locus $t \neq = 0$ .
Closed points: $[a_{0} : a_{1}]$ with residue field $k$ (if $k$ algebraically closed), plus the generic point $(0)$ .

Answer:

P_{k}^{1} = Spec k [t] \cup_{t \mapsto u^{- 1}} Spec k [u], dim = 1, # charts = 2.

Common Misconceptions

Points of $Spec A$ are NOT just maximal ideals. Every prime is a point, so non-closed (generic) points exist. The Zariski topology is not Hausdorff: distinct points need not be separated, and the generic point is dense. Also,

Γ (Spec A, O) = A

exactly, but for a general scheme global sections can be tiny (e.g.

Γ (P_{k}^{n}, O) = k

) — you cannot recover the scheme from its global functions alone. Finally, a morphism of schemes is not merely a continuous map plus any ring maps on opens; the homomorphisms on stalks must be local homomorphisms.

Continuous + ringed map that is NOT a morphism of schemes

Take $X = Spec K$ ( $K$ a field, one point with residue field $K$ ) and $Y = Spec R$ where $R$ is a DVR with maximal ideal $m$ , fraction field $K$ , residue field $k$ . The inclusion $R ↪ K$ induces a scheme morphism sending the point of $X$ to the generic point $(0)$ of $Y$ . If instead one tried to send $X$ to the closed point $m$ while pulling back along $R \to K$ , the induced stalk map $R_{m} = R \to K$ would have to send $m$ into $0$ — but $m \mapsto m K = K$ , which is not contained in the maximal ideal $(0)$ of $K$ . The locality of the stalk homomorphism fails, so this is not a valid morphism of locally ringed spaces. This shows why the locality axiom is essential and not automatic.

Connections

Spec

and

Proj

provide the affine and projective building blocks for all of Scheme Theory and feed directly into Quasi-coherent Sheaves, where modules over

A

become sheaves

M

Spec A

and graded modules over

S

become sheaves on

Proj S

(yielding the twisting sheaves

O (n)

). The functorial perspective links to Representable Functors and the Functor of Points via [4] and the Yoneda Lemma. Properties of morphisms — separated, proper, flat, smooth, étale — are studied in Morphisms of Schemes and are the inputs to Sheaf Cohomology and Intersection Theory. The special case

Spec Z

and its function-field analogues underlie Arithmetic Geometry and the Weil Conjectures.

Real-World Applications

Number theory / arithmetic geometry: $Spec Z$ and $Spec O_{K}$ are the geometric homes of integers and rings of integers; reduction mod $p$ is base change to $Spec F_{p}$ , central to Wiles's proof of Fermat's Last Theorem.
Coding theory: algebraic-geometric (Goppa) codes are built from divisors on smooth projective curves $Proj$ over finite fields, beating the Gilbert–Varshamov bound.
Cryptography: elliptic curves over $F_{q}$ , defined as $Proj$ schemes, supply the groups in ECC and pairing-based protocols.
Moduli problems: Hilbert and Quot schemes, the moduli of curves $\overline{M}_{g}$ , and GIT quotients are constructed schematically and used in string-theory compactifications.
Singularity theory & robotics: non-reduced fibers ( $Spec$ with nilpotents) model coincident roots and configuration-space degeneracies.
Computational algebra: Gröbner bases compute in coordinate rings, realizing $Spec$ / $Proj$ concretely in software (Macaulay2, Singular, SageMath).
Mathematical physics: mirror symmetry and derived categories of coherent sheaves on $Proj$ Calabi–Yau threefolds organize counts of curves and D-brane charges.

Historical Development

1882 — Dedekind & Weber: algebraic function fields treated geometrically; primes of a ring as "points," a conceptual ancestor of $Spec$ .
1930s–40s — Krull, van der Waerden, Zariski: valuation theory, the Zariski topology, and generic points sharpen the local algebra of varieties.
1946 — Weil, Foundations: abstract varieties via gluing, setting the stage for relative geometry.
1955 — Serre, FAC: Faisceaux algébriques cohérents introduces coherent sheaves and sheaf cohomology on varieties using the Zariski topology.
1956 — Chevalley & others: notions of "scheme" over a field circulate; Cartier and Chevalley formulate early prime-spectrum spaces.
1958 — Grothendieck (Edinburgh ICM): announces the program reorganizing geometry around $Spec$ of arbitrary rings.
1960–1967 — EGA (Grothendieck–Dieudonné): the definitive treatment of schemes, morphisms, $Proj$ , cohomology, and the relative point of view.
1960s — SGA seminars: étale cohomology, descent, fundamental groups — tools that resolved the Weil conjectures (Deligne, 1974).
1977 — Hartshorne, Algebraic Geometry: the standard graduate text crystallizing scheme theory for generations of students.

Subtleties & Limits

Scheme theory is foundationally complete, yet conceptual extensions remain active. Derived and spectral algebraic geometry (Lurie, Toën–Vezzosi) replace commutative rings by simplicial/connective ring spectra so that $Spec$ records higher homotopical data, essential for cotangent complexes and virtual fundamental classes; the right " $Proj$ " and intersection theory in this setting are still being developed. Non-commutative geometry seeks a $Spec$ for non-commutative rings where prime ideals are too few to topologize usefully. In arithmetic, the search for the field with one element $F_{1}$ and a geometry under $Spec Z$ aims to realize a curve-over-a-field analogue that would attack the Riemann Hypothesis; competing frameworks (Connes–Consani, Borger's $λ$ -rings, Durov) have not yet converged. Effective bounds and decidability for natural schematic questions (e.g. existence of rational points, the Hilbert scheme's components) remain genuinely open.