04.02.04 · algebraic-geometry / schemes

Morphism of schemes

shipped3 tiersLean: partial

Anchor (Master): Hartshorne §II–§III; Vakil; Grothendieck-Dieudonné EGA I, IV; Mumford *Red Book*

Intuition [Beginner]

A morphism of schemes $f : X \to Y$ is the algebraic-geometric notion of a "map between geometric objects." Where a continuous map between topological spaces is a function preserving open sets, a morphism of schemes is more: it is a continuous map of underlying topological spaces together with a compatible map of structure sheaves, going in the opposite direction. The structure sheaf data records "which functions are regular" on each open set, and a morphism must respect this — pulling back regular functions on $Y$ to regular functions on $X$ .

For affine schemes, morphisms reverse: a morphism $Spec (A) \to Spec (B)$ corresponds to a ring map $B \to A$ . So affine schemes embody the equivalence "rings = geometry, with arrows reversed." More general morphisms are built up from this affine case by gluing.

The richness of scheme theory comes from the properties of morphisms: finite type, separated, proper, flat, smooth, étale, affine, projective, quasi-finite, immersion, and many more. Each property names a precise geometric or algebraic condition, and the systematic classification of morphism properties is a major theme of EGA IV (the longest of Grothendieck's foundational treatises). A morphism of schemes is not just a map; it is a map of a particular type, and the type encodes how it behaves geometrically.

Visual [Beginner]

A morphism between two schemes, with the underlying continuous map of topological spaces and a structure-sheaf map going in the opposite direction.

Worked example [Beginner]

The morphism of affine schemes $f : A_{k}^{1} = Spec (k [x]) \to A_{k}^{1} = Spec (k [y])$ corresponding to $y \mapsto x^{2}$ is the squaring map. The ring map $φ : k [y] \to k [x]$ , $y \mapsto x^{2}$ , induces a map of spectra reversing the direction.

Pointwise: a point of $A_{k}^{1}$ over an algebraically closed $k$ is an element of $k$ . The squaring morphism sends $a \in A_{k}^{1}$ (corresponding to the prime $(x - a) \subset k [x]$ ) to $a^{2} \in A_{k}^{1}$ (corresponding to $(y - a^{2}) \subset k [y]$ ). The preimage of $b \in A_{k}^{1}$ is ${a : a^{2} = b} = {\pm b}$ , two points for $b \neq = 0$ and one (with multiplicity 2) for $b = 0$ .

Another example: the embedding of a closed point $i : Spec (k) ↪ A_{k}^{1} = Spec (k [t])$ corresponding to the quotient $k [t] ↠ k [t] / (t) = k$ . The ring map sends $f (t) \mapsto f (0)$ . The induced morphism is the inclusion of the closed point $0 \in A_{k}^{1}$ into the line.

A non-affine example: the projection $π : P_{k}^{1} \times A_{k}^{1} \to A_{k}^{1}$ from the product onto the second factor. This is not induced by a single ring map — $P_{k}^{1}$ is not affine — but it is a well-defined morphism of schemes, built up from morphisms on affine charts and glued.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X, Y$ be schemes (locally ringed spaces locally isomorphic to spectra). A morphism of schemes $f : X \to Y$ is a morphism of locally ringed spaces:

(M1) A continuous map $f : X \to Y$ of underlying topological spaces.

(M2) A morphism of sheaves of rings $f^{♯} : O_{Y} \to f_{*} O_{X}$ on $Y$ — equivalently, by adjunction, a sheaf-of-rings morphism $f^{- 1} O_{Y} \to O_{X}$ on $X$ .

(M3) The induced map on stalks $f_{x}^{♯} : O_{Y, f (x)} \to O_{X, x}$ is a local ring homomorphism for every $x \in X$ — i.e., it sends the maximal ideal $m_{f (x)} \subseteq O_{Y, f (x)}$ into the maximal ideal $m_{x} \subseteq O_{X, x}$ .

The locality condition (M3) is essential: it is what distinguishes a scheme morphism from an arbitrary ringed-space morphism.

Affine case (anti-equivalence). $Hom_{Sch} (Spec (A), Spec (B)) = Hom_{CRing} (B, A)$ . The functors $Spec : CRing^{op} \to AffSch$ and $Γ : AffSch \to CRing^{op}$ are mutually inverse equivalences. Every morphism of affine schemes is determined by a ring map.

General morphisms via affine cover. A morphism $f : X \to Y$ between general schemes is determined by its restriction to an affine cover ${U_{i}}$ of $X$ and a compatible affine cover ${V_{j}}$ of $Y$ with $f (U_{i}) \subseteq V_{j (i)}$ . The morphism $f ∣_{U_{i}} : U_{i} \to V_{j (i)}$ is then a morphism of affine schemes, given by a ring map $Γ (V_{j (i)}; O_{Y}) \to Γ (U_{i}; O_{X})$ .

Properties of morphisms. Major classes (developed in Hartshorne §II.3 and EGA I, IV):

(P1) Finite type. $f$ is locally of finite type if for every $y \in Y$ there is an affine $V = Spec (B) ∋ y$ in $Y$ such that $f^{- 1} (V)$ admits an affine cover ${U_{i} = Spec (A_{i})}$ with each $A_{i}$ a finitely generated $B$ -algebra. Of finite type if additionally the cover is finite.

(P2) Affine. $f$ is affine if $f^{- 1} (V)$ is affine for every affine open $V \subseteq Y$ .

(P3) Closed / open immersion. $f$ is a closed immersion if it is a homeomorphism onto a closed subset $Z \subseteq Y$ and the induced map $O_{Y} \to f_{*} O_{X}$ is surjective. An open immersion is a homeomorphism onto an open subset $U \subseteq Y$ with $O_{X} ≅ O_{Y} ∣_{U}$ .

(P4) Separated. $f$ is separated if the diagonal morphism $Δ_{f} : X \to X \times_{Y} X$ is a closed immersion.

(P5) Proper. $f$ is proper if it is separated, of finite type, and universally closed (every base change is closed). Equivalently (valuative criterion): for every discrete valuation ring $R$ with field of fractions $K$ , every commutative diagram with $Spec (K) \to X$ and $Spec (R) \to Y$ extends uniquely to $Spec (R) \to X$ .

(P6) Flat. $f$ is flat at $x$ if $O_{X, x}$ is a flat $O_{Y, f (x)}$ -module via $f_{x}^{♯}$ . Flat if flat at every point.

(P7) Smooth. $f$ is smooth of relative dimension $n$ if it is locally of finite presentation, flat, and the geometric fibres are smooth $n$ -dimensional varieties.

(P8) Étale. $f$ is étale if it is smooth of relative dimension 0, equivalently flat and unramified — locally a quasi-finite separable morphism.

(P9) Finite. $f$ is finite if it is affine and for every affine open $V = Spec (B) \subseteq Y$ , $f^{- 1} (V) = Spec (A)$ with $A$ a finite $B$ -algebra (finitely generated as a $B$ -module).

(P10) Quasi-compact. $f$ is quasi-compact if $f^{- 1} (V)$ is quasi-compact for every quasi-compact open $V \subseteq Y$ .

These properties are stable under composition, base change, and locality (with appropriate qualifications). The systematic classification is the central subject of EGA IV.

Functor of points. A scheme $X$ is determined by its functor of points $h_{X} : Sch^{op} \to Set$ , $T \mapsto Hom_{Sch} (T, X)$ . A morphism $f : X \to Y$ is determined by the natural transformation $h_{X} \to h_{Y}$ , $φ \mapsto f \circ φ$ . This functorial perspective (Grothendieck) is often the most natural way to specify and reason about morphisms.

Diagonal and base change. For $f : X \to Y$ and $g : Z \to Y$ , the fibre product $X \times_{Y} Z$ is the scheme satisfying the universal property:

Hom (T, X \times_{Y} Z) = {(φ, ψ) : φ : T \to X, ψ : T \to Z, f \circ φ = g \circ ψ} .

Existence of fibre products in $Sch$ is a foundational theorem (Grothendieck EGA I §3). The diagonal $Δ_{f} : X \to X \times_{Y} X$ , $x \mapsto (x, x)$ , is the morphism testing separatedness.

Key theorem with proof [Intermediate+]

Theorem (anti-equivalence of affine schemes and commutative rings). The functor $Spec : CRing^{op} \to AffSch$ is an anti-equivalence of categories. Equivalently, for commutative rings $A, B$ ,

Hom_{Sch} (Spec (A), Spec (B)) ≅ Hom_{CRing} (B, A) .

Proof. Forward direction. A ring homomorphism $φ : B \to A$ induces a morphism $Spec (φ) : Spec (A) \to Spec (B)$ as follows.

Map of underlying spaces. Send $p \in Spec (A)$ to $φ^{- 1} (p) \in Spec (B)$ . The preimage of a prime under a ring homomorphism is a prime, so this is well-defined. Continuity: the preimage of the basic open $D (b) \subseteq Spec (B)$ is ${p : φ (b) \in / p} = D (φ (b)) \subseteq Spec (A)$ , an open set. So the map is continuous.
Sheaf map $\varphi^\sharp : \mathcal{O}{\mathrm{Spec}(B)} \to \mathrm{Spec}(\varphi) \mathcal{O}_{\mathrm{Spec}(A)} $. * O n t h e ba s i co p e n$ D(b) \subseteq \mathrm{Spec}(B) $,$ \mathcal{O}(D(b)) = B[b^{-1}] $, an d$ \mathrm{Spec}(\varphi)^{-1}(D(b)) = D(\varphi(b)) $ha s$ \mathcal{O}(D(\varphi(b))) = A[\varphi(b)^{-1}] $. T h er in g ma p$ \varphi $e x t e n d s t o a l oc a l i se d ma p$ B[b^{-1}] \to A[\varphi(b)^{-1}] $,$ b'/b^n \mapsto \varphi(b')/\varphi(b)^n $. T hi s d e f in es$ \varphi^\sharp$ on basic opens; it extends to a sheaf map on all opens.
Locality on stalks. At $p \in Spec (A)$ , the stalk map $φ_{p}^{♯} : B_{φ^{- 1} (p)} \to A_{p}$ sends $b / g \mapsto φ (b) / φ (g)$ . The maximal ideal $φ^{- 1} (p) B_{φ^{- 1} (p)}$ goes to $φ (φ^{- 1} (p)) A_{p} \subseteq p A_{p}$ (since $φ (φ^{- 1} (p)) \subseteq p$ ). So $φ_{p}^{♯}$ is local.

This shows $Spec (φ) : Spec (A) \to Spec (B)$ is a morphism of schemes.

Reverse direction (faithful). Two distinct ring maps $φ, ψ : B \to A$ give distinct $Spec$ morphisms. If $φ (b) \neq = ψ (b)$ for some $b \in B$ , then $Spec (φ)^{♯}$ and $Spec (ψ)^{♯}$ act differently on $b \in O (D (1)) = B$ , hence are different sheaf morphisms.

Reverse direction (full). Given a morphism $f : Spec (A) \to Spec (B)$ , recover a ring map. Let $φ := f^{♯} (Spec (B)) : B = O_{Spec (B)} (Spec (B)) \to O_{Spec (A)} (f^{- 1} (Spec (B))) = A$ — using that global sections of the structure sheaf on $Spec (R)$ recover $R$ .

This $φ : B \to A$ satisfies $f = Spec (φ)$ as a morphism of schemes: the underlying continuous map matches because the locality condition (M3) on stalks forces $f^{- 1} (q) = φ^{- 1} (q)$ for every $q \in Spec (B)$ (the local ring map respects maximal ideals); the sheaf maps match because both equal the localised version of $φ$ on basic opens.

So $Spec : CRing^{op} \to AffSch$ is fully faithful. Together with essential surjectivity (every affine scheme is by definition a $Spec$ ), this is the anti-equivalence. $□$

This anti-equivalence — commutative algebra dual to affine algebraic geometry — is the foundational theorem of scheme theory and the basis for every algebraic-geometric calculation. Every property of affine morphisms translates to a property of ring maps, and vice versa. This dictionary is what makes scheme theory powerful.

Bridge. The construction here builds toward later units of the strand, where the same pattern is taken up at higher structure. 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 1 (easy, proof).

Show that the composition of two morphisms of schemes is a morphism of schemes.

Hint

Composition of continuous maps is continuous; composition of sheaf maps is functorial; locality on stalks is preserved.

Answer

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. The composition $g \circ f : X \to Z$ has:

Continuous map of underlying spaces: composition of two continuous maps.
Sheaf map: the composition $O_{Z} g^{♯} g_{*} O_{Y} g_{*} f^{♯} g_{*} f_{*} O_{X} = (g \circ f)_{*} O_{X}$ — using $(g \circ f)_{*} = g_{*} \circ f_{*}$ .
Locality on stalks: the stalk map at $x \in X$ is $O_{Z, g (f (x))} \to O_{Y, f (x)} \to O_{X, x}$ , the composition of two local ring maps. The composition of local ring maps is local: $m_{g (f (x))} \mapsto m_{f (x)} \cdot O_{Y, f (x)} \mapsto m_{x} \cdot O_{X, x}$ , all containments preserved.

So $g \circ f : X \to Z$ is a morphism of schemes. The category $Sch$ has compositions, identities, and associativity — it is an honest category. $□$

Exercise 3 (medium, proof).

Show that a morphism $f : X \to Y$ is an isomorphism iff it is bijective on underlying spaces and induces isomorphisms on all stalks.

Hint

The local-ring stalk isomorphisms recover the sheaf isomorphism on opens.

Answer

$(\Rightarrow)$ An isomorphism of schemes is a homeomorphism on underlying spaces (with continuous inverse), and the inverse sheaf-of-rings map gives stalk inverses, hence stalk isomorphisms.

$(\Leftarrow)$ Suppose $f$ is bijective on underlying spaces with continuous inverse $f^{- 1}$ (a homeomorphism), and the stalk maps $f_{x}^{♯} : O_{Y, f (x)} \to O_{X, x}$ are all isomorphisms.

By stalkwise determination of sheaf isomorphisms (a sheaf morphism inducing isomorphisms on all stalks is an isomorphism), the sheaf map $f^{♯} : O_{Y} \to f_{*} O_{X}$ is an isomorphism.

The inverse $g := f^{- 1}$ on underlying spaces, equipped with the inverse sheaf map $g^{♯} = (f^{♯})^{- 1}$ , gives a morphism $g : Y \to X$ of locally ringed spaces with $g \circ f = id_{X}$ and $f \circ g = id_{Y}$ . Locality of $g^{♯}$ on stalks follows from locality of $f^{♯}$ (inverse of a local ring iso is local).

So $f$ is an isomorphism in $Sch$ . $□$

This is the foundational stalkwise criterion for isomorphism: a morphism is an isomorphism iff it is a homeomorphism with stalk-isomorphisms. This generalises to arbitrary properties detected stalkwise (flat, smooth, étale all admit such characterisations).

Exercise 4 (medium, numeric).

Compute the fibre product $A_{k}^{1} \times_{A_{k}^{1}} A_{k}^{1}$ where the two maps to $A_{k}^{1}$ are $x \mapsto x^{2}$ (squaring).

Hint

Fibre product of affine schemes corresponds to tensor product of rings.

Answer

The fibre product corresponds to the tensor product of rings. With $A_{k}^{1} = Spec (k [x])$ and both squaring maps $φ : k [y] \to k [x]$ , $y \mapsto x^{2}$ :

A_{k}^{1} \times_{A_{k}^{1}} A_{k}^{1} = Spec (k [x_{1}] \otimes_{k [y]} k [x_{2}]) = Spec (\frac{k [ x _{1} , x _{2} ]}{( x _{1}^{2} - x _{2}^{2} )}) .

The relation $x_{1}^{2} = x_{2}^{2}$ factors as $(x_{1} - x_{2}) (x_{1} + x_{2}) = 0$ . So

Spec (\frac{k [ x _{1} , x _{2} ]}{( x _{1}^{2} - x _{2}^{2} )}) = Spec (\frac{k [ x _{1} , x _{2} ]}{( x _{1} - x _{2} )}) \cup Spec (\frac{k [ x _{1} , x _{2} ]}{( x _{1} + x _{2} )}) = A_{k}^{1} ⊔_{{0}} A_{k}^{1},

two copies of $A_{k}^{1}$ glued along the origin (where $x_{1} = x_{2} = 0$ ).

Geometrically: the fibre product detects pairs $(a, b)$ with $a^{2} = b^{2}$ , equivalently $a = \pm b$ . The two components correspond to the two cases. The intersection is the origin.

If $char k = 2$ , the analysis differs: $x_{1}^{2} - x_{2}^{2} = (x_{1} - x_{2})^{2}$ , giving a non-reduced scheme $Spec (k [x_{1}, x_{2}] / ((x_{1} - x_{2})^{2}))$ — a thickened diagonal, supported on the line $x_{1} = x_{2}$ but with nilpotent structure. This non-reducedness encodes the failure of the squaring map to be étale in characteristic 2.

Exercise 5 (medium, conceptual).

Why is the locality condition (M3) on stalk maps essential for the definition of a morphism of schemes?

Hint

Without locality, there are too many ringed-space morphisms, including some that don't correspond to ring maps for affine schemes.

Answer

Without locality, ringed-space morphisms $X \to Y$ would include any sheaf-of-rings map $O_{Y} \to f_{*} O_{X}$ , regardless of behaviour on stalks. Two issues:

Loss of the affine anti-equivalence. Consider $f : Spec (A) \to Spec (B)$ in the ringed-space category. If $A$ has more than one prime ideal, there are non-local ring maps $B \to A_{p}$ for various $p$ , and these would correspond to extra morphisms not coming from ring maps $B \to A$ . The bijection $Hom_{Sch} (Spec (A), Spec (B)) = Hom_{CRing} (B, A)$ would fail.
Loss of geometric meaning. The point $f (x) \in Y$ should be the "image of $x$ under $f$ " — geometrically, the prime of $O_{Y}$ that corresponds to $x$ . The locality condition $f_{x}^{♯} (m_{f (x)}) \subseteq m_{x}$ encodes this: a function vanishing at $f (x)$ pulls back to a function vanishing at $x$ . Without locality, there's no consistent "image of a point" notion.

For schemes, the correct definition is morphism of locally ringed spaces: continuous map plus sheaf-of-rings map plus locality on stalks. This restores the affine anti-equivalence and gives the geometric notion of "scheme morphism."

The category of locally ringed spaces is itself larger than the category of schemes (general LRS need not be locally affine), but locality is preserved by all natural operations. The category of schemes is a full subcategory of LRS, and the morphism notion is inherited.

Exercise 6 (hard, proof).

State the valuative criterion for separatedness and outline a proof.

Hint

A morphism is separated iff every diagram with a DVR has at most one extension.

Answer

Valuative criterion for separatedness (EGA II §7). A morphism $f : X \to Y$ of finite type between Noetherian schemes is separated iff for every discrete valuation ring $R$ with field of fractions $K$ and every commutative diagram

Spec (K) u X

↓_{η} ↓_{f}

Spec (R) v Y

there is at most one morphism $Spec (R) \to X$ extending $u$ (i.e., making both triangles commute).

Outline of proof.

$(\Rightarrow)$ Suppose $f$ is separated. Two extensions $u_{1}, u_{2} : Spec (R) \to X$ define a morphism $u_{1} \times u_{2} : Spec (R) \to X \times_{Y} X$ . The two extensions agree at the generic point $Spec (K) \subset Spec (R)$ , so the image at $K$ lies in the diagonal $Δ (X) \subset X \times_{Y} X$ .

The diagonal is closed (separatedness). Since $Spec (R)$ is integral and the image of the generic point lies in a closed subscheme, the entire image lies in the closed subscheme. So the image of $u_{1} \times u_{2}$ is in $Δ (X)$ , meaning $u_{1} = u_{2}$ as morphisms.

$(\Leftarrow)$ Suppose the valuative property holds. We show $Δ : X \to X \times_{Y} X$ is closed. The image of $Δ$ is a constructible subset (general theorem of Chevalley). To show closedness, by the constructibility, it suffices to show that for every irreducible component $Z \subseteq \overline{Δ (X)}$ of the closure, the generic point of $Z$ is in $Δ (X)$ .

Pull back to a DVR test by taking the local ring at the generic point of $Z$ and resolving its singularity by a DVR (Noether normalisation / Krull-Akizuki). The valuative criterion gives uniqueness of an extension, which translates to the generic point of $Z$ lying in $Δ (X)$ .

So $Δ (X)$ is closed, $Δ$ is a closed immersion, and $f$ is separated. $□$

There is also a valuative criterion for properness, replacing "at most one" with "exactly one" extension. Properness is separatedness + universally closed + finite type, and the valuative criterion captures all three together: a unique extension over every DVR.

These valuative criteria are foundational tools throughout algebraic and arithmetic geometry: they reduce checking properties of morphisms to a single test (extensions over DVRs), much easier than checking universal closedness directly.

Exercise 7 (hard, conceptual).

Explain the systematic perspective of EGA IV on properties of morphisms.

Hint

EGA IV classifies properties by their stability under composition, base change, and local properties, organising the properties into a coherent framework.

Answer

EGA IV (1965–67), the longest of Grothendieck's Éléments de Géométrie Algébrique, is a systematic study of properties of morphisms. The structure:

EGA IV-1 (1964): Local properties. Locally Noetherian, locally of finite type, locally of finite presentation, etc. — properties characterised stalkwise or on affine charts. Catenary, universally catenary, regular, Cohen-Macaulay.

EGA IV-2 (1965): Étale and smooth morphisms. Definitions, infinitesimal lifting criteria, formal smoothness/étaleness/unramifiedness. Foundational for étale cohomology.

EGA IV-3 (1966): Flat morphisms and going-down. Flatness, faithfully flat descent, flat dimension, openness of flat locus, fibrewise criteria for flatness.

EGA IV-4 (1967): Various properties. Quasi-finite, finite presentation, unramified, étale (advanced), Zariski's main theorem, geometrically connected/integral/reduced fibres.

The systematic perspective. Each property $P$ is studied for:

Definition. Precise condition (often local on source / target / both).
Stability under composition. Is $g \circ f$ in $P$ when $f, g$ are? Many properties (finite type, finite, separated, proper, flat, smooth, étale, finite presentation) are stable under composition.
Stability under base change. Is $f \times_{Y} T : X \times_{Y} T \to T$ in $P$ when $f$ is, for any base change $T \to Y$ ? Most properties are.
Locality on the source / target / fppf-cover. Can $P$ be checked on an open cover of $X$ , or an open cover of $Y$ , or after a faithfully flat base change? Different properties have different locality behaviour.
Cancellation. If $g \circ f \in P$ and $g \in P^{'}$ , is $f \in P^{''}$ ? Cancellation theorems organise the properties.
Permanence under fibre product / equaliser. Properties of the components, properties of the constructed scheme.
Spreading-out. For a morphism over a "limit" scheme, when can one find the same property over an "approximating" scheme? (Foundational for moduli problems and arithmetic.)

This systematic classification is one of the great achievements of 20th-century mathematics. It transforms the study of morphisms from ad-hoc into a unified theory. Subsequent work — Deligne's six-functor formalism, Faltings's p-adic Hodge theory, Scholze's perfectoid spaces — builds on the EGA IV framework, often by extending it to new geometric settings (étale topology, fppf topology, perfectoid topology).

The "systematic perspective" is best summarised: every reasonable property of morphisms admits a clean statement, criterion, and stability theory, organised into a consistent framework. Working algebraic geometers consult EGA IV daily because of this systematic completeness.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has AlgebraicGeometry.Scheme.Hom, the affine-Spec adjunction, and a substantial library of properties of morphisms (closed immersion, finite, proper, separated, ...).

[object Promise]

Advanced results [Master]

Anti-equivalence and gluing. $CRing^{op} ≅ AffSch$ as full subcategory of $Sch$ . General schemes are obtained by gluing affines along open subschemes; morphisms of schemes are obtained by gluing morphisms of affines along compatible open subschemes. This is the local-to-global principle for the category of schemes.

Functor of points and Yoneda. A scheme $X$ is determined by its functor of points $h_{X} (T) = Hom (T, X)$ . Conversely, a representable functor $Sch^{op} \to Set$ comes from a unique scheme. This functorial perspective is foundational for moduli theory, where many "moduli functors" have representable solutions (Hilbert schemes, Picard schemes, moduli of curves under additional conditions).

Properties stable under base change. Properties stable under arbitrary base change include: finite type, finite presentation, separated, proper, flat, smooth, étale, finite, immersion, affine, projective. Properties NOT stable under arbitrary base change but stable under flat base change (or equivalent): integral fibres, geometric connectedness. These finer behaviours are systematically classified in EGA IV.

Faithfully flat descent. A morphism $f : X \to Y$ that is faithfully flat (flat plus surjective) and quasi-compact is an effective epimorphism in $Sch$ : properties of $Y$ can often be detected after base change to $X$ . The classical statement: a quasi-coherent module on $Y$ is uniquely determined (with descent data) by its pullback to $X$ . Grothendieck's faithfully flat descent (FGA exposé 190, 1959) is one of the most powerful tools in algebraic geometry.

Six-functor formalism for morphisms. Direct image $f_{*}$ , inverse image $f^{*}$ , proper-pushforward $f_{!}$ , exceptional inverse $f^{!}$ on derived categories of (quasi-)coherent sheaves. Verdier duality gives $Hom (R f_{!} F, G) = Hom (F, f^{!} G)$ — the algebraic-geometric incarnation of Poincaré-Serre duality.

Smooth and étale: deformation-theoretic characterisations. $f : X \to Y$ is smooth iff every infinitesimal extension lifts uniquely (the formal smoothness criterion). Étale iff smooth of relative dimension 0. The deformation-theoretic characterisations connect EGA IV to the cotangent complex and modern derived deformation theory.

Stable algebraic geometry. Higher categorical morphisms — morphisms of derived schemes, $\infty$ -stacks, and prismatic schemes — extend the EGA IV framework. Lurie's Spectral Algebraic Geometry formalises the $\infty$ -categorical version, with derived versions of finite type / smooth / étale, and a full six-functor formalism in the derived setting.

Galois descent and Galois groups. A finite étale morphism $f : X \to Y$ corresponds (when $Y$ is connected with a geometric point $\overset{y}{ˉ}$ ) to a continuous action of the étale fundamental group $π_{1}^{\overset{e}{ˊ} t} (Y, \overset{y}{ˉ})$ on the fibre $f^{- 1} (\overset{y}{ˉ})$ — Grothendieck's Galois theory of schemes (SGA 1). This unifies classical Galois theory of fields with topological covering theory of spaces.

Synthesis. This construction generalises the pattern fixed in 04.02.01 (scheme), 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]

Detailed proofs of: the affine anti-equivalence (sketched in the key-theorem section); existence of fibre products in $Sch$ ; valuative criteria for separatedness and properness; faithfully flat descent; the deformation-theoretic characterisations of smooth and étale; six-functor formalism for morphisms — these are deferred to companion units in the schemes, derived-categories, and étale-cohomology strands. The basic anti-equivalence is proved in the key-theorem section.

Connections [Master]

Scheme 04.02.01 — schemes are the objects; morphisms are the arrows.
Affine scheme 04.02.02 — affine morphisms correspond to ring maps via $Spec /Γ$ anti-equivalence.
Projective scheme 04.02.03 — projective morphisms admit closed embeddings into projective bundles; specialisations of EGA IV properties.
Sheaf 04.01.01 — morphisms of schemes carry a sheaf-of-rings component; the locality condition makes them well-behaved.
Direct and inverse image 04.01.04 — every morphism induces $f_{*}, f^{*}$ on sheaves.
Quasi-coherent sheaf 04.06.01 — quasi-coherence is preserved by $f^{*}$ always; by $f_{*}$ for quasi-compact-quasi-separated morphisms.
Coherent sheaf 04.06.02 — coherence is preserved by $f^{*}$ for finite-type morphisms; by $f_{*}$ for proper morphisms (Grothendieck's finiteness theorem, EGA III).
Riemann-Roch theorem for curves 04.04.01 — Riemann-Roch is a statement about morphisms to a point and properties of the corresponding pushforward.

Historical & philosophical context [Master]

The notion of morphism of schemes was formalised by Alexander Grothendieck and Jean Dieudonné in Éléments de Géométrie Algébrique I (EGA I, 1960), specifically §1 (definitions) and §6 (separated and proper morphisms). Grothendieck's foundational insight was that algebraic geometry should be the study of schemes and their morphisms, with morphisms being the maps of locally ringed spaces respecting the local-ring structure. This was a categorical reformulation of the classical varieties-and-rational-maps language, and it had transformative consequences.

Before Grothendieck, algebraic geometers worked primarily with varieties (irreducible reduced schemes of finite type over a field) and rational maps (partial maps defined on dense open subsets). The pre-Grothendieck framework — Weil's Foundations of Algebraic Geometry (1946), van der Waerden, Severi — used a complicated mixture of birational geometry, intersection theory, and ad-hoc constructions. The functorial language was missing, the relative perspective was missing, and many natural questions could not be cleanly stated.

Grothendieck's reformulation in EGA I §1 introduced morphisms of schemes as morphisms of locally ringed spaces (the locality condition on stalks being the essential addition over arbitrary ringed-space morphisms). The anti-equivalence $CRing^{op} ≅ AffSch$ was the foundational theorem, and general schemes were built by gluing affines via morphisms. This made every commutative ring — Noetherian or not, integral or not, of arbitrary characteristic — an algebraic-geometric object, and every morphism of rings a corresponding morphism of schemes.

The systematic theory of properties of morphisms was then developed in EGA II–IV (1961–67). EGA II (1961) covered ample line bundles and projective morphisms. EGA III (1961–63) covered cohomology of coherent sheaves, including Grothendieck's finiteness theorem for proper morphisms. EGA IV (1965–67), in four parts, was a tour-de-force classification of morphism properties: locally Noetherian, locally of finite type, locally of finite presentation, étale, smooth, flat, faithfully flat, finite, quasi-finite, separated, proper, immersion, finite presentation. Each property received the full systematic treatment: definition, stability under composition / base change / locality, criteria, examples, counterexamples, fibre-wise behaviour. The total length of EGA IV approaches 2000 pages, and it remains the canonical reference for the algebraic-geometric working mathematician.

The conceptual achievement of EGA I–IV was the relative perspective: algebraic geometry is fundamentally relative, with every scheme being a scheme over some base $S$ , and every property of schemes generalising to a property of morphisms. This perspective enabled enormous generalisations: the étale fundamental group (SGA 1), étale cohomology (SGA 4), the proof of the Weil conjectures (Deligne, 1973–74), the finiteness theorems for moduli (Faltings, Mochizuki), and modern arithmetic geometry (perfectoid spaces, Scholze 2012; the geometrisation of local Langlands, Fargues-Scholze 2021). The relative-morphism perspective also extended into derived algebraic geometry (Lurie, Töen-Vezzosi), where morphisms of derived schemes carry homotopical data that recovers cohomology and obstruction theory naturally.

Pierre Deligne, Luc Illusie, Michel Raynaud, and the entire Bourbaki community contributed to extending and applying the EGA I–IV framework. David Mumford's The Red Book of Varieties and Schemes (Lectures on Curves on an Algebraic Surface, 1966; expanded into the Red Book in 1988) provides the most accessible introduction to scheme-theoretic morphisms and is the canonical pedagogical reference. Robin Hartshorne's Algebraic Geometry (1977) gives the standard textbook treatment, and Ravi Vakil's The Rising Sea: Foundations of Algebraic Geometry (the contemporary canonical online textbook) presents the full modern perspective.

In contemporary research, morphisms of schemes remain central: in arithmetic (heights, equidistribution, Mordell-Weil, BSD), in derived algebraic geometry (Lurie's Spectral Algebraic Geometry), in $p$ -adic Hodge theory (perfectoid morphisms, prismatic morphisms), in moduli theory (representability of moduli functors, Artin's representability theorem), and in the homotopical geometry of $\infty$ -stacks. The single concept — a morphism of locally ringed spaces respecting local rings — has thus traversed an enormous landscape, from Grothendieck's 1960 definition through 70 years of foundational and applied work.

Bibliography [Master]

Hartshorne, Algebraic Geometry — §II.3 covers morphisms of schemes.
Vakil, The Rising Sea: Foundations of Algebraic Geometry — §6–§9 (morphisms and their properties).
Grothendieck-Dieudonné, Éléments de Géométrie Algébrique I (EGA I, 1960) — §1 and §6 introduce morphisms of schemes.
Grothendieck-Dieudonné, EGA IV (1965–67) — the systematic classification of morphism properties (~2000 pages).
Mumford, The Red Book of Varieties and Schemes — Ch. II covers morphisms.
Eisenbud-Harris, The Geometry of Schemes — geometric introduction with attention to morphisms.
Liu, Algebraic Geometry and Arithmetic Curves — morphisms with arithmetic emphasis.
Görtz-Wedhorn, Algebraic Geometry I — comprehensive modern treatment of morphisms.
Deligne, La conjecture de Weil II (1980) — applications of EGA IV in proof of the Weil conjectures.
Lurie, Spectral Algebraic Geometry — derived / $\infty$ -categorical morphisms of derived schemes.
Stacks Project — (https://stacks.math.columbia.edu) — exhaustive online reference for properties of morphisms.

Prerequisites

04.02.01
04.02.02

Tier anchors

beginner: Strogatz analogous level for ring-theoretic reversal of geometric maps
intermediate: Hartshorne §II.3; Vakil §6–§9
master: Hartshorne §II–§III; Vakil; Grothendieck-Dieudonné EGA I, IV; Mumford *Red Book*

References

TODO_REF
Hartshorne — Algebraic Geometry · §II.3, morphisms of schemes
TODO_REF
Vakil — The Rising Sea · §6–§9, morphisms and properties
TODO_REF
Grothendieck-Dieudonné — EGA I · §1, morphisms of schemes; §6, separated and proper morphisms
TODO_REF
Grothendieck-Dieudonné — EGA IV · 1965–67, properties of morphisms (smooth, étale, flat, finite type)
TODO_REF
Mumford — The Red Book of Varieties and Schemes · Ch. II, morphisms

Lean module

Codex.AlgebraicGeometry.Schemes.Morphism

Reviewer

TBD

Estimated time

beginner: 12m
intermediate: 40m
master: 55m