For rings without a dualizing module it is sometimes possible to use the dualizing complex as a substitute.
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A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring then it has a dualizing module. In particular any complete local Cohen–Macaulay ring has a dualizing module.
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A dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m, the R/m vector space vanishes if n ≠ height(m) and is 1-dimensional if n = height(m). A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module. In particular if the ring is local the dualizing module is unique up to isomorphism.
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If R is a Gorenstein ring, then R considered as a module over itself is a dualizing module. If R is an Artinian local ring then the Matlis module of R (the injective hull of the residue field) is the dualizing module. The Artinian local ring R = k[x,y]/(x2,y2,xy) has a unique dualizing module, but it is not isomorphic to R. The ring Z[] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals. The local ring k[x,y]/(y2,xy) is not Cohen–Macaulay so does not have a dualizing module.
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Dual graphs have several roles in meshing. One can make a polyhedral Voronoi diagram mesh by dualizing a Delaunay triangulation simplicial mesh. One can create a cubical mesh by generating an arrangement of surfaces and dualizing the intersection graph; see spatial twist continuum. Sometimes both the primal mesh and its dual mesh are used in the same simulation; see Hodge star operator.
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This follows since dualizing each statement in the proof "in " gives a corresponding statement of the proof "in ". The principle of plane duality says that dualizing any theorem in a self-dual projective plane produces another theorem valid in . The above concepts can be generalized to talk about space duality, where the terms "points" and "planes" are interchanged (and lines remain lines). This leads to the principle of space duality.
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This right adjoint sends the injective hull E(k) mentioned above to k, which is a dualizing object in D(k). This abstract fact then gives rise to the above-mentioned equivalence.
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In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.
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In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality.
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If R is a discrete valuation ring with quotient field K then the Matlis module is K/R. In the special case when R is the ring of p-adic numbers, the Matlis dual of a finitely-generated module is the Pontryagin dual of it considered as a locally compact abelian group. If R is a Cohen–Macaulay local ring of dimension d with dualizing module Ω, then the Matlis module is given by the local cohomology group H(Ω). In particular if R is an Artinian local ring then the Matlis module is the same as the dualizing module.
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In coherent sheaf theory, pushing to the limit of what could be done with Serre duality without the assumption of a non-singular scheme, the need to take a whole complex of sheaves in place of a single dualizing sheaf became apparent. In fact the Cohen–Macaulay ring condition, a weakening of non-singularity, corresponds to the existence of a single dualizing sheaf; and this is far from the general case. From the top-down intellectual position, always assumed by Grothendieck, this signified a need to reformulate. With it came the idea that the 'real' tensor product and Hom functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational devices.
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The most direct application of Panca-kritya (the observation of the five actions of consciousness) is Vikalpa Kshaya, literally meaning "dissolution of thoughts".The Yoga of Kashmir Shaivism – S.Shankarananda, p. 305 It is an activity by which the dualizing content of cognitions is dissolved into Atman, which is nondual by excellence.The Pratyabhijna Philosophy – G.V. Tagare, p.
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Constructivist Foundations is an international triannual peer-reviewed academic journal that focuses on constructivist approaches to science and philosophy, including radical constructivism, enactive cognitive science, second order cybernetics, biology of cognition and the theory of autopoietic systems, and non-dualizing philosophy. It was established in 2005 and the editor-in-chief is Alexander Riegler (Free University of Brussels).
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Vaha-cheda (cutting the two vital currents, prana and apana) leads to illumination by resting the ascending and descending vayus in the heart.The Pratyabhijna Philosophy – G.V. Tagare, p. 103 By bringing a cessation to the dualizing activity of prana and apana, equilibrium is reached, and in this superior condition the true nature of the heart shines forth.Introduction to Kasmir Shaivism, p.
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One meaning of the Cohen–Macaulay condition can be seen in coherent duality theory. A variety or scheme X is Cohen–Macaulay if the "dualizing complex", which a priori lies in the derived category of sheaves on X, is represented by a single sheaf. The stronger property of being Gorenstein means that this sheaf is a line bundle. In particular, every regular scheme is Gorenstein.
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For a Gorenstein scheme X of finite type over a field, f: X → Spec(k), the dualizing complex f!(k) on X is a line bundle (called the canonical bundle KX), viewed as a complex in degree −dim(X).Hartshorne (1966), Proposition V.9.3. If X is smooth of dimension n over k, the canonical bundle KX can be identified with the line bundle Ωn of top-degree differential forms.
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174 By focusing on the pure awareness substrate of cognition instead of the external objects, the practitioner reaches illumination. Dualizing thought constructs must be eliminated and in their place the light and ecstasy of pure awareness shines as the real nature of cognition. Repeating the gesture of vikalpa-ksaya with all thoughts, as they appear, there is a gradual transformation at the subconscious level (causal body), leading towards identity with Śiva. Thus, the process resembles the pruning of the weeds in a garden.
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With this terminology, a coherent sheaf on an integral normal scheme is reflexive if and only if it is torsion-free and normal in the sense of Barth. A reflexive sheaf of rank one on an integral locally factorial scheme is invertible. A divisorial sheaf on a scheme X is a rank-one reflexive sheaf that is locally free at the generic points of the conductor DX of X. For example, a canonical sheaf (dualizing sheaf) on a normal projective variety is a divisorial sheaf.
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In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". Forming the plane dual of a statement is known as dualizing the statement. If a statement is true in a projective plane , then the plane dual of that statement must be true in the dual plane .
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These principles provide a good reason for preferring to use a "symmetric" term for the incidence relation. Thus instead of saying "a point lies on a line" one should say "a point is incident with a line" since dualizing the latter only involves interchanging point and line ("a line is incident with a point"). The validity of the principle of plane duality follows from the axiomatic definition of a projective plane. The three axioms of this definition can be written so that they are self-dual statements implying that the dual of a projective plane is also a projective plane.
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The dual notion of a uniform module is that of a hollow module: a module M is said to be hollow if, when N1 and N2 are submodules of M such that N_1+N_2=M, then either N1 = M or N2 = M. Equivalently, one could also say that every proper submodule of M is a superfluous submodule. These modules also admit an analogue of uniform dimension, called co-uniform dimension, corank, hollow dimension or dual Goldie dimension. Studies of hollow modules and co-uniform dimension were conducted in , , , and . The reader is cautioned that Fleury explored distinct ways of dualizing Goldie dimension.
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The essential idea is that a Calabi-Yau manifold with complex dimension three should be foliated by "special Lagrangian" tori, which are certain types of three-dimensional minimal submanifolds of the six-dimensional Riemannian manifold underlying the Calabi-Yau structure. Given one three- dimensional Calabi-Yau manifold, one constructs its "mirror" by looking its torus foliation, dualizing each torus, and reconstructing the three- dimensional Calabi-Yau manifold, which will now have a new structure. The Strominger-Yau-Zaslow (SYZ) proposal, although not stated very precisely, is now understood to be overly optimistic. One must allow for various degenerations and singularities; even so, there is still no single precise form of the SYZ conjecture.
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The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets: : If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets. If a statement or definition is equivalent to its dual then it is said to be self-dual. Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol.
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We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the same: one converts from one to the other by dualizing, i.e. by changing the direction of all arrows, replacing injective objects with projective ones, and so on. Suppose that A is an abelian category with enough injectives and F a left exact functor to another abelian category B. If C is a complex of objects of A bounded on the left, the hypercohomology :Hi(C) of C (for an integer i) is calculated as follows: # Take a quasi-isomorphism Φ : C → I, here I is a complex of injective elements of A. # The hypercohomology Hi(C) of C is then the cohomology Hi(F(I)) of the complex F(I).
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A geometric solution to this is to intersect the curve not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating the curve in some direction, but in general one talks about taking a curve that is linearly equivalent to , and counting the intersection , thus obtaining an intersection number, denoted . Note that unlike for distinct curves and , the actual points of intersection are not defined, because they depend on a choice of , but the “self intersection points of can be interpreted as generic points on , where . More properly, the self- intersection point of is the generic point of , taken with multiplicity . Alternatively, one can “solve” (or motivate) this problem algebraically by dualizing, and looking at the class of – this both gives a number, and raises the question of a geometric interpretation.
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