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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'' (such that the diagram commutes) has to be assumed from the start; two objects ''B'' and ''B′'' that simply have isomorphic sub- and fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
