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Open restrictions of the canonical desingularization

Statement

Assume AC (The Axiom of Choice).

In the setting of Independence of the embedded desingularization from the ambient embedding, let $V \to U$ be an open immersion of integral affine $K$ -varieties (Open immersions of schemes) and let ${res}_{U} ⁣ : \tilde{U} \to U$ , ${res}_{V} ⁣ : \tilde{V} \to V$ be the canonical desingularizations. Then there is an open immersion $\tilde{V} ↪ \tilde{U}$ lifting $V \to U$ such that $\tilde{V} \to {res}_{U}^{- 1} (V)$ is an isomorphism over $V$ .

Facts & Assumptions

Given: An open embedding $V ↪ U$ of affine $K$ -varieties of finite type over a field $K$ of characteristic zero, with canonical desingularizations $\tilde{V} \to V$ and $\tilde{U} \to U$ .

[A1]

The Axiom of Choice: AC is assumed for the canonical-resolution consumer clauses and their cited AC-dependent construction suppliers.

[F1]

Weak embedded desingularization in characteristic zero, Canonical resolutions commute with smooth morphisms: embedded desingularization commutes with smooth ambient morphisms; an open immersion is étale, hence smooth of relative dimension zero.

[F2]

Weak embedded desingularization in characteristic zero, Independence of the embedded desingularization from the ambient embedding: the canonical desingularization of an affine variety is obtained by an embedded desingularization in any smooth affine ambient variety and is independent of that choice.

[F3]

Open immersions of schemes, Locally finite type and finite type morphisms: in the principal case $V = U_{f}$ for a regular function $f$ ; one may choose a closed embedding $U ↪ X$ into a smooth affine $X$ and a function $F$ on $X$ restricting to $f$ , so that $U_{f} ↪ X_{F}$ is a closed immersion and $X_{F} ↪ X$ is an open immersion of smooth affine schemes.

Proof

1.1A1F1F2F3

The principal case. Assume first $V = U_{f}$ with $f \in K [U]$ . By [F3] choose a smooth affine ambient $X$ and $F \in K [X]$ with $F ∣_{U} = f$ ; the open immersions $X_{F} ↪ X$ and $U_{f} ↪ U$ are étale, so [F1] gives a canonical identification of the embedded desingularization of $U_{f}$ in $X_{F}$ with the restriction of the embedded desingularization of $U$ in $X$ ; by [F2] this is the required open embedding of the canonical desingularizations over $U_{f}$ .

2.1A1F1F2step 1.1∎

The general case. For an arbitrary affine open $V ↪ U$ , the complement is defined by finitely many functions; covering $V$ by principal opens $U_{g}$ contained in $V$ and applying step 1.1 to each, the identifications glue because the desingularizations are canonical and their restrictions to the intersections agree by the same argument. This yields an open embedding $\tilde{V} ↪ \tilde{U}$ lifting $V ↪ U$ , with $\tilde{V} \to {res}_{U}^{- 1} (V)$ an isomorphism over $V$ .

Depends on

Used by

Resolution of singularities in characteristic zero Theorem

Dependency tree · two levels

47 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)