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 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.
In mathematics, coherentduality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex...
In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre...
In mathematics, Grothendieck duality may refer to: Coherentduality of coherent sheaves Grothendieck local duality of modules over a local ring This disambiguation...
Grothendieck's coherentduality) one instance of Grothendieck's six operations formalism. Verdier duality generalises the classical Poincaré duality of manifolds...
duality and explains a very general instance thereof in detail. Probably the most general duality that is classically referred to as "Stone duality"...
uses the dualizing sheaf on the ambient Pn to construct the dualizing sheaf on X. coherentduality reflexive sheaf Gorenstein ring Dualizing module Hartshorne...
came from the need to find a suitable formulation of Grothendieck's coherentduality theory. Derived categories have since become indispensable also outside...
if the dual of A is B, then the dual of B is A. Alexander duality Alvis–Curtis duality Artin–Verdier duality Beta-dual space Coherentduality Conjugate...
types of coherentism: the coherence theory of truth, and the coherence theory of justification (also known as epistemic coherentism). Coherent truth is...
frame Projective transformation Fundamental theorem of projective geometry Duality (projective geometry) Real projective plane Real projective space Segre...
applications of derived categories included coherentduality and Verdier duality, which extends Poincaré duality to singular spaces. A shift or translation...
results on the vanishing of cohomology in various cases, duality theorems such as Serre duality, relations between topology and algebraic geometry such...
Two theories related by a duality need not be string theories. For example, Montonen–Olive duality is an example of an S-duality relationship between quantum...
Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. Suppose that...
The dual of a family F ⊆ ℘(C) is the family F ⊥ of all subsets S ⊆ C orthogonal to every member of F, i.e., such that S ⊥ T for all T ∈ F. A coherent space...
a number of properties that a risk measure might or might not have. A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity...
_{j}(-1)^{j}\dim _{k}(H^{j}(X,{\mathcal {O}}_{X})).} Serre duality is an analog of Poincaré duality for coherent sheaf cohomology. In this analogy, the canonical...
commutative ring. It is sometimes also called a coherent space because of the connection to coherent topoi. Let X be a topological space and let K ∘ {\displaystyle...
Coherent optical module refers to a typically hot-pluggable coherent optical transceiver that uses coherent modulation (BPSK/QPSK/QAM) rather than amplitude...
pair (Av, P). This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré...
(2009). "Notes on derived categories and Grothendieck duality" (PDF). Foundations of Grothendieck Duality for Diagrams of Schemes. Lecture Notes in Mathematics...
In mathematics, Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by Pierre Cartier (1962). Given any...
philosophical and spiritual traditions that emphasize the absence of fundamental duality or separation in existence. This viewpoint questions the boundaries conventionally...
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