Dévissage¶
Reduce a statement about algebraic objects to finite filtrations of simpler subquotients and a justified transfer rule.
Core Idea¶
Dévissage reduces a statement about algebraic objects or categories to simpler subquotients in a finite filtration and then uses a justified transfer rule to recover the intended conclusion. A filtration alone is not a proof: the property or invariant must pass through the relevant exact sequences under the theorem's stated hypotheses.[ref-d334ed7659aa][ref-4cb8e303795f]
Scope of Application¶
On a Noetherian scheme, coherent sheaves admit finite filtrations by specified pieces supported on integral closed subschemes. An extension-stable property verified for all those pieces holds for every coherent sheaf. A separate K-theory dévissage theorem compares suitable abelian categories when objects of the larger category have finite filtrations by objects of an exact subcategory.[ref-d334ed7659aa][ref-4cb8e303795f]
Clarity¶
The method separates which pieces are enough from why they determine the whole. Different theorems have different premises. In particular, Stacks' extension-closure lemma and its one-rank-generic-witness lemma are not interchangeable; the latter needs stronger two-out-of-three and support conditions.[^ref-d334ed7659aa]
Manages Complexity¶
Rather than analyze an arbitrary object at once, a proof establishes finite reduction to a controlled class and checks how information is reconstructed. The technique simplifies the carrier without pretending extension data never matters.
Abstract Reasoning¶
State the target property or invariant, construct the finite filtration, check each quotient belongs to the admitted piece class, and identify the exact transfer theorem. If the intended property is not extension-stable, or a K-theory subcategory lacks the theorem's closure hypotheses, the proposed dévissage cannot justify the conclusion as written.[ref-d334ed7659aa][ref-4cb8e303795f]
Knowledge Transfer¶
The finite-filtration-plus-transfer test works across coherent-sheaf and module/category settings, although the pieces and theorems differ. A generic “break a problem into parts” analogy lacks the algebraic subquotients and reconstruction warrant needed for literal dévissage.
[^ref-d334ed7659aa]: The Stacks Project, §30.12 “Devissage of coherent sheaves”, Lemmas 30.12.3–30.12.6. [^ref-4cb8e303795f]: Charles A. Weibel, The K-Book: An Introduction to Algebraic K-Theory, AMS, 2013, ISBN 978-0-8218-9132-2. Author-hosted manuscript: https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf, Chapter V §4, Dévissage Theorem 4.1; manuscript pagination may differ from the published book.
Relationships to Other Abstractions¶
Current abstraction Dévissage Domain-specific
Parents (1) — more general patterns this builds on
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Dévissage presupposes Filtration (algebra) Domain-specific
Dévissage requires finite subobject filtrations before justified reconstruction.
Hierarchy paths (3) — routes to 3 parentless roots
- Dévissage → Filtration (algebra) → Order → Comparison → Self Checking
- Dévissage → Filtration (algebra) → Order → Relation
- Dévissage → Filtration (algebra) → Order → Set and Membership
Neighborhood in Abstraction Space¶
Dévissage sits in a moderately populated region (56th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Algebraic Substructures & Closures (12 abstractions)
Nearest neighbors
- Resolution Theorem (Algebraic K-Theory) — 0.90
- Auslander–Reiten theory — 0.86
- Closed Immersion — 0.85
- Initial and terminal objects — 0.85
- Filtration (algebra) — 0.84
Computed from structural-signature embeddings · 2026-10-08