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Derivative ideals under semilinear ground-field isomorphisms

Statement

Let $K$ and $K^{'}$ be fields of characteristic zero and let $σ ⁣ : K \tilde{⟶} K^{'}$ be a field isomorphism; both fields have prime subfield $Q$ (Field, A field's prime subfield is isomorphic to $Q$ in characteristic zero and to $F_{p}$ in characteristic $p$ ). Let $X$ be a smooth $K$ -scheme and $X^{'}$ a smooth $K^{'}$ -scheme (Smooth morphism of schemes). Suppose $φ ⁣ : X^{'} \to X$ is a $σ$ -semilinear isomorphism, meaning that it is an isomorphism of $Q$ -schemes and its pullback acts on the ground-field constants by $σ$ (Morphisms of schemes).

Then for every coherent ideal sheaf $I \subseteq O_{X}$ and every $i \geq 0$ , $φ^{*} (D_{K}^{i} (I)) = D_{K^{'}}^{i} (φ^{*} I),$ where the derivative ideals on $X$ and $X^{'}$ are formed using $K$ - and $K^{'}$ -derivations, respectively (Derivative ideals of an ideal sheaf and of a marked ideal).

Facts & Assumptions

Given: Fields $K, K^{'}$ of characteristic zero, a field isomorphism $σ ⁣ : K \to K^{'}$ , smooth schemes $X / K$ and $X^{'} / K^{'}$ , a $σ$ -semilinear isomorphism $φ ⁣ : X^{'} \to X$ , and a coherent ideal sheaf $I \subseteq O_{X}$ .

[F1]

Derivative ideals of an ideal sheaf and of a marked ideal: $D_{K} (I)$ is generated locally by $I$ and the sections $D (f)$ for $K$ -derivations $D$ ; $D_{K}^{i}$ is the $i$ -fold iterate, and similarly over $K^{'}$ .

[F2]

Derivation of an algebra: a derivation is additive, satisfies the Leibniz rule, and is linear over the indicated ground field.

[F3]

Morphisms of schemes: on corresponding open sets the isomorphism induces inverse ring isomorphisms $φ^{*}$ and $(φ^{*})^{- 1}$ , and semilinearity means $φ^{*} (c) = σ (c)$ for $c \in K$ .

[F4]

Field, A field's prime subfield is isomorphic to $Q$ in characteristic zero and to $F_{p}$ in characteristic $p$ : the prime subfields of $K$ and $K^{'}$ are both $Q$ , and $σ$ fixes that prime field.

Proof

1.1F2F3F4

Transport derivations. Let $D$ be a $K$ -derivation of $O_{X} (U)$ and define $D^{'}$ on $O_{X^{'}} (φ^{- 1} U)$ by $D^{'} (g) = φ^{*} (D ((φ^{*})^{- 1} g))$ . If $c^{'} = σ (c) \in K^{'}$ , then semilinearity gives $D^{'} (c^{'} g) = φ^{*} (c D ((φ^{*})^{- 1} g)) = c^{'} D^{'} (g)$ because $D (c) = 0$ ; the Leibniz rule follows by conjugating the Leibniz rule for $D$ . Thus $D^{'}$ is a $K^{'}$ -derivation. Conjugation by $φ^{*}$ is bijective, with inverse conjugation by $(φ^{*})^{- 1}$ .

2.1F1step 1.1∎

Derivative ideals agree. For each local section $f \in I (U)$ , one has $D^{'} (φ^{*} f) = φ^{*} (D (f))$ . As $D$ varies, the bijection in step 1.1 identifies all $K$ -derivative generators with all $K^{'}$ -derivative generators, so $φ^{*} (D_{K} (I)) = D_{K^{'}} (φ^{*} I)$ . Applying this identity successively to each derivative ideal gives $φ^{*} (D_{K}^{i} (I)) = D_{K^{'}}^{i} (φ^{*} I)$ for every $i \geq 0$ .

Remarks

Włodarczyk's Lemma 4.3.1 states this for varieties over one characteristic-zero field and an isomorphism over $Q$ ; the semilinear formulation above also permits relabelling the ground field along an isomorphism $K ≃ K^{'}$ .
In particular, this applies to the automorphisms of an algebraic closure in Galois descent; those automorphisms need not be linear over the algebraic closure.

Depends on

Used by

Canonical resolution under isomorphisms of the ground field Lemma

Dependency tree · two levels

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Sources

Jaroslaw Wlodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005) 779-822; author's arXiv version math/0401401 (28 pp., dated October 25, 2018) (standard reference, not scraped)