04.05.04 · algebraic-geometry / divisors

Cartier divisor

shipped3 tiersLean: partial

Anchor (Master): Hartshorne §II; Vakil; Cartier 1957–58 Bourbaki seminars

Intuition [Beginner]

A Cartier divisor is the "right" notion of divisor for general schemes — locally given by a single rational function. Where Weil divisors record formal sums of codimension-1 subvarieties, Cartier divisors record local data: a system of local equations ${f_{i} \in K (X)^{\times}}$ on a cover of $X$ , with consistency on overlaps.

The advantage: Cartier divisors correspond bijectively to line bundles (with a chosen rational section), even on schemes where Weil divisors and line bundles diverge. Pierre Cartier introduced this notion in 1957 to handle non-locally-factorial schemes — schemes where some Weil divisors fail to be locally cut out by a single equation. On the cone over a smooth conic, for example, the line through the cone point is a Weil divisor that is not Cartier: it's not locally principal at the cone point (it requires two equations).

For locally factorial schemes (every local ring a UFD), Weil and Cartier divisors agree. For more general schemes — singular varieties, non-normal schemes, arithmetic schemes — the Cartier-divisor notion is the one that controls line bundles and projective embeddings.

Visual [Beginner]

A scheme with local rational-function data on each chart, glued together by invertible-rational-function transitions on overlaps.

Worked example [Beginner]

On the projective line $P_{k}^{1}$ , every Weil divisor is Cartier (since $P^{1}$ is locally factorial). The point $[0]$ is cut out locally by the equation $x = 0$ on the affine chart $D (y)$ . The data of "Cartier divisor of $[0]$ " is the single function $x \in K (P^{1})^{\times}$ on $D (y)$ , with the empty equation on $D (x)$ where $[0]$ is not present.

A more elaborate example: on the affine cone $X = V (x z - y^{2}) \subset A^{3}$ over a smooth conic, the line $L = V (x, y) = {(0, 0, t) : t \in k}$ is a Weil divisor. But $L$ is not Cartier: at the cone point $(0, 0, 0)$ , the local ring $O_{X, 0}$ is not a UFD, and $L$ requires two equations $(x, y)$ to define — there's no single rational function whose vanishing cuts out $L$ near the cone point. So $L$ is Weil but not Cartier on $X$ .

This phenomenon — Weil $\neq =$ Cartier — is the central reason for distinguishing the two notions on singular schemes.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a scheme. A Cartier divisor on $X$ is a global section of the sheaf $K_{X}^{\times} / O_{X}^{\times}$ , where:

$K_{X}$ is the sheaf of total quotient rings — locally, on $Spec (R) \subseteq X$ , $K_{X} = (R ∖ zero-divisors)^{- 1} R$ , the total quotient ring.
$K_{X}^{\times}$ is the sheaf of invertible elements, i.e., non-zero-divisors.
$O_{X}^{\times} \subseteq K_{X}^{\times}$ is the sheaf of invertible regular functions.

So a Cartier divisor is a system of local data ${(U_{i}, f_{i})}$ where:

(C1) ${U_{i}}$ is an open cover of $X$ . (C2) $f_{i} \in K_{X}^{\times} (U_{i})$ is a non-zero-divisor in the total quotient ring. (C3) On overlaps $U_{i} \cap U_{j}$ , $f_{i} / f_{j} \in O_{X}^{\times} (U_{i} \cap U_{j})$ — the ratio is a unit.

Two such systems define the same Cartier divisor iff they refine to a common cover with matching local data.

Group structure. Cartier divisors form an abelian group $CaDiv (X) = H^{0} (X; K_{X}^{\times} / O_{X}^{\times})$ under pointwise multiplication of local equations.

Principal Cartier divisors. Given a global rational function $f \in H^{0} (X; K_{X}^{\times}) = Frac (Γ (X, O_{X}))$ (when $X$ is integral), $div (f)$ is the Cartier divisor with $f_{i} = f$ on every chart. These form a subgroup $CaDiv^{princ} (X)$ .

Cartier divisor class group.

CaCl (X) := CaDiv (X) / CaDiv^{princ} (X) = H^{0} (X; K_{X}^{\times} / O_{X}^{\times}) / Pic (Γ (X, O_{X})) .

Theorem (Cartier divisors and line bundles). The Cartier divisor class group $CaCl (X)$ is naturally isomorphic to the Picard group $Pic (X) = H^{1} (X; O_{X}^{\times})$ .

The isomorphism: a Cartier divisor ${(U_{i}, f_{i})}$ defines a line bundle $O (D)$ via the cocycle $g_{ij} = f_{i} / f_{j}$ on overlaps. Two Cartier divisors give isomorphic line bundles iff they differ by a principal Cartier divisor. So Cartier divisor classes ↔ Picard group, exactly.

Effective Cartier divisors. A Cartier divisor is effective if all $f_{i} \in O_{X} (U_{i})$ are regular (not just rational). An effective Cartier divisor corresponds to a closed subscheme $Y \subseteq X$ of codimension 1 — the zero locus of the local equations.

Cartier vs Weil. On a Noetherian normal scheme $X$ :

Cartier divisors form a subgroup of Weil divisors. Every Cartier divisor ${(U_{i}, f_{i})}$ defines a Weil divisor $\sum_{Y} v_{Y} (f_{i}) \cdot [Y]$ where the order is computed in any chart containing $Y$ .
On locally factorial schemes, the inclusion is an equality. Every prime divisor is locally principal — generated by a single equation in the local UFD. So all Weil divisors are Cartier.
On non-locally-factorial normal schemes, the inclusion is strict. The class group $Cl (X)$ contains $Pic (X)$ as a subgroup, with quotient measuring the failure of local factoriality.

Examples.

Smooth varieties (in particular smooth projective curves and smooth surfaces): $Pic = Cl$ .
Affine cone over a smooth conic: $Pic ⊊ Cl$ . The cone point is the obstruction.
Singular projective curves: line bundles ( $Pic$ ) and divisor classes ( $Cl$ ) can differ at the singularities.
Number rings: $Pic (Spec (O_{K})) = Cl (O_{K})$ — the ideal class group, since Dedekind domains are locally factorial (every local ring is a DVR).

Key theorem with proof [Intermediate+]

Theorem (Cartier ↔ line bundle correspondence). There is a natural exact sequence

0 \to H^{0} (X; O_{X}^{\times}) \to H^{0} (X; K_{X}^{\times}) \to H^{0} (X; K_{X}^{\times} / O_{X}^{\times}) \to H^{1} (X; O_{X}^{\times}) \to H^{1} (X; K_{X}^{\times}),

identifying:

$H^{0} (X; O_{X}^{\times}) = Γ (X, O_{X})^{\times}$ , the units of global functions.
$H^{0} (X; K_{X}^{\times}) =$ global non-zero-divisor rational functions.
$H^{0} (X; K_{X}^{\times} / O_{X}^{\times}) = CaDiv (X)$ , Cartier divisors.
$H^{1} (X; O_{X}^{\times}) = Pic (X)$ , Picard group.

From the sequence: Cartier divisors modulo principal divisors map injectively into $Pic (X)$ . The image is the kernel of $H^{1} (X; O_{X}^{\times}) \to H^{1} (X; K_{X}^{\times})$ . When $X$ is connected and $H^{1} (X; K_{X}^{\times}) = 0$ (e.g., for connected reduced schemes), the map is an isomorphism: $CaCl (X) ≅ Pic (X)$ .

Proof sketch. The sequence is the long exact sequence in sheaf cohomology associated to the short exact sequence of sheaves of abelian groups

0 \to O_{X}^{\times} \to K_{X}^{\times} \to K_{X}^{\times} / O_{X}^{\times} \to 0.

The identifications follow:

$H^{0}$ of $O^{\times}$ : global units.
$H^{0}$ of $K^{\times}$ : the multiplicative group of global non-zero-divisor rational functions.
$H^{0}$ of the quotient: Cartier divisors by definition.
$H^{1}$ of $O^{\times}$ : line bundles up to isomorphism, i.e., the Picard group.

The injectivity of $CaCl (X) \to Pic (X)$ comes from the snake-lemma argument extracting the divisor class.

When $H^{1} (K_{X}^{\times}) = 0$ (for example, on integral schemes — where $K_{X}$ is the constant sheaf with value the function field $K (X)$ , hence flasque — and the sheaf is acyclic): $CaCl (X) = Pic (X)$ exactly. $□$

This identification — Cartier divisor classes equal the Picard group, exactly — is the foundational theorem of Cartier-divisor theory, and the reason Cartier divisors are the divisor notion controlling line bundles.

Bridge. The construction here builds toward 04.05.02 (picard group), where the same data is upgraded, and the symmetry side is taken up in 04.05.05 (ample and very ample line bundle). 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 3 (medium, proof).

Show that effective Cartier divisors on $X$ correspond bijectively to closed subschemes of $X$ that are locally principal (locally given by a single non-zero-divisor equation).

Hint

A locally principal ideal sheaf has invertible inverse.

Answer

An effective Cartier divisor ${(U_{i}, f_{i})}$ defines a closed subscheme $Y_{i} = V (f_{i}) \subset U_{i}$ on each chart. On overlaps $U_{i} \cap U_{j}$ , $f_{i} / f_{j} \in O^{\times}$ , so $V (f_{i}) = V (f_{j})$ on the overlap. The pieces glue to a global closed subscheme $Y \subseteq X$ , locally given by a single equation $f_{i}$ on each $U_{i}$ .

Conversely, given a closed subscheme $Y \subset X$ with locally principal ideal sheaf $I_{Y}$ (each stalk generated by one element), choose generators $f_{i} \in I_{Y} (U_{i})$ on a cover. The transitions $f_{i} / f_{j}$ live in $O_{X}^{\times}$ (since both generate the same locally principal ideal), giving Cartier-divisor data ${(U_{i}, f_{i})}$ .

So effective Cartier divisors = closed subschemes with locally principal ideal sheaf = "locally principal closed subschemes." $□$

This is the geometric content of "effective Cartier divisor": exactly the class of codimension-1 closed subschemes that are well-behaved (locally cut out by one equation), avoiding pathologies at singularities.

Exercise 4 (medium, conceptual).

Why does Cartier-divisor theory work cleanly on arbitrary schemes, while Weil-divisor theory is sometimes problematic?

Hint

Weil divisors require codimension-1 prime structure; Cartier divisors avoid this hypothesis.

Answer

Weil divisor theory requires:

Integrality — to define a function field $K (X)$ and rational functions.
Normality — so each local ring at a height-1 prime is a DVR (defines the order $v_{Y}$ ).
Noetherianness — for the formal-sum construction to converge.

Without these, the Weil-divisor formalism breaks: rational functions may not be well-defined, the order at a prime may not be an integer, etc.

Cartier divisors avoid these hypotheses: the data is just local equations with overlap consistency. As long as the scheme has a sheaf $K_{X}^{\times}$ of invertible rational data — which exists on essentially all schemes — Cartier divisors are well-defined.

So Cartier divisors work on:

Non-integral schemes (with multiple components or generic points).
Non-normal schemes.
Schemes over arbitrary bases (including $Z$ , formal schemes, perfectoid spaces).

This generality is exactly what's needed for arithmetic geometry, modern intersection theory, and the relative perspective of Grothendieck's algebraic geometry.

Exercise 5 (medium, numeric).

On the affine cone $X = V (x z - y^{2})$ , compute $Cl (X) / Pic (X)$ — the obstruction to local factoriality.

Hint

The cone has class group $Z /2$ generated by the line; Pic is zero.

Answer

For the affine cone $X = V (x z - y^{2})$ over a smooth conic in $P^{2}$ :

$Cl (X) = Z /2$ , generated by the class of the line $L$ . (Two copies of $L$ — say $L$ and $L^{'} = V (x, z - 1) \cap X$ — sum to a Weil divisor that is principal: $L + L^{'} = div (appropriate function)$ .)
$Pic (X) = 0$ — the affine cone has zero Picard group (every line bundle on an affine variety is structure-sheaf isomorphic when the variety has rational singularities).

So $Cl (X) / Pic (X) = Z /2$ — the cone has a 2-torsion obstruction at the cone point. This $Z /2$ is the local class group at the cone, measuring the failure of local factoriality.

In contrast, the projective cone (its compactification in $P^{3}$ ) has $Cl = Z^{2}$ (one for each line family) and $Pic = Z$ (only the hyperplane class is Cartier).

Exercise 6 (hard, proof).

Show that the first Chern class of a line bundle is well-defined as the class of the associated effective Cartier divisor: $c_{1} (L) = [D]$ for any global rational section of $L$ .

Hint

A rational section gives a Cartier divisor; different sections give linearly equivalent divisors.

Answer

Let $L$ be a line bundle on $X$ admitting a non-zero rational section $s$ — equivalent to $L \otimes K_{X}$ having a non-zero global section. The divisor $div (s)$ is a Cartier divisor: locally, $s$ trivialises $L ∣_{U_{i}} ≅ O_{U_{i}}$ via a local generator, and $s$ corresponds to a rational function $f_{i} \in K_{X}^{\times} (U_{i})$ . Overlap data: $f_{i} / f_{j}$ is the transition function of $L$ , i.e., in $O_{X}^{\times}$ . So ${(U_{i}, f_{i})}$ is Cartier.

Two rational sections $s_{1}, s_{2}$ give Cartier divisors $div (s_{1}), div (s_{2})$ differing by $div (s_{1} / s_{2})$ — a principal Cartier divisor. So the class $[div (s)] \in CaCl (X) = Pic (X)$ is independent of the section.

This class equals $c_{1} (L) \in Pic (X) ≅ H^{2} (X; Z)$ via the standard identification of Cartier divisor classes with line bundles. So the first Chern class of $L$ is the divisor class of any rational section. $□$

This connects Cartier divisors directly to the topological-cohomological invariant: $c_{1} (L)$ is computed by the (Poincaré dual of the) zeros and poles of any rational section.

Exercise 7 (hard, conceptual).

Explain why Cartier divisors are the natural divisor notion for the relative setting of algebraic geometry over an arbitrary base.

Hint

Cartier divisors transform well under base change.

Answer

For $X \to S$ a flat morphism of schemes and $D \subset X$ a Cartier divisor, the base change $D \times_{S} T \subset X \times_{S} T$ is again a Cartier divisor on the base-changed family. This flat base change property fails for Weil divisors in general — the codimension-1 subvariety structure can change under base change.

This is critical for:

Families of varieties. A Cartier divisor relative to a base $S$ is "the same divisor varying over $S$ ," in a precise sense.
Picard scheme. The functor " $T \mapsto Pic (X \times_{S} T)$ " is representable by the Picard scheme $Pic_{X / S}$ . Cartier divisors are the right input.
Moduli problems. Hilbert schemes, moduli of varieties with line bundles, mirror symmetry — all use Cartier divisors over relative bases.
Arithmetic schemes. For $X / Spec (Z)$ , fibre divisors at primes $p$ should base-change cleanly. Cartier divisors give the right notion.

Weil divisors are conceptually fundamental on a single integral normal scheme. Cartier divisors are the functorial / relative / modern version, working uniformly across families and base changes. This is why Grothendieck's foundations and modern algebraic geometry universally use Cartier divisors.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has Cartier-divisor-related infrastructure (sheaves of rational functions, line bundles); full Cartier divisor theory is partially formalised.

[object Promise]

Advanced results [Master]

Picard scheme. The Picard functor " $T \mapsto Pic (X \times_{S} T)$ " is representable by an algebraic scheme $Pic_{X / S}$ over $S$ , the Picard scheme (Grothendieck). Cartier divisors over a base provide the natural input.

Mumford's regularity for Cartier divisors. Effective Cartier divisors on a projective scheme have regularity controlled by the Castelnuovo-Mumford regularity of their ideal sheaves. This gives effective bounds for line-bundle properties.

Cartier divisors on stacks. On Deligne-Mumford and algebraic stacks, the notion extends with coefficients in $Q$ — $Q$ -Cartier divisors. These are essential in higher-dimensional birational geometry (Mori program, Kawamata-Viehweg vanishing).

Cartier vs Weil on log-canonical and klt singularities. In modern minimal model program, the failure of "Cartier = Weil" is measured by the index of a Weil divisor — the smallest $n$ such that $n D$ is Cartier. Weil divisors with small index are "close to Cartier."

Cartier dual of a group scheme. The Cartier dual is a duality on commutative finite group schemes: for a finite commutative group scheme $G$ , $Hom (G, G_{m})$ is again a finite commutative group scheme — the Cartier dual. This is a different but related Cartier construction (Pierre Cartier developed both).

Adelic interpretation in arithmetic. Cartier divisors on an arithmetic scheme $X / Spec (O_{K})$ correspond, via the adelic perspective, to systems of valuations across all places (finite and infinite). This is the foundation of Arakelov geometry.

Synthesis. This construction generalises the pattern fixed in 04.05.01 (weil divisor), 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 long exact sequence relating $CaDiv, Pic$ via $K^{\times} / O^{\times}$ ; the Cartier-divisor base change formula; the equivalence with locally principal closed subschemes; failure of Weil = Cartier on the affine cone — these are deferred to companion units. The key theorem is sketched in the formal-definition section.

Connections [Master]

Weil divisor 04.05.01 — Weil and Cartier divisors agree on locally factorial schemes; differ in general.
Line bundle on a scheme 04.05.03 — Cartier divisor classes = Picard group exactly.
Picard group 04.05.02 — defined as Cartier divisors modulo principal divisors (or equivalently as line bundle isomorphism classes).
Coherent sheaf 04.06.02 — ideal sheaves of effective Cartier divisors are coherent and locally principal.
Sheaf cohomology 04.03.01 — the Cartier-Picard correspondence is a sheaf-cohomology calculation.
Riemann-Roch theorem for curves 04.04.01 — Riemann-Roch is naturally stated in terms of (effective) Cartier divisors.
Ample line bundle 04.05.05 — ampleness is a Cartier-divisor positivity property.

Historical & philosophical context [Master]

The notion of Cartier divisor was introduced by Pierre Cartier in a series of Bourbaki-seminar talks during 1957–58, in the context of Grothendieck's nascent scheme theory. Cartier was solving the problem: how to define divisors on non-classical schemes (non-locally-factorial, non-normal, non-Noetherian) so that the correspondence with line bundles persists.

The Weil-divisor formalism, while geometrically natural, breaks down on:

Singular varieties: prime divisors at singularities may not be locally principal.
Non-normal varieties: orders $v_{Y}$ may not be integer-valued.
Arbitrary base schemes in the relative setting (Grothendieck's perspective).

Cartier's insight: define divisors via local equations, not via formal sums of subvarieties. This local-equation formulation:

Always corresponds to line bundles — Cartier divisor classes equal $Pic (X)$ .
Behaves well under base change — relative Cartier divisors are the right notion for families.
Generalises Weil divisors — every Weil divisor on a locally factorial scheme is Cartier.
Captures effective subschemes correctly — effective Cartier divisors = locally principal codimension-1 closed subschemes.

Cartier's 1957–58 Bourbaki seminars established the correspondence with line bundles and the key cohomological identification $CaCl (X) = Pic (X)$ . The notion was systematically developed by Grothendieck in EGA IV (1967), where Cartier divisors became the standard divisor notion in scheme theory.

Today Cartier divisors are universal in algebraic geometry:

Birational geometry (Mori program): Cartier divisors and their $Q$ -extensions drive the classification of higher-dimensional varieties.
Arakelov geometry: arithmetic Cartier divisors on schemes over $Spec (O_{K})$ encode arithmetic intersection numbers.
Moduli theory: families of Cartier divisors over a base give the universal Picard scheme.
Mirror symmetry: Cartier divisors and line bundles play the role of "lattices" on both sides of the mirror.

Pierre Cartier himself (1932–2024) was one of the great mathematicians of the 20th century, contributing not only to algebraic geometry but to combinatorics, formal groups, and the theory of motives. His name is attached to several fundamental constructions (Cartier dual of a group scheme, Cartier operator on de Rham cohomology in characteristic $p$ ), and his Bourbaki-seminar talks set the standard for clarity and depth of exposition.

Bibliography [Master]

Hartshorne, Algebraic Geometry — §II.6 is the standard introduction.
Vakil, The Rising Sea: Foundations of Algebraic Geometry — §14, modern treatment.
Cartier, Bourbaki Seminar (1957–58) — the original introduction.
Eisenbud-Harris, The Geometry of Schemes — §III, geometric perspective.
Grothendieck-Dieudonné, EGA IV — systematic scheme-theoretic treatment.
Lazarsfeld, Positivity in Algebraic Geometry I — Cartier divisors and positivity.
Kollár-Mori, Birational Geometry of Algebraic Varieties — Cartier divisors in the minimal model program.
Lang, Introduction to Arakelov Theory — arithmetic Cartier divisors.