Coherent duality
Encyclopedia
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
, 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 (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 an accessible 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 manifold
s, 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 1992 Greenlees-May duality is part of the continuing consideration of this area.
or invertible sheaf
as a dualizing sheaf, the general theory (it turns out) cannot be quite so simple. (More precisely, it can, but at the cost of the Gorenstein ring
condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint functor f !, called twisted or exceptional inverse image functor
, to a higher direct image with compact support
functor Rf!.
Higher direct images are a sheafified form of sheaf cohomology
in this case with proper (compact) support; they are bundled up into a single functor by means of the derived category
formulation of homological algebra
(introduced with this case in mind). In case f is proper Rf ! = Rf ∗ is itself a right adjoint, to the inverse image functor f ∗. 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 Trf (Hartshorne) or ∫f (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, f ! exists as an exact functor
from a derived category of quasi-coherent sheaves on Y, to the analogous category on X, 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 f !, 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 F⋅ a bounded above complex of OX-modules with quasi-coherent cohomology and G⋅ a bounded below complex of OY-modules with coherent cohomology. Here the Homs are the sheaf of homomorphisms.
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
, applying to coherent sheaves, in algebraic geometry
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...
and complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
theory, as well as some aspects of commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on 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
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....
in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
more apparent. Then according to a general principle, Grothendieck's relative point of view
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, rather than a single such object...
, the theory of Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (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
Robin Hartshorne
Robin Cope Hartshorne is an American mathematician. Hartshorne is an algebraic geometer who studied with Zariski, Mumford, J.-P. Serre and Grothendieck....
, became an accessible 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 manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
s, requires some version of the compact support concept. This was addressed in SGA2 in terms of local cohomology
Local cohomology
In mathematics, local cohomology is a chapter of homological algebra and sheaf theory introduced into algebraic geometry by Alexander Grothendieck. He developed it in seminars in 1961 at Harvard University, and 1961-2 at IHES. It was later written up as SGA2...
, and Grothendieck local duality; and subsequently. The 1992 Greenlees-May duality is part of the continuing consideration of this area.
Adjoint functor point of view
While Serre duality uses a line bundleLine 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 organising these...
or invertible sheaf
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
as a dualizing sheaf, the general theory (it turns out) cannot be quite so simple. (More precisely, it can, but at the cost of the Gorenstein ring
Gorenstein ring
In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition....
condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint functor f !, called twisted or exceptional inverse image functor
Exceptional inverse image functor
In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form.-Definition:Let f: X →...
, to a higher direct image with compact support
Direct image with compact support
In mathematics, in the theory of sheaves the direct image with compact support is an image functor for sheaves.-Definition:Let f: X → Y be a continuous mapping of topological spaces, and Sh the category of sheaves of abelian groups on a topological space...
functor Rf!.
Higher direct images are a sheafified form of sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
in this case with proper (compact) support; they are bundled up into a single functor by means of the derived category
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
formulation of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
(introduced with this case in mind). In case f is proper Rf ! = Rf ∗ is itself a right adjoint, to the inverse image functor f ∗. 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
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
- Rf !f ! → id,
which is denoted by Trf (Hartshorne) or ∫f (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, f ! exists as an exact functor
Exact functor
In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
from a derived category of quasi-coherent sheaves on Y, to the analogous category on X, whenever
- f: X → Y
is a proper or quasi projective morphism of noetherian schemes, of finite Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
. From this the rest of the theory can be derived: dualizing complexes pull back via f !, 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- Rf ∗RHomX(F⋅, f ! G⋅) → R HomY(Rf ∗ F⋅, G⋅)
for F⋅ a bounded above complex of OX-modules with quasi-coherent cohomology and G⋅ a bounded below complex of OY-modules with coherent cohomology. Here the Homs are the sheaf of homomorphisms.