Duality

1 equals 1: it is a true statement asserting that "true and true results to true". (d) 0 + 0 equals 0 : (d) is the dual of (c): it is a true statement asserting, correctly, that "false or false will always evaluate to false".

A duality in mathematics is a one-to-one translation of concepts, theorems, or mathematical structures into other concepts, theorems, or structures, frequently (but not always) using an involution operation: if B is the dual of A, then the dual of B is A. Such involutions can have fixed points, resulting in A's dual being A. Desargues' theorem, for example, is self-dual in this sense under projective geometry's standard duality.

Duality has a variety of interpretations in mathematics. It has been regarded as a pervasive and important notion in mathematics, as well as a major general theme with applications in practically every branch of the subject.

Many mathematical dualities between objects of two kinds relate to pairings, bilinear functions from one type to another type to a family of scalars. For example, linear algebra duality corresponds to bilinear maps from pairs of vector spaces to scalars, distributions, and associated test functions duality corresponds to the pairing of integrating a distribution against a test function, and Poincaré duality corresponds to intersection number, which is viewed as a pairing between submanifolds of a given manifold. For more information, click here.

It is feasible to identify geometric transformations in some projective planes that map each point of the projective plane to a line, and each line of the projective plane to a point, while keeping the incidence. A general principle of duality in projective planes emerges for such planes: given any theorem in such a plane projective geometry, replacing the terms "point" and "line" everywhere results in a new, equally valid theorem. The statement "two points determine a unique line, the line passing through these points," for example, has the dual statement "two lines determine a unique point, the intersection point of these two lines."

The dual vector space provides a conceptual explanation for this phenomenon in some planes (most notably field planes). In fact, the projective plane's points correspond to one-dimensional subvector spaces, whereas the plane's lines belong to two-dimensional subvector spaces W. The duality in such projective geometries arises from giving to a one-dimensional V the subspace of linear maps that fulfill. This space is two-dimensional as a result of the linear algebra dimension formula, i.e. it becomes corresponding to a line in the projective plane.

Using l-adic cohomology with Ql-coefficients instead, the same duality pattern applies for a smooth projective variety over a separably closed field. This is extended to possibly singular varieties by employing intersection cohomology instead, resulting in a duality known as Verdier duality. Serre duality or coherent duality is identical to the previous claims, but they apply to the cohomology of coherent sheaves.

The new age formulation of these dualities can be seen using derived categories and certain direct and inverse image functors of sheaves (with respect to the classical analytical topology on manifolds for Poincaré duality, l-adic sheaves and the étale topology in the second case, and with respect to coherent sheaves for cohesion duality).

In arithmetic, étale cohomology of finite, local, and global fields (also known as Galois cohomology, because étale cohomology over a field is equivalent to group cohomology over the field's (absolute) Galois group) admits comparable pairings. The profinite completion of Z, the integers, is isomorphic to the absolute Galois group G(Fq) of a finite field. As a result, it's the ideal match.

A duality between commutative C*-algebras A and compact Hausdorff spaces is known as Gelfand duality. The space of continuous functions (which disappear at infinity) from X to C, the complex numbers, is assigned to X. In contrast, the spectrum of A may be reconstructed from space X. Gelfand and Pontryagin’s duality can both be derived in a large formal, category-theoretic manner.

In algebraic geometry, there is a duality between commutative rings and affine schemes: there is an affine spectrum, Spec A, for every commutative ring A. By taking global sections of the structural sheaf OS, one can receive back a ring given an affine scheme S. Furthermore, ring homomorphisms have a one-to-one relationship with affine scheme morphisms, implying equivalence.

In a number of cases, the two categories that are dual to each other are really derived from partially ordered sets, i.e., there is a sense that one object is "smaller" than another. A Galois link is a duality that respects the orderings in question. The typical duality in Galois theory given in the introduction is an example: a larger field extension corresponds to a smaller group under the mapping that assigns the Galois group Gal (/ L) to any extension L K (within any fixed bigger field).

A full Heyting algebra is formed by the collection of all open subsets of a topological space X. There is a duality that connects sober environments and spatial places, known as Stone duality. Head to the Vedantu app and website for gaining access to free study materials.