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Schemes, Morphisms & the Functor of Points

Guide 2 built one affine scheme out of a ring; now we glue affines into general schemes, learn that a morphism of schemes is a map of locally ringed spaces, and discover the slogan that tames the whole abstraction: a scheme IS its functor of points, the rule X(R) that says what an R-valued solution is.

Gluing affines: what a scheme actually is

Guide 2 took a commutative ring R and built a single geometric object: the spectrum Spec R, a topological space of prime ideals carrying the structure sheaf O, the whole package being an affine scheme. That is the atom. But a manifold is not one coordinate chart, and a variety is not one affine piece — projective space, an elliptic curve, the line with a doubled origin, all of these are stitched together from affine patches. A scheme is exactly the result of that stitching: a topological space X with a sheaf of rings O_X such that every point has an open neighborhood which, with the restricted sheaf, is isomorphic to some affine scheme Spec R.

The definition mirrors the manifold definition from Volume I almost word for word. There a smooth manifold was a space locally modeled on R^n via charts; here a scheme is a space locally modeled on Spec R via affine opens. The crucial upgrade is that the local model carries not just functions to R but an entire sheaf of rings — and the rings need not be reduced, need not be over a field, may have nilpotents and many components. That extra bookkeeping is exactly what lets schemes see things a variety cannot: the fat point Spec(k[x]/(x^2)) is a single point whose ring remembers a 'first-order direction', a tangent vector glued into the geometry itself.

Morphisms: maps of locally ringed spaces

A category is useless without its arrows, and here the arrows are subtler than continuous maps. The right setting is the locally ringed space of Guide 1: a space with a sheaf of rings every stalk of which is a local ring. A morphism of schemes f: X -> Y is a continuous map f together with a comparison of structure sheaves f#: O_Y -> f_* O_X — a way of pulling back functions — subject to one essential condition: the induced map on stalks O_{Y, f(p)} -> O_{X, p} must be a LOCAL homomorphism, sending the maximal ideal into the maximal ideal. That last clause is not decoration; it is the whole point.

Why insist on locality? Because the maximal ideal of a stalk is precisely 'functions vanishing at the point', so a local homomorphism is the algebraic shadow of the geometric demand 'a function vanishing at f(p) pulls back to a function vanishing at p'. Drop that condition and you admit maps that are nonsense geometrically. Here is the payoff theorem, the bridge that makes the whole apparatus usable: for affine schemes, morphisms Spec A -> Spec B correspond EXACTLY, and contravariantly, to ring homomorphisms B -> A. Hom(Spec A, Spec B) = Hom(B, A) with the arrow reversed. Geometry of affines is algebra of rings, read backwards.

Relative geometry: working over a base S

Schemes are almost never studied in isolation; they are studied OVER a base. An S-scheme is just a morphism X -> S, and a morphism of S-schemes is a triangle that commutes over S. When S = Spec k this recovers 'varieties over the field k', but the relative viewpoint is far more powerful: take S = Spec Z and a single scheme X -> Spec Z is a family of geometries, one over each prime, the engine of arithmetic geometry. The base records the ground you are allowed to stand on, and Grothendieck's revolution was to insist that almost every property worth having is a property of the MORPHISM X -> S, not of X alone.

The basic relative operation is the fiber product X x_S Y, the scheme-theoretic pullback that simultaneously generalizes intersection, preimage, and base change. On affine pieces it is dead simple — it is Spec of the tensor product: Spec A x_{Spec R} Spec B = Spec(A tensor_R B) — and the general fiber product is glued from these affine pieces. Two uses dominate. First, the FIBER of f: X -> S over a point s is X x_S Spec(k(s)), literally the part of X sitting above s. Second, BASE CHANGE: given X -> S and a new base S' -> S, the pullback X x_S S' transports X to live over S', the way you reduce a Z-scheme mod p by base-changing along Spec(F_p) -> Spec Z.

AFFINE FIBER PRODUCT  (the one formula behind everything)

    Spec A  x        Spec B   =   Spec ( A (x)  B )
            Spec R                          R

Example -- fiber of  y^2 = x^3 - x   over a prime p :

    X = Spec Z[x,y] / (y^2 - x^3 + x)   ->   Spec Z

    fiber over (p) :   X  x        Spec F_p  =  Spec  F_p[x,y]/(y^2 - x^3 + x)
                          Spec Z

    a smooth elliptic curve for most p ;  SINGULAR at the few bad primes
    p = 2 and p | disc -- the family degenerates exactly there.
The fiber product on affines is just Spec of a tensor product; specializing the base to a residue field extracts the fiber. One arithmetic family X -> Spec Z bundles infinitely many curves, with bad reduction visible as singular fibers.

The functor of points: a scheme is what it tests

Here is the idea that finally tames the abstraction, and it is worth slowing down for. A scheme has 'points' in the topological sense — prime ideals — but those are a strange and bloated set: Spec Z has a point for every prime AND a generic point, and Spec of a polynomial ring has points that are whole subvarieties. The repair is to stop asking 'what are the points of X' and start asking 'what maps INTO X'. For any ring R, define X(R) = Hom(Spec R, X), the set of R-valued points. This rule, sending each R to the set X(R), is the functor of points of X.

Why is this a revelation rather than a repackaging? Because X(R) is the set of SOLUTIONS over R, in the most literal sense. If X = Spec(Z[x, y]/(x^2 + y^2 - 1)) is the circle, then for any ring R the set X(R) is exactly the pairs (a, b) in R x R with a^2 + b^2 = 1 — the R-points of the circle, the thing a number theorist actually wants. The scheme that looked like an inscrutable space of prime ideals is, viewed through its functor, nothing but the assignment 'here are your solutions over every ring at once', varying functorially as you change R. The geometry IS the family of solution sets.

The theorem that licenses this entire move is Yoneda's lemma: a scheme X is determined, up to unique isomorphism, by its functor R -> X(R) (more precisely by the functor on the category of all schemes). So no information is lost in passing from X to its functor of points — you may DEFINE a scheme by specifying which solution sets it represents, then check the functor is representable. This is how moduli problems are posed: 'the functor sending R to {families of elliptic curves over R}' is a thing you can write down before you know whether a scheme represents it; proving it is representable is proving the moduli space exists.

Separated and proper: the scheme-theoretic Hausdorff and compact

The Zariski topology of a scheme is far too coarse to be Hausdorff — distinct points are rarely separated by disjoint opens, and topological compactness is nearly useless because every scheme's space is quasi-compact for trivial reasons. So the good notions of 'Hausdorff' and 'compact' have to be reinvented relative to the base, as properties of a morphism, and this is one of the cleanest illustrations of why the relative viewpoint earns its keep. The replacements are separated and proper morphisms, the topic that completes this guide and sets up the cohomology of Guide 5.

Separatedness is the analogue of Hausdorff. Recall a topological space is Hausdorff exactly when its diagonal X -> X x X is closed. Copy that verbatim: a morphism X -> S is separated when the diagonal morphism X -> X x_S X is a CLOSED immersion. This is precisely the condition that kills pathologies like the line with a doubled origin — two copies of the affine line glued along everything except the origin. That object is a perfectly good scheme, but its diagonal is not closed: the two origins cannot be told apart by closed conditions, the algebraic echo of a non-Hausdorff space where a sequence has two limits. Separatedness forbids exactly this.

Properness is the analogue of compactness — more precisely of 'compact relative to the base', the way a continuous map is proper when preimages of compact sets are compact. A morphism is proper when it is separated, of finite type, and UNIVERSALLY closed (closed, and stays closed after any base change). The slogan to carry: proper schemes are the ones over which integration and cohomology behave, where global sections are finite-dimensional and Serre duality of Guide 5 holds. Projective schemes — closed subschemes of P^n_S — are always proper, which is why projective varieties are the workhorses; affine space A^n for n > 0 is the opposite, NOT proper, since the projection A^1 -> Spec k is not closed (the image of the hyperbola xy = 1 misses the origin).

  1. Build the local model: a ring R gives the affine scheme (Spec R, O), the atom from Guide 2 — a space of primes with its structure sheaf.
  2. Glue atoms into a scheme: a space X with O_X that is locally Spec R; the projective line P^1 (two affine lines glued by y = 1/x) is the first non-affine example.
  3. Add arrows: morphisms are maps of locally ringed spaces; on affines Hom(Spec A, Spec B) = Hom(B, A), reversed — geometry of affines IS ring theory backwards.
  4. Go relative: work over a base S; the fiber product X x_S Y (Spec of a tensor product on affines) delivers fibers and base change at once.
  5. Switch to solutions: read X through its functor R -> X(R) = Hom(Spec R, X); by Yoneda this loses nothing and turns moduli problems into representability questions.
  6. Impose good behavior: separated (closed diagonal = Hausdorff analogue) and proper (universally closed = compact analogue) are the hypotheses Guide 5's cohomology and Serre duality will require.