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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 (to the two bottom rows in the first case, and to the two top rows in the second case). Linderholm (p. 201) offers a satirical view of the nine lemma: :"Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but ... [...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 (to the two bottom rows in the first case, and to the two top rows in the second case). Linderholm (p. 201) offers a satirical view of the nine lemma: :"Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but ... [...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] |
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 \exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagram Chasing
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 \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tic-tac-toe
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row is the winner. It is a solved game, with a forced draw assuming best play from both players. Gameplay Tic-tac-toe is played on a three-by-three grid by two players, who alternately place the marks X and O in one of the nine spaces in the grid. In the following example, the first player (''X'') wins the game in seven steps: There is no universally-agreed rule as to who plays first, but in this article the convention that X plays first is used. Players soon discover that the best play from both parties leads to a draw. Hence, tic-tac-toe is often played by young children who may not have discovered the optimal strategy. Because of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Made Difficult
''Mathematics Made Difficult'' is a book by Carl E. Linderholm that uses advanced mathematical methods to prove results normally shown using elementary proofs. Although the aim is largely satirical, it also shows the non-trivial mathematics behind operations normally considered obvious, such as numbering, counting, and factoring integers. Linderholm discusses these seemingly-obvious ideas using concepts like categories and monoids. As an example, the proof that 2 is a prime number starts: It is easily seen that the only numbers between 0 and 2, including 0 but excluding 2, are 0 and 1. Thus the remainder left by any number on division by 2 is either 0 or 1. Hence the quotient ring ''Z''/2''Z'', where 2''Z'' is the ideal in Z generated by 2, has only the elements and where these are the images of 0 and 1 under the canonical quotient map. Since must be the unit of this ring, every element of this ring except is a unit, and the ring is a field Field may refer to: Expan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other 'tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
