Coherent Duality
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In mathematics, coherent duality is any of a number of generalisations of
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Al ...
, applying to
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 ...
, in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
theory, as well as some aspects of
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
that are part of the 'local' theory. The historical roots of the theory lie in the idea of the
adjoint linear system In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
of a
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. These arose first in the fo ...
in classical algebraic geometry. This was re-expressed, with the advent of
sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
, in a way that made an analogy with
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. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
was extended to a
proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field '' ...
; Serre duality was recovered as the case of the morphism of a
non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
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 In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism :X \times Y \to Y is a closed map (i.e. maps closed sets onto closed sets). Thi ...
) 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 __NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
, 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 manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example ...
s, requires some version of the ''
compact support In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
'' concept. This was addressed in
SGA2 SGA may refer to: * Old Irish language (ISO 639-3 code) * ''Schwarz-Gelbe Allianz'' (Black-Yellow Alliance), an Austrian political party * Second-generation antipsychotics * Séminaire de Géométrie Algébrique du Bois Marie, an influential mat ...
in terms of
local cohomology In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a fu ...
, and
Grothendieck local duality In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. Statement Suppose that ''R'' is a Cohen–Macaulay local ring of dimension ''d'' ...
; and subsequently. The Greenlees–May duality, first formulated in 1976 by
Ralf Strebel Ralph (pronounced ; or ,) is a male given name of English, Scottish and Irish origin, derived from the Old English ''Rædwulf'' and Radulf, cognate with the Old Norse ''Raðulfr'' (''rað'' "counsel" and ''ulfr'' "wolf"). The most common forms ...
and in 1978 by
Eben Matlis Eben Matlis (August 28, 1923 - March 27, 2015) was a mathematician known for his contributions to the theory of rings and modules, especially for his work with injective modules over commutative Noetherian rings, and his introduction of Matlis dual ...
, 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 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 ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
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 is ...
condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
functor f^, called ''twisted'' or ''
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 duali ...
'', to a higher
direct image with compact support In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations. Definition Let ...
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 proce ...
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 In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...
for the
comonad In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is an ...
of the sought-for adjunction, namely a
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
: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 In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
from a derived category of quasi-coherent sheaves 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 In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ...
. 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 In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, a ...
(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 Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
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 Avramov, Iyengar and Lipman 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 is ...
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 In mathematics, Verdier duality is a cohomology, cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact space, locally compact topolo ...


Notes


References

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