Short five lemma

Short five lemma

In mathematics, especially homological algebra and other applications of Abelian category theory, the short five lemma is a special case of the five lemma.It states that for the following commutative diagram (in any Abelian category, or in the category of groups), if the rows are short exact sequences, and if "g" and "h" are isomorphisms, then "f" is an isomorphism as well.

It follows immediately from the five lemma.

The essence of the lemma can be summarized as follows: if you have a homomorphism "f" from an object "B" to an object "B′", and this homomorphism induces an isomorphism from a subobject "A" of "B" to a subobject "A′" of "B′" and also an isomorphism from the factor object "B"/"A" to "B′"/"A′", then "f" itself is an isomorphism. Note however that the existence of "f" has to be assumed from the start; two objects "B" and "B′" that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the category of Abelian groups, "B" could be the cyclic group of order four and "B′" the Vierergruppe).


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Five lemma — In mathematics, especially homological algebra and other applications of Abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams.The five lemma is valid not only for abelian categories but also… …   Wikipedia

  • Exact sequence — In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them… …   Wikipedia

  • List of lemmas — This following is a list of lemmas (or, lemmata , i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. 0 to 9 *0/1 Sorting Lemma ( comparison… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Abelian category — In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of… …   Wikipedia

  • List of category theory topics — This is a list of category theory topics, by Wikipedia page. Specific categories *Category of sets **Concrete category *Category of vector spaces **Category of graded vector spaces *Category of finite dimensional Hilbert spaces *Category of sets… …   Wikipedia

  • List of homological algebra topics — This is a list of homological algebra topics, by Wikipedia page. Basic techniques Cokernel Exact sequence Chain complex Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma… …   Wikipedia

  • Outline of category theory — The following outline is provided as an overview of and guide to category theory: Category theory – area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as… …   Wikipedia

  • List of algebraic topology topics — This is a list of algebraic topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Contents 1… …   Wikipedia

  • Spinoza: metaphysics and knowledge — G.H.R.Parkinson The philosophical writings of Spinoza are notoriously obscure, and they have been interpreted in many ways. Some interpreters see Spinoza as (in the words of a contemporary)1 ‘the reformer of the new [sc. Cartesian] philosophy’.… …   History of philosophy

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”