Dévissage

Let X be a noetherian scheme. Let C be a subset of the objects of the category of coherent O_X-modules which contains the zero sheaf and which has the property that, for any short exact sequence ${\displaystyle 0\to A'\to A\to A''\to 0}$ of coherent sheaves, if two of A, A′, and A′′ are in C, then so is the third. Let X′ be a closed subspace of the underlying topological space of X. Suppose that for every irreducible closed subset Y of X′, there exists a coherent sheaf G in C whose fiber at the generic point y of Y is a one-dimensional vector space over the residue field k(y). Then every coherent O_X-module whose support is contained in X′ is contained in C.[2]

In the particular case that X′ = X, the theorem says that C is the category of coherent O_X-modules. This is the setting in which the theorem is most often applied, but the statement above makes it possible to prove the theorem by noetherian induction.

A variation on the theorem is that if every direct factor of an object in C is again in C, then the condition that the fiber of G at x be one-dimensional can be replaced by the condition that the fiber is non-zero.[3]

Suppose that f : X → S is a finitely presented morphism of affine schemes, s is a point of S, and M is a finite type O_X-module. If n is a natural number, then Gruson and Raynaud define an S-dévissage in dimension n to consist of:

1. A closed finitely presented subscheme X′ of X containing the closed subscheme defined by the annihilator of M and such that the dimension of X′ ∩ f⁻¹(s) is less than or equal to n.
2. A scheme T and a factorization X′ → T → S of the restriction of f to X′ such that X′ → T is a finite morphism and T → S is a smooth affine morphism with geometrically integral fibers of dimension n. Denote the generic point of T ×_S k(s) by τ and the pushforward of M to T by N.
3. A free finite type O_T-module L and a homomorphism α : L → N such that α ⊗ k(τ) is bijective.

If n₁, n₂, ..., n_r is a strictly decreasing sequence of natural numbers, then an S-dévissage in dimensions n₁, n₂, ..., n_r is defined recursively as:

1. An S-dévissage in dimension n₁. Denote the cokernel of α by P₁.
2. An S-dévissage in dimensions n₂, ..., n_r of P₁.

The dévissage is said to lie between dimensions n₁ and n_r. r is called the length of the dévissage. The last step of the recursion consists of a dévissage in dimension n_r which includes a morphism α_r : L_r → N_r. Denote the cokernel of this morphism by P_r. The dévissage is called total if P_r is zero.[4]

Gruson and Raynaud prove in wide generality that locally, dévissages always exist. Specifically, let f : (X, x) → (S, s) be a finitely presented morphism of pointed schemes and M be an O_X-module of finite type whose fiber at x is non-zero. Set n equal to the dimension of M ⊗ k(s) and r to the codepth of M at s, that is, to n − depth(M ⊗ k(s)).[5] Then there exist affine étale neighborhoods X′ of x and S′ of s, together with points x′ and s′ lifting x and s, such that the residue field extensions k(x) → k(x′) and k(s) → k(s′) are trivial, the map X′ → S factors through S′, this factorization sends x′ to s′, and that the pullback of M to X′ admits a total S′-dévissage at x′ in dimensions between n and n − r.

• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.
• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20. doi:10.1007/bf02684747. MR 0173675.
• Gruson, Laurent; Raynaud, Michel (1971), "Critéres de platitude et de projectivité", Inventiones Mathematicae (in French), 13: 1–17, doi:10.1007/bf01390094, ISSN 0020-9910