Coherent Duality
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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 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 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 f is proper, then Rf_ = Rf_ is 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 :Rf_f^\rightarrow id, which is denoted by Tr_f (Hartshorne) or \int_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 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 Y, to the analogous category on X, whenever :f : X \rightarrow Y 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 f : X \rightarrow Y of noetherian schemes of finite dimension, found in Hartshorne (''Residues and duality'') is the following quasi-isomorphism :Rf_RHom_X(F^\bullet,f^! G^\bullet)\to RHom_Y(Rf_F^\bullet ,G^\bullet) for F^\bullet a bounded above complex of O_X-modules with quasi-coherent cohomology and G^\bullet a bounded below complex of O_Y-modules with coherent cohomology. Here the Hom's are sheaves of homomorphisms.


Construction of the ''f''! pseudofunctor using rigid dualizing complexes

Over the years, several approaches for constructing the f^ 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 k be a commutative ring, and let A be a commutative k-algebra. There is a functor RHom_(A,M\otimes^L_k M) which takes a cochain complex M to an object RHom_(A,M\otimes^L_k M) in the derived category over A. Asumming A is noetherian, a rigid dualizing complex over A relative to k is by definition a pair (R,\rho) where R is a dualizing complex over A which has finite flat dimension over k, and where \rho: R\to RHom_(A,R\otimes^L_k R) is an isomorphism in the derived category D(A). If such a rigid dualizing complex exists, then it is unique in a strong sense. Assuming A is a localization of a finite type k-algebra, existence of a rigid dualizing complex over A relative to k was first proved by Yekutieli and Zhang assuming k is a regular noetherian ring of finite Krull dimension, and by
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, Iyengar and
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assuming k 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 A is of finite flat dimension over k. If X is a scheme of finite type over k, one can glue the rigid dualizing complexes that its affine pieces have, and obtain a rigid dualizing complex R_X. Once one establishes a global existence of a rigid dualizing complex, given a map f:X\to Y of schemes over k, one can define f^ := D_X \circ Lf^* \circ D_Y, where for a scheme X, we set D_X:= RHom_(-,R_X).


Dualizing Complex Examples


Dualizing Complex for a Projective Variety

The dualizing complex for a projective variety X \subset \mathbb^n is given by the complex :\omega_X^\bullet = \mathrm_(\mathcal_X,\omega_ n


Plane Intersecting a Line

Consider the projective variety :X = \text\left( \frac \right) = \text\left( \frac \right) We can compute \mathrm_(\mathcal_X,\omega_ 3 using a resolution \mathcal^\bullet \to \mathcal_X by locally free sheaves. This is given by the complex : 0 \to \mathcal(-3) \xrightarrow \mathcal(-2) \oplus \mathcal(-2) \xrightarrow \mathcal \to \mathcal_X \to 0 Since \omega_ \cong \mathcal(-4) we have that : \omega_X^\bullet = \mathrm_(\mathcal^\bullet, \mathcal(-4) 3 = \mathrm_(\mathcal^\bullet\otimes \mathcal(4) 3 \mathcal) This is the complex : mathcal(-4) \xrightarrow \mathcal(-2) \oplus \mathcal(-2) \xrightarrow \mathcal(-1)-3]


See also

* Verdier duality


Notes


References

* * * * * {{DEFAULTSORT:Coherent Duality Topological methods of algebraic geometry Sheaf theory Duality theories