Exact sequence

Summary

An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an ) such that the of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0;\xrightarrow{\ f_1\ }; G_1 ;\xrightarrow{\ f_2\ }; G_2 ;\xrightarrow{\ f_3\ }; \cdots ;\xrightarrow{\ f_n\ }; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname{im}(f_i)=\ker(f_{i+1}). The sequence is called exact if it is exact at each G_i for all 1\leq i

Official source

https://en.wikipedia.org/wiki/Exact_sequence

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (21)

Related publications

Related people

Related units

Related concepts (55)

In mathematics, an abelian category is a in which morphisms and can be added and in which s and cokernels exist and have desirable properties. The motivating prototypical example of an abelian cate

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, sin

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped wit

Related concepts

Related courses (15)

Related courses

Related lectures (33)

Related lectures

Characteristic classes as complete obstructions

Martina Rovelli

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group.

SPRINGER HEIDELBERG2019

Towards new invariants for principal bundles

Martina Rovelli

We investigate the theory of principal bundles from a homotopical point of view. In the first part of the thesis, we prove a classification of principal bundles over a fixed base space, dual to the well-known classification of bundles with a fixed structure group. This leads to an adjointness property in a homotopical context between the classifying space and the loop space. We then focus on characteristic classes, which are invariants for principal bundles that take values in the cohomology of the base space. Each characteristic class captures different geo- metric features of principal bundles. We propose a uniform treatment to interpret most of known characteristic classes as obstructions to group reduction and to the extension of a universal cocycle. By plugging in the correct parameters, the method recovers several classical theorems. Afterwards, we construct a long exact sequence of abelian groups for any principal bundle. This sequence involves the cohomology of the base space and the group cohomology of the structure group. Moreover the connecting map is deeply related with the characteristic classes of the bundle.

EPFL2017

Cellular Homotopy Excision

Kay Remo Werndli

There is a classical "duality" between homotopy and homology groups in that homotopy groups are compatible with homotopy pullbacks (every homotopy pullback gives rise to a long exact sequence in homotopy), while homology groups are compatible with homotopy pushouts (every homotopy pushout gives rise to a long exact sequence in homology). This last statement is sometimes referred to as the Mayer-Vietoris or excision axiom. The classical Blakers-Massey theorem (or homotopy excision theorem) asks to what extent the excision property for homotopy pushouts remains true if we replace homology groups by homotopy groups and gives a range in which the excision property holds. It does so by estimating the connectivity of a certain comparison map, which is a rather crude measure, as it is just a single number. Since connectivity is a special case of a cellular inequality, the hope is that there is a stronger statement hidden behind the connectivity result in terms of such inequalities. This process of generalising the homotopy excision theorem has been initiated by Chachólski in the 90s, where he proved a more general version for homotopy pushout squares. The caveat was that one had to suspend the comparison map in question first and the goal of our project -- which we obtained -- was to lose this suspension and then move on to cubical diagrams, rather than squares. To do so, there are a few basic ingredients that are necessary. We first talk about our abstract approach to derived functors, then construct left Bousfield localisations of combinatorial model categories and finally, generalise the foundational concepts in the theory of closed classes to non-connected spaces.

EPFL2016