Coherent duality in mathematics refers to 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. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ...

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. In mathematics, the concept of a linear system of divisors arose first in the form of a linear system of algebraic curves in the projective plane. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... This is a glossary of scheme theory. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... This article is about algebraic varieties. ... In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ...

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. For other uses, see Manifold (disambiguation). ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

Adjoint functor point of view

While Serre duality uses a 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 In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... 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. ... In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

f^!,

now called twisted inverse image, to a higher direct image functor

Rf_*.

Higher direct images are a sheafified form of sheaf cohomology; they are bundled up into a single functor by means of the derived category formulation of homological algebra (introduced with this case in mind). It should be noted that 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 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. This is the main step, in numerous areas, from sheaf theory as a description of a geometric... In mathematics, the derived category D(C) of a category C is a (highly) abstract construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ... In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

Rf_*f^! → I

which is denoted by Tr_f (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 In homological algebra, an exact functor is one which preserves exact sequences. ...

f: X → Y

is a proper morphism of noetherian schemes, of finite Krull dimension. (Verdier, Bombay 1968 paper). 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 commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ...

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.