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The unit equation defines the empty effective Cartier divisor

Example

On every scheme $X$ the constant meromorphic function $1$ is a global unit of $K_{X}$ , and its class in $K_{X}^{\times} / O_{X}^{\times}$ is the zero element $0$ of the Cartier group $CaDiv (X)$ . This empty divisor is effective: the single chart $X$ with the single equation $f = 1$ is a local-equation representation by a regular section, since multiplication by $1$ is the identity. Its associated closed subscheme is empty, and its associated invertible sheaf is $O_{X}$ itself: $O_{X} (\emptyset) : = O_{X} (0) = O_{X}, Z_{0} = \emptyset .$ Here $\emptyset$ in the notation $O_{X} (\emptyset)$ denotes the zero element of $CaDiv (X)$ , whose vanishing locus is empty; it is the only effective Cartier divisor on the empty scheme.

Facts & Assumptions

Given: An arbitrary scheme $X$ , the constant meromorphic function $1 \in Γ (X, K_{X}^{\times})$ , and the Cartier divisor $0 \in CaDiv (X)$ that is its class.

[F1]

The group law of $CaDiv (X)$ is induced by the quotient sheaf $K_{X}^{\times} / O_{X}^{\times}$ , so the zero element $0$ is the class of the constant equation $1$ , which is a global meromorphic unit; a Cartier divisor is principal exactly when it admits a representation by a single global equation (Cartier divisor, Sheaf total quotient rings).

[F2]

A Cartier divisor $D$ is effective if it has a local-equation representation $(U_{i}, f_{i})$ with $f_{i} \in O_{X} (U_{i})$ whose germs are regular sections, that is, multiplication by each germ $(f_{i})_{x}$ is injective; a unit equation, in particular $f_{i} = 1$ , gives the zero Cartier divisor, and the empty scheme has only this effective divisor (Effective cartier divisor).

[F3]

An effective Cartier divisor $D$ determines a closed subscheme $Z_{D} ↪ X$ with ideal sheaf $I_{D} ∣_{U_{i}} = f_{i} O_{U_{i}}$ for every local-equation datum; the construction depends only on $D$ (Effective Cartier divisors are closed subschemes cut out by regular equations).

[F4]

The invertible sheaf associated to a Cartier divisor $D$ with local equations $f_{i}$ satisfies $O_{X} (D) ∣_{U_{i}} = f_{i}^{- 1} O_{U_{i}}$ , and for the zero divisor the equation is $1$ , so $O_{X} (0) = O_{X}$ ; on the empty scheme the formula gives the unique module sheaf, which is locally free of rank one vacuously (Invertible sheaf of cartier divisor).

Verification

1.1F1

The zero Cartier divisor is the class of the unit equation: the global section $1 \in Γ (X, K_{X}^{\times})$ maps to the identity element $0$ of the quotient group $Γ (X, K_{X}^{\times} / O_{X}^{\times}) = CaDiv (X)$ , and the single chart $X$ with equation $1$ represents it.

1.2F2

The zero divisor is effective: taking the trivial cover ${X}$ and the equation $f = 1 \in O_{X} (X)$ , multiplication by the germ $1_{x}$ is the identity map on $O_{X, x}$ for every $x \in X$ , hence injective; by [F2] the zero Cartier divisor is an effective Cartier divisor.

1.3F2F4

The empty scheme. If $X = \emptyset$ , then $K_{X} = O_{X} = 0$ , the quotient sheaf is the zero sheaf, and $CaDiv (\emptyset) = 0$ : the zero divisor is the only Cartier divisor and, by [F2], the only effective Cartier divisor. Its vanishing subscheme is the empty scheme and its associated sheaf is the unique $O_{\emptyset}$ -module sheaf, which is $O_{\emptyset}$ ; the claims about its vanishing subscheme and its associated sheaf formulated below hold there vacuously.

2.1F2F3step 1.2

Its closed subscheme is empty: by [F3] the ideal sheaf is $I_{0} ∣_{X} = 1 \cdot O_{X} = O_{X}$ , the unit ideal; on an affine chart $U = Spec A$ the quotient $A / A = 0$ is the zero ring, whose spectrum is empty, and the glued vanishing subscheme is therefore empty, $Z_{0} = \emptyset$ .

2.2F4step 1.1

Its invertible sheaf is the structure sheaf: by [F4] the associated sheaf satisfies $O_{X} (0) ∣_{U_{i}} = f_{i}^{- 1} O_{U_{i}} = 1^{- 1} O_{X} = O_{X}$ on each chart of the trivial cover, and these identifications agree on overlaps; hence $O_{X} (0) ≅ O_{X}$ , the isomorphism being multiplication by the unit $1$ .

3.1step 1.2step 1.3step 2.1step 2.2∎

Conclusion. On every scheme $X$ the unit equation defines the empty effective Cartier divisor $0 = \emptyset$ , whose vanishing subscheme is empty and whose associated invertible sheaf is $O_{X}$ ; thus $O_{X} (\emptyset) = O_{X}$ in the notation of the Example. The only input is the unit equation $1$ , so no choice principle, no Noetherian or finiteness hypothesis, and no separatedness or reducedness is used.

The example shows that effectiveness of the zero divisor is not a vacuous or convention-dependent statement: the equation $1$ is regular, the ideal sheaf it generates is the unit ideal, and the scheme it cuts out is empty. The convention $O_{X} (\emptyset) = O_{X}$ is consistent with the sign rule of Invertible sheaf of cartier divisor, under which a regular equation becomes a zero-scheme of an effective divisor; for the unit equation the zero scheme is empty and no poles are introduced.

Depends on

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Sources

The Stacks Project, Divisors, Definition 31.14.1 and Definition 31.15.1 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Ch. 15 §§15.1–15.2 (standard reference, not scraped)