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Reduced induced closed subscheme construction
Statement
Let be a scheme and a closed subset. There is a unique closed subscheme supported on whose local rings have no nonzero nilpotent elements. This is the reduced induced scheme structure on .
Statement references
External input
Import the existence and uniqueness assertion of Stacks Lemma 26.12.4, with the terminology of Definition 26.12.5. The hypotheses are exactly a scheme and a closed subset. The displayed construction proof was read; its cited foundations were not re-proved here. The residue-field argument needed by the Nisnevich definition is supplied locally there.
Depends on
Used by · 9
- Nisnevich covers and distinguished squares
- Motivic and filtered K theory with support
- Principal divisors are endpoint boundaries
- Bottom supported Bloch groups on a codimension two curve
- Finite subfamilies of Nisnevich covers on the finite-type site
- A rank-two family avoids each prescribed codimension-two locus
- Completely decomposed proper maps are abstract-blowup covers
- Projective-line projection dichotomy and endpoint coordinates
- Comparison of the MVW and cdp cdh topologies
Dependency tree · 2 levels
Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.
External dependencies
Every closed subset T of a scheme X supports a unique reduced closed subscheme of X.
Hypotheses: X is any scheme and T is a closed subset of its underlying space; no finiteness assumption.
Reading limits: Read the lemma statement and displayed proof, and Definitions 26.12.1 and 26.12.5 at https://stacks.math.columbia.edu/tag/01IZ. The proof constructs the vanishing ideal sheaf and affine radical-ideal quotients; its earlier cited algebra and scheme lemmas were not independently reviewed. No residue-field or base-change theorem is imported from this lemma.
Sources
- The Stacks Project, reduced induced closed subschemes · Lemma 26.12.4, statement and displayed proof; reduced schemes in Definition 26.12.1 and terminology in Definition 26.12.5 (Section 26.12)