In
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, dévissage is a technique introduced by
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
for proving statements about
coherent sheaves on
Noetherian scheme
In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, where each A_i is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noe ...
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
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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''.
Grothendieck's dévissage theorem
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
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
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over the
residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ri ...
''k''(''y''). Then every coherent ''O''
''X''-module whose support is contained in ''X''′ is contained in C.
In the particular case that , 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.
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 ''O''
''X''-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 a 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 ''O''
''T''-module ''L'' and a homomorphism such that is bijective.
If ''n''
1, ''n''
2, ..., ''n''
''r'' is a strictly decreasing sequence of natural numbers, then an ''S''-dévissage in dimensions ''n''
1, ''n''
2, ..., ''n''
''r'' is defined recursively as:
# An ''S''-dévissage in dimension ''n''
1. Denote the cokernel of α by ''P''
1.
# An ''S''-dévissage in dimensions ''n''
2, ..., ''n''
''r'' of ''P''
1.
The dévissage is said to lie between dimensions ''n''
1 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 . Denote the cokernel of this morphism by ''P''
''r''. The dévissage is called total if ''P''
''r'' 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 ''O''
''X''-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 .
[EGA 0IV, Définition 16.4.9] 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 .
References
Bibliography
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{{DEFAULTSORT:Devissage
Algebraic geometry
Theorems in geometry