In mathematics, coherent duality is any of a number of generalisations of
Serre duality, applying to
coherent sheaves, in
algebraic geometry and
complex manifold theory, as well as some aspects of
commutative algebra that are part of the 'local' theory.
The historical roots of the theory lie in the idea of the
adjoint linear system of a
linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of
sheaf theory, in a way that made an analogy with
Poincaré duality more apparent. Then according to a general principle,
Grothendieck's relative point of view
Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rathe ...
, the theory of
Jean-Pierre Serre was extended to a
proper morphism; Serre duality was recovered as the case of the morphism of a
non-singular projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
(or
complete variety) to a point. The resulting theory is now sometimes called Serre–Grothendieck–Verdier duality, and is a basic tool in algebraic geometry. A treatment of this theory, ''Residues and Duality'' (1966) by
Robin Hartshorne, became a reference. One concrete spin-off was the
Grothendieck residue.
To go beyond proper morphisms, as for the versions of Poincaré duality that are not for
closed manifolds, requires some version of the ''
compact support'' concept. This was addressed in
SGA2 in terms of
local cohomology, and
Grothendieck local duality; and subsequently. The
Greenlees–May duality, first formulated in 1976 by
Ralf Strebel and in 1978 by
Eben Matlis, is part of the continuing consideration of this area.
Adjoint functor point of view
While Serre duality uses a
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
or
invertible sheaf as a
dualizing sheaf
In algebraic geometry, the dualizing sheaf on a proper scheme ''X'' of dimension ''n'' over a field ''k'' is a coherent sheaf \omega_X together with a linear functional
:t_X: \operatorname^n(X, \omega_X) \to k
that induces a natural isomorphism of ...
, the general theory (it turns out) cannot be quite so simple. (More precisely, it can, but at the cost of imposing the
Gorenstein ring In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring ...
condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a
right adjoint functor
, called ''twisted'' or ''
exceptional inverse image functor'', to a higher
direct image with compact support functor
.
''Higher direct images'' are a sheafified form of
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
in this case with proper (compact) support; they are bundled up into a single functor by means of the
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pro ...
formulation of
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
(introduced with this case in mind). If
is proper, then
is a right adjoint to the ''inverse image'' functor
. The ''existence theorem'' for the twisted inverse image is the name given to the proof of the existence for what would be the
counit for the
comonad of the sought-for adjunction, namely a
natural transformation
:
,
which is denoted by
(Hartshorne) or
(Verdier). It is the aspect of the theory closest to the classical meaning, as the notation suggests, that duality is defined by integration.
To be more precise,
exists as an
exact functor from a derived category of
quasi-coherent sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on
, to the analogous category on
, whenever
:
is a proper or quasi projective morphism of noetherian schemes, of finite
Krull dimension. From this the rest of the theory can be derived: dualizing complexes pull back via
, the
Grothendieck residue symbol, the dualizing sheaf in the
Cohen–Macaulay case.
In order to get a statement in more classical language, but still wider than Serre duality, Hartshorne (''Algebraic Geometry'') uses the
Ext functor of sheaves; this is a kind of stepping stone to the derived category.
The classical statement of Grothendieck duality for a projective or proper morphism
of noetherian schemes of finite dimension, found in Hartshorne (''Residues and duality'') is the following quasi-isomorphism
:
for
a bounded above complex of
-modules with quasi-coherent cohomology and
a bounded below complex of
-modules with coherent cohomology. Here the
's are sheaves of homomorphisms.
Construction of the ''f''! pseudofunctor using rigid dualizing complexes
Over the years, several approaches for constructing the
pseudofunctor emerged. One quite recent successful approach is based on the notion of a rigid dualizing complex. This notion was first defined by Van den Bergh in a noncommutative context. The construction is based on a variant of derived
Hochschild cohomology (Shukla cohomology): Let
be a commutative ring, and let
be a commutative
algebra. There is a functor
which takes a cochain complex
to an object
in the derived category over
.
Asumming
is noetherian, a rigid dualizing complex over
relative to
is by definition a pair
where
is a dualizing complex over
which has finite flat dimension over
, and where
is an isomorphism in the derived category
. If such a rigid dualizing complex exists, then it is unique in a strong sense.
Assuming
is a
localization of a finite type
-algebra, existence of a rigid dualizing complex over
relative to
was first proved by
Yekutieli and Zhang assuming
is a regular noetherian ring of finite Krull dimension, and by
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, Iyengar and
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assuming
is a
Gorenstein ring In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring ...
of finite Krull dimension and
is of finite flat dimension over
.
If
is a scheme of finite type over
, one can glue the rigid dualizing complexes that its affine pieces have, and obtain a rigid dualizing complex
. Once one establishes a global existence of a rigid dualizing complex, given a map
of schemes over
, one can define
, where for a scheme
, we set
.
Dualizing Complex Examples
Dualizing Complex for a Projective Variety
The dualizing complex for a projective variety
is given by the complex
:
Plane Intersecting a Line
Consider the projective variety
:
We can compute
using a resolution
by locally free sheaves. This is given by the complex
:
Since
we have that
:
This is the complex
:
See also
*
Verdier duality
Notes
References
*
*
*
*
*
{{DEFAULTSORT:Coherent Duality
Topological methods of algebraic geometry
Sheaf theory
Duality theories