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Snake Lemma
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called ''connecting homomorphisms''. Statement In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram: : where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernels and cokernels of ''a'', ''b'', and ''c'': :\ker a ~~ \ker b ~~ \ker c ~\overset~ \operatornamea ~~ \operatornameb ~~ \operatornamec where ''d'' is a homomorphism, known as the ''connecting homomorphism''. Furthermore, if the morphism ''f'' is a monomorphism, then so is the morphism \ker a ~~ \ker b, and if ''g is an epimorphism, then so is \ope ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snake
Snakes are elongated, limbless, carnivorous reptiles of the suborder Serpentes . Like all other squamates, snakes are ectothermic, amniote vertebrates covered in overlapping scales. Many species of snakes have skulls with several more joints than their lizard ancestors, enabling them to swallow prey much larger than their heads (cranial kinesis). To accommodate their narrow bodies, snakes' paired organs (such as kidneys) appear one in front of the other instead of side by side, and most have only one functional lung. Some species retain a pelvic girdle with a pair of vestigial claws on either side of the cloaca. Lizards have evolved elongate bodies without limbs or with greatly reduced limbs about twenty-five times independently via convergent evolution, leading to many lineages of legless lizards. These resemble snakes, but several common groups of legless lizards have eyelids and external ears, which snakes lack, although this rule is not universal (see Amphisbaen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with index 2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian if and only if and simple if and only if or . A5 is the smallest non-abelian simple group, having order 60, and the smallest non- solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent cubic to a quartic, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation for thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nine Lemma
right In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well. The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance ... (to the two bottom rows in the first case, and to the two top rows in the second case). Linderh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 not only valid for abelian categories but also works in the category of groups, for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other. Statements Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups. : file:5 lemma.svg The five lemma states that, if the rows are exact, ''m'' and ''p'' are isomorphisms, ''l'' is an epimorphism, and ''q'' is a monomorphism, then ''n'' is also an isomorphism. The two four-lemmas state: Proof The method of proof we shall use is commonly referred to as diagram chasing. We shall prove the five lemma by individually proving each of the two fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snake Lemma Nat2
Snakes are elongated, limbless, carnivorous reptiles of the suborder Serpentes . Like all other squamates, snakes are ectothermic, amniote vertebrates covered in overlapping scales. Many species of snakes have skulls with several more joints than their lizard ancestors, enabling them to swallow prey much larger than their heads (cranial kinesis). To accommodate their narrow bodies, snakes' paired organs (such as kidneys) appear one in front of the other instead of side by side, and most have only one functional lung. Some species retain a pelvic girdle with a pair of vestigial claws on either side of the cloaca. Lizards have evolved elongate bodies without limbs or with greatly reduced limbs about twenty-five times independently via convergent evolution, leading to many lineages of legless lizards. These resemble snakes, but several common groups of legless lizards have eyelids and external ears, which snakes lack, although this rule is not universal (see Amphisbaenia, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snake Lemma Nature
Snakes are elongated, limbless, carnivorous reptiles of the suborder Serpentes . Like all other squamates, snakes are ectothermic, amniote vertebrates covered in overlapping scales. Many species of snakes have skulls with several more joints than their lizard ancestors, enabling them to swallow prey much larger than their heads (cranial kinesis). To accommodate their narrow bodies, snakes' paired organs (such as kidneys) appear one in front of the other instead of side by side, and most have only one functional lung. Some species retain a pelvic girdle with a pair of vestigial claws on either side of the cloaca. Lizards have evolved elongate bodies without limbs or with greatly reduced limbs about twenty-five times independently via convergent evolution, leading to many lineages of legless lizards. These resemble snakes, but several common groups of legless lizards have eyelids and external ears, which snakes lack, although this rule is not universal (see Amphisbaenia, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X : F( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mitchell's Embedding Theorem
Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd. Details The precise statement is as follows: if A is a small abelian category, then there exists a ring ''R'' (with 1, not necessarily commutative) and a full, faithful and exact functor ''F'': A → ''R''-Mod (where the latter denotes the category of all left ''R''-modules). The functor ''F'' yields an equivalence between A and a full subcategory of ''R''-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in ''R''-Mod. Such an equivalence is necessarily additive. The theorem thus essentially says that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Diagram
350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. Description A commutative diagram often consists of three parts: * objects (also known as ''vertices'') * morphisms (also known as ''arrows'' or ''edges'') * paths or composites Arrow symbols In algebra texts, the type of morphism can be denoted with different arrow usages: * A monomorphism may be labeled with a \hookrightarrow or a \rightarrowtail. * An epimorphism may be labeled with a \twoheadrightarrow. * An isomorphism may be labeled with a \overset. * The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as \exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
