Dévissage
Encyclopedia
In algebraic geometry
, dévissage is a technique introduced by Alexander Grothendieck
for proving statements about coherent sheaves
on noetherian scheme
s. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness
and the proof that higher direct images of coherent sheaves under proper morphisms are coherent.
Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme. They did this by defining an object called a relative dévissage which is well-suited to certain kinds of inductive arguments. They used this technique to give a new criterion for a module to be flat
. As a consequence, they were able to simplify and generalize the results of EGA IV 11 on descent of flatness.
The word dévissage is French for unscrewing.
In the particular case that X′ = X, the theorem says that C is the category of OX-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-empty.
If n1, n2, ..., nr is a strictly decreasing sequence of natural numbers, then an S-dévissage in dimensions n1, n2, ..., nr is defined recursively as:
The dévissage is said to lie between dimensions n1 and nr. r is called the length of the dévissage. The last step of the recursion consists of a dévissage in dimension nr which includes a morphism . Denote the cokernel of this morphism by Pr. The dévissage is called total if Pr is zero.
Gruson and Raynaud prove in wide generality that locally, dévissages always exist. Specifically, let be a finitely presented morphism of pointed schemes and M be an OX-module of finite type whose fiber at x is non-zero. Set n equal to the dimension of and r to the codepth of M at s, that is, to . 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 and are trivial, the map 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 .
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, dévissage is a technique introduced by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
for proving statements about coherent sheaves
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
on noetherian scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness
Generic flatness
In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free...
and the proof that higher direct images of coherent sheaves under proper morphisms are coherent.
Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme. They did this by defining an object called a relative dévissage which is well-suited to certain kinds of inductive arguments. They used this technique to give a new criterion for a module to be flat
Flat module
In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
. As a consequence, they were able to simplify and generalize the results of EGA IV 11 on descent of flatness.
The word dévissage is French for unscrewing.
Grothendieck's dévissage theorem
Let X be a noetherian scheme. Let C be a full abelian subcategory of the category of coherent OX-modules, and let X′ be a closed subspace of the underlying topological space of X. Suppose that for every point x of X′, there exists a coherent sheaf G in C whose fiber at x is a one-dimensional vector space over the residue field k(x). Then every coherent OX-module whose support is contained in X′ is contained in C.In the particular case that X′ = X, the theorem says that C is the category of OX-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-empty.
Gruson and Raynaud's relative dévissages
Suppose that is a finitely presented morphism of affine schemes, s is a point of S, and M is a finite type OX-module. If n is a natural number, then Gruson and Raynaud define an S-dévissage in dimension n to consist of:- A closed finitely presented subscheme X′ of X containing the closed subscheme defined by the annihilator of M and such that the dimension of is less than or equal to n.
- A scheme T and a factorization of the restriction of f to X′ such that is a finite morphism and is an smooth affine morphism with geometrically integral fibers of dimension n. Denote the generic point of by τ and the pushforward of M to T by N.
- A free finite type OT-module L and a homomorphism such that is bijective.
If n1, n2, ..., nr is a strictly decreasing sequence of natural numbers, then an S-dévissage in dimensions n1, n2, ..., nr is defined recursively as:
- An S-dévissage in dimension n1. Denote the cokernel of α by P1.
- An S-dévissage in dimensions n2, ..., nr of P1.
The dévissage is said to lie between dimensions n1 and nr. r is called the length of the dévissage. The last step of the recursion consists of a dévissage in dimension nr which includes a morphism . Denote the cokernel of this morphism by Pr. The dévissage is called total if Pr is zero.
Gruson and Raynaud prove in wide generality that locally, dévissages always exist. Specifically, let be a finitely presented morphism of pointed schemes and M be an OX-module of finite type whose fiber at x is non-zero. Set n equal to the dimension of and r to the codepth of M at s, that is, to . 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 and are trivial, the map 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 .