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A locally principal subscheme need not be an effective Cartier divisor

Statement refuted

The claim refuted is: every closed subscheme that is locally cut out by principal ideals is an effective Cartier divisor. Let $k$ be a field and let $X = Spec A$ with $A = k [ϵ] / (ϵ^{2})$ be the dual-numbers scheme. The closed subscheme $Z = V (ϵ)$ is cut out on the single affine chart $X$ by the principal ideal $(ϵ)$ , but $ϵ$ is a zero divisor, $ϵ \cdot ϵ = 0$ with $ϵ \neq 0$ ; every generator of $(ϵ)$ is such a multiple, hence a zero divisor, so the ideal has no regular generator and $Z$ is not an effective Cartier divisor.

Facts & Assumptions

Given: A field $k$ , the ring $A = k [ϵ] / (ϵ^{2})$ with $ϵ^{2} = 0$ and $ϵ \neq 0$ , the affine scheme $X = Spec A$ , and the closed subscheme $Z = V (ϵ)$ cut out by the principal ideal $(ϵ) \subseteq A$ .

[F1]

$X = Spec A$ is the dual-numbers scheme of $k$ (The affine scheme of dual numbers); its global sections are $O_{X} (X) = A$ , whose elements are the classes $a + b ϵ$ with $a, b \in k$ and with $ϵ^{2} = 0$ and $ϵ \neq 0$ .

[F2]

Closed subschemes of $X = Spec A$ are, up to unique isomorphism over $X$ , exactly the morphisms $Spec (A / I) ↪ Spec A$ for ideals $I \subseteq A$ ; in particular $Z = V (ϵ)$ is the closed subscheme $Spec (A / (ϵ))$ cut out by the principal ideal $(ϵ)$ (Closed immersions into affine schemes are quotient spectra).

[F3]

A Cartier divisor on $X$ is a section of the quotient sheaf $K_{X}^{\times} / O_{X}^{\times}$ of Sheaf total quotient rings; an effective Cartier divisor admits 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 on $O_{X, x}$ . In particular, on the affine chart $X$ a local equation is an element $f \in A$ such that multiplication by $f$ on $A$ is injective (Effective cartier divisor).

[F4]

Part 1 of Effective Cartier divisors are closed subschemes cut out by regular equations attaches to every effective Cartier divisor $D$ on $X$ a closed subscheme $Z_{D}$ whose ideal sheaf $I_{D}$ is the kernel of $O_{X} \to (i_{D})_{*} O_{Z_{D}}$ , and for every local-equation datum ${(U_{i}, f_{i})}$ of $D$ one has $I_{D} ∣_{U_{i}} = f_{i} O_{U_{i}}$ . Part 2 states the converse: a closed subscheme locally cut out by nonzerodivisors is of the form $Z_{D}$ for an effective Cartier divisor $D$ .

Counterexample

1.1F1

The ring $A = k [ϵ] / (ϵ^{2})$ is local with maximal ideal $(ϵ)$ , and its units are exactly the elements $a + b ϵ$ with $a \neq 0$ . Indeed $A / (ϵ) ≅ k$ is a field, so $(ϵ)$ is maximal; if $a \neq 0$ then $(a + b ϵ) (a^{- 1} - b a^{- 2} ϵ) = 1 + b a^{- 1} ϵ - b a^{- 1} ϵ = 1$ , so $a + b ϵ$ is a unit; conversely an element $b ϵ$ of $(ϵ)$ is not a unit because $A / (ϵ) ≅ k$ kills it.

1.2F1

The element $ϵ$ is a zero divisor of $A$ : $ϵ \neq 0$ by hypothesis and $ϵ \cdot ϵ = ϵ^{2} = 0$ , so multiplication by $ϵ$ on $A$ sends the nonzero element $ϵ$ to $0$ and is not injective. The same computation gives $a ϵ \cdot ϵ = 0$ for every $a \in k$ .

2.1step 1.1step 1.2

The generators of the ideal $(ϵ)$ are exactly the elements $a ϵ$ with $a \neq 0$ : an element of $(ϵ)$ is $c + d ϵ$ times $ϵ$ , which equals $c ϵ$ because $ϵ^{2} = 0$ ; if $a \neq 0$ then $ϵ = a^{- 1} (a ϵ)$ lies in $(a ϵ)$ , so $(a ϵ) = (ϵ)$ , and conversely a generator $f = (c + d ϵ) ϵ = c ϵ$ of $(ϵ)$ satisfies $ϵ \in (f) = (c ϵ)$ only if $c \neq 0$ . By step 1.2 every such generator is a zero divisor.

3.1F2F3F4step 2.1

The closed subscheme $Z = V (ϵ)$ is not an effective Cartier divisor on $X$ . Suppose it were, say $Z = Z_{D}$ for an effective Cartier divisor $D$ . By [F4] the ideal sheaf $I_{D}$ equals the ideal sheaf of $Z$ , which on the single chart $X$ is $(ϵ)$ ; since $X$ has only one point, its only nonempty open is $X$ , so an effective datum supplies an equation $f \in A = O_{X} (X)$ ; by [F4] we have $I_{D} ∣_{X} = f A$ , so $f$ generates $(ϵ)$ . By step 2.1 the element $f$ is a zero divisor, so multiplication by $f$ on $A$ is not injective and $f$ is not a regular section; this contradicts the effectiveness requirement of [F3].

4.1step 1.1step 1.2step 2.1step 3.1∎

Therefore $Z = V (ϵ)$ is locally principal, being cut out on its unique affine chart by the principal ideal $(ϵ)$ , yet it is not an effective Cartier divisor, because no generator of that ideal is a nonzerodivisor. This refutes the claim that local principality alone makes a closed subscheme an effective Cartier divisor.

The obstruction is the nilpotent structure of $A$ : the vanishing scheme $Z = Spec k$ is a single reduced point, but the scheme $X$ is non-reduced and the equation of the point is a zero divisor. On an integral scheme a nonzero regular function on a nonempty open has a nonzero germ in every local domain, so it is a nonzerodivisor (Effective cartier divisor). The example shows that the nonzerodivisor hypothesis cannot be dropped in the converse construction of an effective Cartier divisor from a locally principal closed subscheme (Effective Cartier divisors are closed subschemes cut out by regular equations). It does not test dropping effectiveness for an existing Cartier divisor: on this $X$ every nonzerodivisor is a unit, so $K_{X} = O_{X}$ and every Cartier divisor is zero.

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Sources

The Stacks Project, Divisors, Definition 31.14.1 and Lemma 31.14.2 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Ch. 15 §15.1 (standard reference, not scraped)