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 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 1992 Greenlees-May duality is part of the continuing consideration of this area.
v· line bundle 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
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 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.[1] 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 Hom's are the sheaf of homomorphisms.
^ Verdier 1969, an elegant and more general approach was found by Amnon Neeman, by using methods from algebraic topology notably Brown representability, see Neeman 1996
Coherent sheaf — In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space … Wikipedia
Duality (mathematics) — In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one to one fashion, often (but not always) by means of an involution operation: if the dual… … Wikipedia
Coherent states in mathematical physics — Coherent states have been introduced in a physical context, first as quasi classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also [1]). However … Wikipedia
Coherent risk measure — In the field of financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function … Wikipedia
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 (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a… … Wikipedia
Stone duality — In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they… … Wikipedia
List of algebraic geometry topics — This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 … Wikipedia
List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… … Wikipedia
Derived category — In mathematics, the derived category D(C) 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. The construction proceeds on the… … Wikipedia
Séminaire Nicolas Bourbaki (1950–1959) — Continuation of the Séminaire Nicolas Bourbaki programme, for the 1950s. 1950/51 series *33 Armand Borel, Sous groupes compacts maximaux des groupes de Lie, d après Cartan, Iwasawa et Mostow (maximal compact subgroups) *34 Henri Cartan, Espaces… … Wikipedia
of noetherian schemes of finite dimension, found in Hartshorne (Residues and duality) is the following quasi-isomorphism