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Fiber Product of Schemes

The universal scheme of pairs of maps compatible over a common base, realized affinely by a tensor product and serving as the engine of base change, scheme-theoretic fibers, and intersections.

Version
v3 · 2026-09-06 · History
Domain-specific #
1831
Origin domain
mathematics
Subdomain
algebraic geometry
Aliases
Fibre product of schemes, Fibred product of schemes, Scheme-theoretic fiber product

Core Idea

Given two morphisms of schemes with a common target,

\[ f:X\longrightarrow S, \qquad g:Y\longrightarrow S, \]

their fiber product is a scheme \(X\times_S Y\) equipped with projection morphisms \(p:X\times_S Y\to X\) and \(q:X\times_S Y\to Y\) such that \(f\circ p=g\circ q\), and universal among all schemes carrying such a compatible pair of maps. Concretely, for every scheme \(T\), composition with the projections gives a natural bijection

\[ \operatorname{Hom}(T,X\times_S Y) \cong \operatorname{Hom}(T,X) \times_{\operatorname{Hom}(T,S)} \operatorname{Hom}(T,Y). \]

Thus a map from \(T\) into the fiber product is exactly a pair of maps \(T\to X\) and \(T\to Y\) whose composites to \(S\) agree. The unique-mediating-map clause determines the result up to unique isomorphism. The Stacks Project proves that the category of schemes has fiber products—indeed all finite limits—and states this universal property directly.[1][2]

The construction has an algebraic engine. If

\[ X=\operatorname{Spec}(A),\quad Y=\operatorname{Spec}(B),\quad S=\operatorname{Spec}(R), \]

with the scheme maps induced contravariantly by ring maps \(R\to A\) and \(R\to B\), then

\[ X\times_S Y\cong\operatorname{Spec}(A\otimes_R B). \]

The tensor product is a pushout of rings, but \(\operatorname{Spec}\) reverses arrows, so it becomes a pullback of affine schemes. General fiber products are obtained by gluing these affine pieces.[2] This affine formula is not merely a convenient implementation: it explains why the result can retain nilpotents, residue-field extensions, and intersection structure invisible in a set of compatible topological points.

The locked identity is two schemes mapped to a common base + a commutative projection square + universal representation of compatible test-scheme maps + affine tensor-product realization + gluing -> the scheme-theoretic pullback. Base extension, the fiber of a morphism over a point, scheme-theoretic inverse image, intersection of closed subschemes, and the domain of a diagonal morphism are recurring special cases. A general categorical pullback supplies the skeleton; Fiber Product of Schemes is the scheme-theoretic realization with its own local algebra and geometric consequences.

Structural Signature

  • the common base scheme — \(S\), the object over which compatibility is measured rather than an ignorable ambient label;
  • the two structure morphisms — \(f:X\to S\) and \(g:Y\to S\), fixing how each input depends on the base;
  • the pullback scheme — \(P=X\times_S Y\), required to exist in the category of schemes;
  • the two projections — \(p:P\to X\) and \(q:P\to Y\);
  • the commutativity invariant — \(f\circ p=g\circ q\), so both projections describe the same base datum;
  • the test scheme — an arbitrary scheme \(T\) with maps \(a:T\to X\) and \(b:T\to Y\);
  • the compatibility predicate — \(f\circ a=g\circ b\);
  • the unique mediating morphism — exactly one \(u:T\to P\) satisfies \(p\circ u=a\) and \(q\circ u=b\);
  • the up-to-unique-isomorphism guarantee — any two schemes satisfying the same square and universal clause are canonically isomorphic;
  • the affine chart engine — \(\operatorname{Spec}(A)\times_{\operatorname{Spec}(R)}\operatorname{Spec}(B)=\operatorname{Spec}(A\otimes_R B)\);
  • the gluing step — compatible affine fiber products cover and assemble the global scheme;
  • the residue-field point data — points above \((x,y)\) over \(s\) are controlled by prime ideals of \(\kappa(x)\otimes_{\kappa(s)}\kappa(y)\), so the topological carrier is not generally a naive set fiber product;[2]
  • the base-change projection — viewing \(P\to Y\) as the pullback of \(X\to S\) along \(Y\to S\);
  • the scheme-structure payload — functions, nilpotents, residue fields, and infinitesimal intersection data retained by the tensor product and structure sheaf.

Recognition test. A construction qualifies only if it starts from two scheme morphisms into one specified base, produces a scheme with the commuting projections, satisfies the universal condition for every test scheme, and locally agrees with the tensor-product formula. A set of point pairs, a Cartesian product with no shared-base constraint, or an arbitrary commutative square is insufficient.

What It Is Not

  • Not an unrestricted Cartesian product. \(X\times Y\) over a terminal object combines independent inputs. \(X\times_S Y\) retains only maps compatible over \(S\), and its structure depends on both maps to the base.
  • Not the set-theoretic pullback of underlying topological spaces. A compatible pair of points can lift to several points or carry extra residue-field data. Stacks describes points using primes of \(\kappa(x)\otimes_{\kappa(s)}\kappa(y)\), not just pairs \((x,y)\).[2]
  • Not merely the tensor product. \(A\otimes_R B\) is the affine coordinate ring. The scheme is its spectrum, and a nonaffine fiber product is glued from many such spectra.
  • Not any commutative square. Commutativity is necessary but not sufficient. The square must be terminal among compatible cones: every other compatible pair factors through it uniquely.
  • Not a single fiber. The scheme-theoretic fiber \(X_s=X\times_S\operatorname{Spec}\kappa(s)\) is one special case obtained by choosing a point of the base.[3]
  • Not synonymous with base change. Base change is the use of a fiber product to transport a scheme or morphism along \(S'\to S\). The fiber product is the underlying universal construction and has other uses.
  • Not automatically a reduced intersection. A fiber product of closed subschemes is their scheme-theoretic intersection and can retain nilpotents at tangencies or embedded structure.[4]
  • Not the derived fiber product. The ordinary scheme fiber product uses \(A\otimes_R B\). Derived algebraic geometry replaces it by a derived tensor product to retain higher \(\operatorname{Tor}\) information; that refinement is outside this node.
  • Not descent. Pulling an object to a cover is base change. Reconstructing it from compatible local data is descent and needs effectivity and cover hypotheses.

Scope of Application

Fiber products are the working syntax of relative algebraic geometry. A scheme “over \(S\)” is a scheme with a structure morphism to \(S\); changing the base from \(S\) to \(S'\) means forming \(X_{S'}=S'\times_S X\). The Stacks Project treats this as the precise language for studying relative properties and families.[3] Field extension is the familiar instance: for \(k\subset K\), a \(k\)-scheme \(X\) becomes the \(K\)-scheme \(X_K=X\times_{\operatorname{Spec}k}\operatorname{Spec}K\).

Fibers of families use the same operation. For \(f:X\to S\) and \(s\in S\), the scheme-theoretic fiber is \(X_s=X\times_S\operatorname{Spec}\kappa(s)\). Its local rings are obtained by tensoring with the residue field, so it records more than a raw inverse-image set.[3] This makes a morphism of schemes a disciplined version of a parameterized family: changing \(s\) changes the geometric member.

Closed subschemes \(Z,V\subset X\) intersect by a fiber product \(Z\times_X V\). If affine-locally they are cut out by ideals \(I,J\subset A\), their intersection is \(\operatorname{Spec}(A/(I+J))\), because \((A/I)\otimes_A(A/J)\cong A/(I+J)\).[4] Inverse images of closed subschemes, equalizers, diagonals, and tests for separatedness are built from the same squares.[5]

Properties of morphisms are routinely tested for stability under base change. Closed and open immersions, smooth morphisms, and proper morphisms are among the standard stable classes.[3][6][7] This stability makes the construction central to moduli problems, families of varieties, relative differentials, algebraic groups, descent theory, and algebraic stacks.

The scope boundary is exact. A pullback in sets, groups, topological spaces, or an arbitrary category shares the universal-property skeleton but is not a Fiber Product of Schemes unless the objects and arrows are schemes and scheme morphisms and the affine tensor/gluing semantics apply.

Clarity

A reliable four-part diagnostic separates the construction from its neighbors.

  1. Write the base explicitly. Record \(X\to S\leftarrow Y\). Omitting \(S\) hides the compatibility relation and can turn a relative product into an apparently absolute one.
  2. State the universal property. A test scheme \(T\) maps to the result exactly when it maps compatibly to both inputs. If only a concrete set or ring has been written, universality has not yet been verified.
  3. Check one affine chart. For \(S=\operatorname{Spec}R\), \(X=\operatorname{Spec}A\), \(Y=\operatorname{Spec}B\), compute \(A\otimes_R B\). This catches variance errors and exposes nilpotents or splitting.
  4. Keep scheme points distinct from rational points. For any test scheme \(T\), the Hom-set formula is exact. A simple pair-of-points description is safe for \(T\)-valued points, but the underlying topological points require residue-field tensor products.

For example,

\[ \operatorname{Spec}\mathbb C\times_{\operatorname{Spec}\mathbb R}\operatorname{Spec}\mathbb C \cong \operatorname{Spec}(\mathbb C\otimes_{\mathbb R}\mathbb C) \cong \operatorname{Spec}(\mathbb C\times\mathbb C). \]

The two input topological spaces and the base each have one point, so their set-theoretic fiber product has one pair; the scheme fiber product has two points. This makes the “not just compatible underlying points” boundary concrete.

Manages Complexity

Without fiber products, base extension, fibers, inverse images, intersections, equalizers, and comparison of families would look like unrelated constructions, each needing independent definitions and proofs. Fiber Product of Schemes compresses them into one universal square. The analyst tracks only the common base, two input maps, compatibility, and the unique mediator; the special-case vocabulary then says what the square is being used for.

The construction also turns global geometry into local algebra. Cover the base and inputs by compatible affine opens, replace the geometric square by \(A\otimes_R B\), compute there, and glue the results.[2] This is a large reduction in dimensionality: sheaves and spaces are handled by a standard ring operation on charts, while the universal property guarantees that the pieces assemble into the right global object independent of the chosen cover.

Base-change stability reduces proof duplication. Once a property \(P\) is proved stable under pullback, every square transports \(P\) automatically from \(X\to S\) to \(X\times_S S'\to S'\). Smoothness and properness are canonical examples.[6][7] The fiber product thereby becomes a proof engine: instead of reconstructing a morphism after every change of coordinates, field, or parameter space, one invokes the universal construction and a stability theorem.

Finally, the functor-of-points formula suppresses construction detail. To define or compare a map into \(X\times_S Y\), it suffices to give compatible maps into \(X\) and \(Y\). Existence and uniqueness arrive together. That mechanism is indispensable in moduli theory, where objects are tested against all schemes and where a representing object must encode compatible families rather than merely collect geometric points.

Abstract Reasoning

The universal square licenses several dependable moves.

Map construction. To construct \(T\to X\times_S Y\), construct \(a:T\to X\) and \(b:T\to Y\), prove \(f a=g b\), and invoke unique mediation. To prove two maps into the fiber product equal, it is enough to show both composites with \(p\) and \(q\) agree.

Affine computation. Translate \(R\to A\) and \(R\to B\) into \(A\otimes_R B\). For example, maps \(k[t]\to k[x]\), \(t\mapsto x^2\), and \(k[t]\to k[y]\), \(t\mapsto y^3\), give

\[ \operatorname{Spec}k[x]\times_{\operatorname{Spec}k[t]}\operatorname{Spec}k[y] \cong \operatorname{Spec}\bigl(k[x,y]/(x^2-y^3)\bigr). \]

The compatibility equation is literally imposed in the coordinate ring.

Iterated base change. Universal properties yield canonical isomorphisms

\[ (X\times_S S')\times_{S'}S''\cong X\times_S S'' \]

for \(S''\to S'\to S\). This is associativity up to canonical isomorphism, not literal equality of chosen constructions.

Property transport. If \(P\) is stable under base change and \(X\to S\) has \(P\), then the projection \(X\times_S Y\to Y\) has \(P\). The reverse inference is invalid without a descent or faithful-cover theorem; base change along the empty scheme can erase all counterevidence.

Fiber extraction. Substitute \(Y=\operatorname{Spec}\kappa(s)\) to inspect one parameter value. For an affine family \(X=\operatorname{Spec}A\to\operatorname{Spec}R\), the fiber is \(\operatorname{Spec}(A\otimes_R\kappa(s))\), equivalent to specializing base parameters while preserving scheme structure.

Intersection and equalizer. Pull back two closed immersions to form a scheme-theoretic intersection. Pull back a diagonal along \((a,b):T\to Y\times_S Y\) to obtain the locus where two maps agree.[5] These are consequences of one pullback calculus rather than independent tricks.

Knowledge Transfer

Within algebraic geometry the mechanism transfers without metaphor. The identical role structure appears in scalar extension, scheme-theoretic fibers, inverse images of subschemes, intersections, products over a field, diagonals, equalizers, base-changed morphisms, moduli families, group-scheme operations, and algebraic-stack atlases. In each case the same intervention works: identify the two maps to a base, form the universal compatible object, calculate affinely by tensor product, and read the relevant special case from one projection.

The affine method transfers between geometry and commutative algebra through contravariance. A pushout \(A\otimes_R B\) of coordinate rings becomes a pullback of affine schemes. This is not a loose analogy; it is the exact anti-equivalence between commutative rings and affine schemes. It lets algebraic calculations predict geometric splitting, nonreducedness, intersection structure, and residue fields.

The broader pullback pattern also recurs in sets, groups, topological spaces, categories, databases, and type theory. That portable core belongs to the catalog’s universal_property, relation, and compatibility machinery. The scheme-specific node earns autonomy because it adds representability in schemes, affine spectra and tensor products, Zariski gluing, structure sheaves, residue-field point data, fibers over \(\operatorname{Spec}\kappa(s)\), and geometric properties under base change. Removing those roles leaves a generic categorical pullback, not this abstraction.

Transfer beyond mathematics is therefore routed to the parent. A database join on a shared key resembles a set pullback, but it is not a fiber product of schemes unless a scheme-theoretic structure and universal morphism condition are genuinely present. The encyclopedia should recognize the generic pattern there without exporting the algebraic-geometry name.

Examples

  1. Affine compatibility equation. With \(t\mapsto x^2\) and \(t\mapsto y^3\) as above, the fiber product has coordinate ring \(k[x,y]/(x^2-y^3)\). Its points solve the shared-base equation, while its scheme structure retains the algebra of the cusp.

  2. Extension of scalars. If \(X=\operatorname{Spec}(k[x]/(f))\) and \(K/k\) is a field extension, then \(X_K\cong\operatorname{Spec}(K\otimes_k k[x]/(f))\cong\operatorname{Spec}(K[x]/(f)).\) The same equations are interpreted over the larger field, and factorization can split the resulting scheme into more components.

  3. Fiber of a family. Let \(X=\operatorname{Spec}(k[t,x]/(x^2-t))\to\operatorname{Spec}k[t]\). Over \(t=a\), \(X_a\cong\operatorname{Spec}(k[x]/(x^2-a)).\) The fibers can split, remain irreducible, or become nonreduced depending on \(a\) and the characteristic.

  4. Tangential intersection. In \(\mathbb A_k^2=\operatorname{Spec}k[x,y]\), intersect the line \(y=0\) with the parabola \(y=x^2\). Their fiber product over \(\mathbb A^2\) is \(\operatorname{Spec}\bigl(k[x,y]/(y,y-x^2)\bigr) \cong\operatorname{Spec}(k[x]/(x^2)).\) The underlying intersection is one point, while the nilpotent records nontransverse scheme structure.[4]

  5. Rational points. For schemes over a field \(k\), the universal property gives \((X\times_S Y)(k)\cong X(k)\times_{S(k)}Y(k).\) This formula concerns \(k\)-valued points and should not be mistaken for a formula for the underlying topological point set.

  6. Complex base over the reals. \(\operatorname{Spec}\mathbb C\times_{\operatorname{Spec}\mathbb R}\operatorname{Spec}\mathbb C\) is \(\operatorname{Spec}(\mathbb C\times\mathbb C)\), exhibiting two points above the one compatible topological pair.

  7. Diagonal and separation. For \(X\to S\), the diagonal \(\Delta_{X/S}:X\to X\times_S X\) compares two copies of one point over the same base. Requiring this diagonal to be a closed immersion defines separatedness.[5]

Structural Tensions

T1 — Unrestricted product versus compatibility pullback. A Cartesian product admits every pair; a fiber product admits only pairs of maps agreeing over the named base. Diagnostic: can the base maps be erased without changing the result? If not, the object is relative.

T2 — Concrete tensor construction versus universal identity. \(A\otimes_R B\) constructs affine charts, while the universal property identifies the result independent of charts. Diagnostic: after building a candidate, has the unique-mediating-map condition been proved?

T3 — Rational points versus scheme points. \(T\)-valued points obey a clean pullback formula, but underlying points carry primes in residue-field tensor products. Diagnostic: is the statement about \(X(T)\) for a fixed test scheme, or about the topological carrier?

T4 — Reduced set intersection versus scheme-theoretic intersection. Two subspaces can meet at one visible point while the fiber product carries nilpotents or multiplicity-sensitive structure. Diagnostic: compute \((A/I)\otimes_A(A/J)\), not only \(V(I)\cap V(J)\) as a set.

T5 — Preservation versus descent. Stable properties move forward under arbitrary base change; reflecting them backward requires a sufficiently faithful cover and a descent theorem. Diagnostic: is the argument transporting a known property, or inferring one from a pullback?

T6 — Affine formula versus global gluing. \(\operatorname{Spec}(A\otimes_R B)\) applies directly only when all three schemes are affine. Diagnostic: if an input is nonaffine, has a compatible affine cover been chosen and glued?

T7 — Ordinary versus derived intersection. The ordinary tensor product retains nilpotents but can discard higher \(\operatorname{Tor}\) data. Diagnostic: does the question require only the classical scheme pullback, or homological intersection information demanding a derived fiber product?

T8 — Strict equality versus canonical isomorphism. Fiber products are unique up to unique isomorphism, and iterated base changes associate canonically rather than by literal identity of implementations. Diagnostic: is an equation harmless shorthand for the canonical isomorphism, or is construction-level equality being assumed?

Structural–Framed Character

Fiber Product of Schemes is formal, neutral, and observer-independent. There is no evaluative weight: a pullback square is neither good nor bad, and failure of a square to be cartesian is a mathematical mismatch rather than a social judgment. It is not human-practice-bound or institutionally constituted; once the schemes and morphisms are fixed, the universal property and affine tensor product determine the object regardless of who calculates it.

Its recurrence across algebraic geometry is recognition. Base extension, fibers, inverse images, intersections, and diagonals genuinely are the same scheme-theoretic pullback mechanism. The framed residue lies only in vocabulary travel: scheme, spectrum, structure morphism, residue field, tensor product of coordinate rings, Zariski gluing, and base change do not transfer literally to unrelated substrates.

The grading is therefore structural with a narrow domain accent: vocab_travels = 0.50; evaluative weight, institutional origin, human-practice binding, and analogy-based import are 0.0; aggregate 0.10. This is a domain-specific formal construction, not a framed practice and not a substrate-independent prime.

Structural Core vs. Domain Accent

Structural core. The portable skeleton is the categorical pullback: two arrows with common codomain, a commuting square, and a universal unique mediator from every compatible cone. It is a universal property and a finite limit. That pattern exists in many categories and belongs to the broader catalog node universal_property.

Domain accent. The residual content is exactly what makes the pullback a scheme: guaranteed representability in the category of schemes; the affine formula \(\operatorname{Spec}(A\otimes_R B)\); contravariance of \(\operatorname{Spec}\); gluing of affine charts; residue-field tensor products describing points; structure sheaves and nilpotents; scheme-theoretic fibers and intersections; and transport of geometric morphism properties under base change.

Why the node is autonomous. domain_specific:universal_property gives the defining proof template but not the existence theorem or scheme-specific realization. prime:cartesian_product explicitly models unrestricted independent combinations and therefore misses the shared-base constraint. domain_specific:tensor describes a multilinear algebraic object but does not turn a ring pushout into a geometric pullback or supply gluing. prime:intersection retains common elements but not morphism compatibility, residue fields, or scheme structure. Their conjunction still does not package the central construction used uniformly for fibers, scalar extension, diagonals, and base change.

Why it is not a prime. Outside scheme theory, the categorical skeleton survives but the scheme-specific mechanisms do not. The cross-domain reach belongs to Universal Property or a future generic Pullback node. Fiber Product of Schemes is the canonical algebraic-geometry specialization, with a substantial residual apparatus and therefore a proper domain-specific abstraction.

  • domain_specific:universal_property. Fiber Product of Schemes is a specific universal construction: the unique map from every compatible cone is constitutive, and the result is determined up to unique isomorphism.
  • prime:cartesian_product. The ordinary product is the special pullback over a terminal base, but the catalog prime emphasizes unrestricted independent combinations. Fiber products add compatibility and scheme structure.
  • prime:intersection. The fiber product of two closed subschemes over their ambient scheme is their scheme-theoretic intersection. Generic Intersection lacks the tensor, nilpotent, and universal-morphism payload.
  • domain_specific:tensor. Tensor products compute affine fiber products contravariantly: ring pushout becomes scheme pullback.
  • prime:relation. Compatibility over \(S\) behaves like a relation cutting down pairs, but the universal scheme represents compatible maps at every test scheme and is not merely a subset.
  • prime:local_to_global_aggregation. Affine tensor-product charts glue to the global fiber product. Local computations are coordinated by the universal property.

Relationships to Other Abstractions

Local relationship map for Fiber Product of SchemesParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Fiber Productof SchemesDOMAINDomain-specific abstraction: Universal property — is a kind ofUniversalpropertyDOMAIN

Current abstraction Fiber Product of Schemes Domain-specific

Parents (1) — more general patterns this builds on

  • Fiber Product of Schemes is a kind of Universal property Domain-specific

    domain_specific:universal_property. Fiber Product of Schemes is a specific universal construction: the unique map from every compatible cone is constitutive, and the result is determined up to unique isomorphism.

Hierarchy paths (4) — routes to 4 parentless roots

Neighborhood in Abstraction Space

Fiber Product of Schemes sits in a sparse region of the domain-specific corpus (81st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

  • domain_specific:universal_property — Universal Property. This is the immediate structural parent and proof form. Fiber Product of Schemes is one particular object characterized by that form, with scheme-specific existence and computation.
  • prime:cartesian_product — Cartesian Product. Cartesian Product forms unrestricted tuples from independent axes. Fiber Product of Schemes enforces equality over a common base and can have point and nilpotent structure not present in the naive product.
  • domain_specific:tensor — Tensor. The coordinate ring \(A\otimes_R B\) is an algebraic construction used affinely. The geometric output is \(\operatorname{Spec}(A\otimes_R B)\), glued globally.
  • prime:intersection — Intersection. Intersection is one special case for subobjects of a common scheme. General fiber products combine arbitrary morphisms to a base, not only embeddings.
  • Diagonal Morphism. The diagonal \(X\to X\times_S X\) is defined using a fiber product and diagnoses separatedness. It is a derived construction, not a synonym.
  • Descent (Mathematics). Base change pulls objects forward along \(S'\to S\); descent asks whether compatible pulled-back data reconstructs an object over \(S\). The latter needs additional cover and cocycle conditions.
  • Categorical pullback. Every scheme fiber product is a categorical pullback, but pullbacks exist in many other categories. The candidate is the scheme-theoretic specialization.
  • Derived fiber product. The derived version retains higher homological data through a derived tensor product. It refines rather than aliases the ordinary scheme fiber product.

References

[1] The Stacks Project Authors. “Existence of fibre products of schemes,” Lemma 26.16.1. Proves that schemes have a final object, products, and fiber products, equivalently finite limits. registry ↩

[2] The Stacks Project Authors. “Fibre products of schemes,” Section 26.17. Gives the universal definition, affine tensor-product formula, affine-cover gluing, residue-field description of points, and behavior of immersions. registry ↩a ↩b ↩c ↩d ↩e

[3] The Stacks Project Authors. “Base change in algebraic geometry,” Section 26.18. Defines schemes over a base, scalar/base change, base-changed morphisms, and scheme-theoretic fibers over residue fields. registry ↩a ↩b ↩c ↩d

[4] The Stacks Project Authors. “Closed immersions and quasi-coherent sheaves,” Definition 29.4.4 and Lemma 29.4.5. Identifies the scheme-theoretic intersection with the fiber product and computes it using \(A/(I+J)\). registry ↩a ↩b ↩c

[5] The Stacks Project Authors. “Separation axioms,” Section 26.21. Develops diagonal morphisms, equalizers, and separatedness through fiber-product diagrams. registry ↩a ↩b ↩c

[6] The Stacks Project Authors. “Smooth morphisms,” Section 29.35. Includes preservation of smoothness under base change. registry ↩a ↩b

[7] The Stacks Project Authors. “Proper morphisms,” Section 29.42. Includes preservation of properness under base change. registry ↩a ↩b

[8] Görtz, Ulrich, and Torsten Wedhorn. Algebraic Geometry I: Schemes, 2nd ed.. Springer Spektrum, 2020. Provides a dedicated chapter on fiber products and subsequent treatments of schemes over fields and representable functors. registry

[9] Hartshorne, Robin. Algebraic Geometry. Graduate Texts in Mathematics 52. Springer, 1977. See Chapter II, especially Theorem II.3.3 and the discussion of fibers and base change. registry

[10] Wikipedia contributors. “Fiber product of schemes,” frozen revision 1278472156, 2025-03-02. Discovery provenance only; not the material authority for this draft. registry