The identity element of an additive monoid or group or of any other algebraic structure (e.g., ring, module, abstract vector space, algebra) equipped with an addition. It is also called the additive identity and is denoted 0. The name and the symbol are borrowed from the ring of integers whose additive identity is, of course, number 0.The zero element of a ring has the property that for all and, moreover, for every element of an -module it holds that . Here, the indices distinguish the zero element of the ring from the zero element of the module. The latter also fulfils the rule for all . The notation 0 is sometimes also used for the universal bound of a Boolean algebra . In fact it behaves with respect to the operation like a zero element with respect to multiplication, since for all ...
The direct sum of modules and is the module(1)where all algebraic operations are defined componentwise. In particular, suppose that and are left -modules, then(2)and(3)where is an element of the ring . The direct sum of an arbitrary family of modules over the same ring is also defined. If is the indexing set for the family of modules, then the direct sum is represented by the collection of functions with finite support from to the union of all these modules such that the function sends to an element in the module indexed by .The dimension of a direct sum is the sum of the dimensions of the quantities summed. The significant property of the direct sum is that it is the coproduct in the category of modules. This general definition gives as a consequence the definition of the direct sum of Abelian groups and (since they are -modules, i.e., modules over the integers) and the direct sum of vector spaces (since they are modules over a field). Note that the direct..
The tensor product of two vector spaces and , denoted and also called the tensor direct product, is a way of creating a new vector space analogous to multiplication of integers. For instance,(1)In particular,(2)Also, the tensor product obeys a distributive law with the directsum operation:(3)The analogy with an algebra is the motivation behind K-theory. The tensor product of two tensors and can be implemented in the Wolfram Language as: TensorProduct[a_List, b_List] := Outer[List, a, b]Algebraically, the vector space is spanned by elements of the form , and the following rules are satisfied, for any scalar . The definition is the same no matter which scalar field is used.(4)(5)(6)One basic consequence of these formulas is that(7)A vector basis of and of gives a basis for , namely , for all pairs . An arbitrary element of can be written uniquely as , where are scalars. If is dimensional and is dimensional, then has dimension .Using tensor products,..
The cokernel of a group homomorphism of Abelian groups (modules, or abstract vector spaces) is the quotient group (quotient module or quotient space, respectively) .
A tensor category is strict if the maps , , and are always identities.A related notion is that of a tensor R-category.
A short exact sequence of groups , , and is given by two maps and and is written(1)Because it is an exact sequence, is injective, and is surjective. Moreover, the group kernel of is the image of . Hence, the group can be considered as a (normal) subgroup of , and is isomorphic to .A short exact sequence is said to split if there is a map such that is the identity on . This only happens when is the direct product of and .The notion of a short exact sequence also makes sense for modules and sheaves. Given a module over a unit ring , all short exact sequences(2)are split iff is projective, and all short exact sequences(3)are split iff is injective.A short exact sequence of vector spaces is alwayssplit.
Given a finitely generated -graded module over a graded ring (finitely generated over , which is an Artinian local ring), the Hilbert function of is the map such that, for all ,(1)where denotes the length. If is the dimension of , then there exists a polynomial of degree with rational coefficients (called the Hilbert polynomial of ) such that for all sufficiently large .The power series(2)is called the Hilbert series of . It is a rational function that can be written in a unique way in the form(3)where is a finite linear combination with integer coefficients of powers of and . If is positively graded, i.e., for all , then is an ordinary polynomial with integer coefficients in the variable . If moreover , then , i.e., the Hilbert series is a polynomial.If has a finite graded free resolution(4)then(5)Moreover, if is a regular sequence over of homogeneous elements of degree 1, then the Hilbert function of the -dimensional quotient module is(6)and in particular,(7)These..
There are no fewer than two closely related but somewhat different notions of gerbe in mathematics.For a fixed topological space , a gerbe on can refer to a stack of groupoids on satisfying the properties 1. for subsets open, and 2. given objects , any point has a neighborhood for which there is at least one morphism in . The second definition is due to Giraud (Brylinski 1993). Given a manifold and a Lie group , a gerbe with band is a sheaf of groupoids over satisfying the following three properties: 1. Given any object of , the sheaf of automorphisms of this object is a sheaf of groups on which is locally isomorphic to the sheaf of smooth -valued functions. Such a local isomorphism is unique up to inner automorphisms of . 2. Given two objects and of , there exists a surjective local homeomorphism such that and are isomorphic. In particular, and are locally isomorphic. 3. There exists a surjective local homeomorphism such that the category is non-empty. Clearly,..
A band over a fixed topological space is represented by a cover , , and for each , a sheaf of groups on along with outer automorphisms satisfying the cocycle conditions and . Here, restrictions of the cover to a finer cover should be viewed as defining the exact same band.The collection of all bands over the space with respect to a single cover has a natural category structure. Indeed, if and are two bands over with respect to , then an isomorphism consists of outer automorphisms compatible on overlaps so that . The collection of all such bands and isomorphisms thereof forms a category.The notion of band is essential to the study of gerbes (Moerdijk). In particular, for a gerbe over a topological space , one can choose an open cover of by open subsets , and for each , an object which together form a sheaf of groups on . One can then consider a collection of sheaf isomorphisms between any two groups and which forms a collection of well-defined outer automorphisms.In..
Exterior algebra is the algebra of the wedge product, also called an alternating algebra or Grassmann algebra. The study of exterior algebra is also called Ausdehnungslehre or extensions calculus. Exterior algebras are graded algebras.In particular, the exterior algebra of a vector space is the direct sum over in the natural numbers of the vector spaces of alternating differential k-forms on that vector space. The product on this algebra is then the wedge product of forms. The exterior algebra for a vector space is constructed by forming monomials , , , etc., where , , , , , and are vectors in and is wedge product. The sums formed from linear combinations of the monomials are the elements of an exterior algebra.The exterior algebra of a vector space can also bedescribed as a quotient vector space,(1)where is the subspace of -tensors generated by transpositions such as and denotes the vector space tensor product. The equivalence class is denoted..
Let be a pair consisting of finite, connected CW-complexes where is a subcomplex of . Define the associated chain complex group-wise for each by setting(1)where denotes singular homology with integer coefficients and where denotes the union of all cells of of dimension less than or equal to . Note that is free Abelian with one generator for each -cell of .Next, consider the universal covering complexes of and , respectively. The fundamental group of can be identified with the group of deck transformations of so that each determines a map(2)which then induces a chain map(3)The chain map turns each chain group into a module over the group ring which is -free with one generator for each -cell of and which is finitely generated over due to the finiteness of .Hence, there is a free chain complex(4)over , the homology groups of which are zero due to the fact that deformation retracts onto . A simple argument shows the existence of a so-called preferred basis..
The wedge product is the product in an exterior algebra. If and are differential k-forms of degrees and , respectively, then(1)It is not (in general) commutative, but it is associative,(2)and bilinear(3)(4)(Spivak 1999, p. 203), where and are constants. The exterior algebra is generated by elements of degree one, and so the wedge product can be defined using a basis for :(5)when the indices are distinct, and the product is zero otherwise.While the formula holds when has degree one, it does not hold in general. For example, consider :(6)(7)(8)If have degree one, then they are linearly independent iff .The wedge product is the "correct" type of product to use in computinga volume element(9)The wedge product can therefore be used to calculate determinants and volumes of parallelepipeds. For example, write where are the columns of . Then(10)and is the volume of the parallelepiped spanned by ...
A natural transformation is called unital if the leftmost diagram above commutes. Similarly, a natural transformation is called unital if the diagram on the right-hand side above commutes.Note that in these definitions, , , and are all objects in a tensor category , is the neutral (or identity) object in , and the juxtaposition is shorthand for the tensor product in . What's more, the subscripts attached to the transformations and denote the components of the functors (indexed with respect to the objects in ) in question.
A set in which can be reduced to one of its points, say , by a continuous deformation, is said to be contractible. The transformation is such that each point of the set is driven to through a path with the properties that 1. Each path runs entirely inside the set. 2. Nearby points move on "neighboring" paths. Condition (1) implies that a disconnected set,i.e., a set consisting of separate parts, cannot be contractible.Condition (2) implies that the circumference of a circle is not contractible. The latter follows by considering two near points and lying on different sides of a point . The paths connecting and with are either opposite each other or have different lengths. A similar argument shows that, in general, for all , the -sphere (i.e., the boundary of the -dimensional ball) is not contractible.A gap or a hole in a set can be an obstruction to contractibility. There are, however, examples of contractible sets with holes, for example,..
The homomorphism which, according to the snake lemma, permits construction of an exact sequence(1)from the above commutative diagram with exact rows. The homomorphism is defined by(2)for all , denotes the image, and is obtained through the following construction, based on diagram chasing.1. Exploit the surjectivity of to find such that . 2. Since because of the commutativity of the right square, belongs to , which is equal to due to the exactness of the lower row at . This allows us to find such that . While the elements and are not uniquely determined, the coset is, as can be proven by using more diagram chasing. In particular, if and are other elements fulfilling the requirements of steps (1) and (2), then and , and(3)hence because of the exactness of the upper row at . Let be such that(4)Then(5)because the left square is commutative. Since is injective, it follows that(6)and so(7)..
The interior product is a dual notion of the wedge product in an exterior algebra , where is a vector space. Given an orthonormal basis of , the forms(1)are an orthonormal basis for . They define a metric on the exterior algebra, . The interior product with a form is the adjoint of the wedge product with . That is,(2)for all . For example,(3)and(4)where the are orthonormal, are two interior products.An inner product on gives an isomorphism with the dual vector space . The interior product is the composition of this isomorphism with tensor contraction.
A commutative diagram is a collection of maps in which all map compositions starting from the same set and ending with the same set give the same result. In symbols this means that, whenever one can form two sequences(1)and(2)the following equality holds:(3)Commutative diagrams are usually composed by commutative triangles and commutative squares.Commutative triangles and squares can also be combined to form plane figures or space arrangements.A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets.A looped arrow indicates a map from a set to itself.The above commutative diagram expresses the fact that is the inverse map to , since it is a pictorial translation of the map equalities and .This can also be represented using two separate diagrams.Many other mathematical concepts and properties, especially in algebraic topology, homological algebra, and category theory, can be formulated..
In category theory, a tensor category consists of a category , an object of , a functor , and a natural isomorphism(1)(2)(3)where the data are subject to the following axioms: 1. Given four objects , , , and of , the top diagram above commutes. 2. Given two objects and of , the bottom diagram above commutes. In the above, is called the tensor product, is called the associator, is called the right unit, and is called the left unit of the tensor category. The object is referred to as the neutral element or the identity of the tensor product.If the maps , , and are always identities, the tensor category in question is said to be strict.A related notion is that of a tensor R-category.
In algebraic topology, the Reidemeister torsion is a notion originally introduced as a topological invariant of 3-manifolds which has now been widely adapted to a variety of contexts. At the time of its discovery, the Reidemeister torsion was the first 3-manifold invariant able to distinguish between manifolds which are homotopy equivalent but not homeomorphic. Since then, the notion has been adapted to higher-dimensional manifolds, knots and links, dynamical systems, Witten's equations, and so on. In particular, it has a number of different definitions for various contexts.For a commutative ring , let be a finite acyclic chain complex of based finitely generated free R-modules of the form(1)The Reidemeister torsion of is the value defined by(2)where is the set of units of , is a chain contraction, is the boundary map, and(3)is a map from to . In this context, Reidemeister torsion is sometimes referred to as the torsion of the complex (Nicolaescu..
Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces (Gardner 1984, pp. 9-10). Formally, the th Betti number is the rank of the th homology group of a topological space. The following table gives the Betti number of some common surfaces.surfaceBetti numbercross-cap1cylinder1klein bottle2Möbius strip1plane lamina0projective plane1sphere0torus2Let be the group rank of the homology group of a topological space . For a closed, orientable surface of genus , the Betti numbers are , , and . For a nonorientable surface with cross-caps, the Betti numbers are , , and .The Betti number of a finitely generated Abelian group is the (uniquely determined) number such thatwhere , ..., are finite cyclic..
Let be a compact -dimensional oriented Riemannian manifold without boundary, let be a group representation of by orthogonal matrices, and let be the associated vector bundle. Suppose further that the Laplacian is strictly negative on where is the linear space of differential k-forms on with values in . In this context, the analytic torsion is defined as the positive real root ofwhere the -function is defined byfor the collection of eigenvalues of , the restriction of to the collection of bundle sections of the sheaf .Intrinsic to the above computation is that is a real manifold. However, there is a collection of literature on analytic torsion for complex manifolds, the construction of which is nearly identical to the construction given above. Analytic torsion on complex manifolds is sometimes called del bar torsion...
The th exterior power of an element in an exterior algebra is given by the wedge product of with itself times. Note that if has odd degree, then any higher power of must be zero. The situation for even degree forms is different. For example, if(1)then(2)(3)(4)
Let be functors between categories and . A natural transformation from to consists of a family of morphisms in which are indexed by the objects of so that, for each morphism between objects in , the equalityholds. The elements are called the components of the natural transformation.If all the components are isomorphisms in , then is called a natural isomorphism between and . In this case, one writes .
A natural transformation between functors of categories and is said to be a natural isomorphism if each of the components is an isomorphism in .
An equalizer of a pair of maps in a category is a map such that 1. , where denotes composition. 2. For any other map with the same property, there is exactly one map such that i.e., one has the above commutative diagram. It can be shown that the equalizer is a monomorphism.Moreover, it is unique up to isomorphism.In the category of sets, the equalizer is given by thesetand by the inclusion map of the subset in .The same construction is valid in the categories of additive groups, rings, modules, and vector spaces. For these, the kernel of a morphism can be viewed, in a more abstract categorical setting, as the equalizer of and the zero map.The dual notion is the coequalizer.
Given two additive groups (or rings, or modules, or vector spaces) and , the map such that for all is called the zero map. It is a homomorphism in the category of groups (or rings or modules or vector spaces).
Given a module over a unit ring , the set of its module endomorphisms is a ring with respect to the addition of maps,and the product given by map composition,The endomorphism ring of is, in general, noncommutative, but it is always a unit ring (its unit element being the identity map on ).