# Homological algebra

## Homological algebra Topics

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### Module kernel

The kernel of a module homomorphism is the set of all elements of which are mapped to zero. It is the kernel of as a homomorphism of additive groups, and is a submodule of .

### Module discriminant

Let a module in an integral domain for be expressed using a two-element basis aswhere and are in . Then the different of the module is defined asand the discriminant is defined as the square of the different(Cohn 1980).For imaginary quadratic fields (with ), the discriminants are given in the following table.The discriminants of real quadratic fields () are given in the following table.2346733569553737706387173973731041417411427713134378144679154782171751831953538521558622578723588989265991292961619330629431659533669797

### Divisor theory

A generalization by Kronecker of Kummer's theory of prime ideal factors. A divisor on a full subcategory of mod() is an additive mapping on with values in a semigroup of ideals on .

### Module

A module is a mathematical object in which things can be added together commutatively by multiplying coefficients and in which most of the rules of manipulating vectors hold. A module is abstractly very similar to a vector space, although in modules, coefficients are taken in rings that are much more general algebraic objects than the fields used in vector spaces. A module taking its coefficients in a ring is called a module over , or a R-module.Modules are the basic tool of homological algebra. Examples of modules include the set of integers , the cubic lattice in dimensions , and the group ring of a group. is a module over itself. It is closed under addition and subtraction (although it is sufficient to require closure under subtraction). Numbers of the form for and a fixed integer form a submodule since, for all ,and is still in .Given two integers and , the smallest module containing and is the module for their greatest common divisor, ...

### Divisible module

A module over a unit ring is called divisible if, for all which are not zero divisors, every element of can be "divided" by , in the sense that there is an element in such that . This condition can be reformulated by saying that the multiplication by defines a surjective map from to .It can be shown that every injective -module is divisible, but the converse only holds for particular classes of rings, e.g., for principal ideal domains. Since and are evidently divisible -modules, this allows us to conclude that they are also injective.An additive Abelian group is called divisible if it is so as a -module.

### Modular system basis

A basis of a modular system is any set of polynomials , , ... of such that every polynomial of is expressible in the formwhere , , ... are polynomials.

### Modular system

A set of all polynomials in variables, , ..., such that if , , and are members, then so are and , where is any polynomial in , ..., .

### Direct limit

The direct limit, also called a colimit, of a family of -modules is the dual notion of an inverse limit and is characterized by the following mapping property. For a directed set and a family of -modules , let be a direct system. is some -module with some homomorphisms , where for each , ,(1)such that if there exists some -module with homomorphisms , where for each , ,(2)then a unique homomorphism is induced and the above diagram commutes.The direct limit can be constructed as follows. For a given direct system, ,(3)letting be the -module generated by where and and are the images of and in .

### Trivial module

A module having only one element: the singleton set . It is a module over any ring with respect to the multiplication defined by(1)for every , and the addition(2)which makes it a trivial additive group. The only element is, in particular, its zero element. Therefore, a trivial module is often called the zero module, and written as .The notion of trivial module is a special case of the more general notion of trivial module structure, which can be defined on every additive Abelian group with respect to every ring by setting(3)for all and all .

### Localization

An operation on rings and modules. Given a commutative unit ring , and a subset of , closed under multiplication, such that , and , the localization of at is the ring(1)where the addition and the multiplication of the formal fractions are defined according to the natural rules,(2)and(3)The ring is a subring of via the identification .For an -module , the localization of at is defined as the tensor product , i.e., as the set of linear combinations of the elementary tensors(4)which are also denoted for short.The properties required for the subset are fulfilled by 1. The set of non zero-divisors of ; in this case is the ring of fractions of . 2. The complement of any prime ideal of : in this case the clumsy notation is replaced by . This ring is called the localization of at , and it is a local ring, with maximal ideal . The name given to this operation derives from the geometric meaning it takes when applied to the rings associated with algebraic varieties.The union..

### Diagram chasing

A proving technique in homological algebra which consists in looking for equivalent map compositions in commutative diagrams, and in exploiting the properties of injective, surjective and bijective homomorphisms and of exact sequences.The construction of the connecting homomorphism in the proof of the snake lemma is an example of diagram chasing.

### Large submodule

A submodule of a module such that for any other nonzero submodule of , the intersection is not the zero module. is also called an essential submodule of , whereas is called an essential extension of .

### Descending chain condition

The descending chain condition, commonly abbreviated "D.C.C.," is the dual notion of the ascending chain condition. The descending chain condition for a partially ordered set requires that all decreasing sequences in become eventually constant.A module fulfilling the descending chain condition iscalled Artinian.

### Irreducible submodule

A submodule of a module that is not the intersection of two submodules of in which it is properly contained. In other words, for all submodules and of ,Using a less common terminology, this is equivalent to requiring that the quotient module be meet-irreducible.

### Irreducible module

A nonzero module over a ring whose only submodules are the module itself and the zero module. It is also called a simple module, and in fact this is the name more frequently used nowadays (Rowen, 1988). Behrens' (1972, p. 23) definition includes the additional condition that be not the zero module.Sometimes, the term irreducible is used as an abbreviation for meet-irreducible (Kasch 1982), which means that the intersection of two nonzero submodules is always nonzero.These two irreducibility notions are different: every irreducible module is meet-irreducible, but the converse does not hold. For example, the submodules of are , and , so is not irreducible, whereas it is certainly meet-irreducible.This ambiguity in terminology is solved in the context of rings, since a simple ring is a ring that is irreducible as a module over itself, whereas an irreducible ring is a ring which is meet-irreducible as a module over itself.Irreducible modules..

### Decomposable

A differential k-form of degree in an exterior algebra is decomposable if there exist one-forms such that(1)where denotes a wedge product. Forms of degree 0, 1, , and are always decomposable. Hence the first instance of indecomposable forms occurs in , in which case is indecomposable.If a -form has a form envelope of dimension then it is decomposable. In fact, the one-forms in the (dual) basis to the envelope can be used as the above.Plücker's equations form a system of quadratic equations on the in(2)which is equivalent to being decomposable. Since a decomposable -form corresponds to a -dimensional subspace, these quadratic equations show that the Grassmannian is a projective algebraic variety. In particular, is decomposable if for every ,(3)where denotes tensor contraction and is the dual vector space to ...

### Tensor product functor

For every module over a unit ring , the tensor product functor is a covariant functor from the category of -modules to itself. It maps every -module to and every module homomorphism to the module homomorphismdefined byThe tensor product functor is defined similarly.

### Inverse system

For a family of -modules indexed by a directed set , letbe an -module homomorphism. Call an inverse system over when 1. for all . 2. When , .

### Inverse limit

The inverse limit of a family of -modules is the dual notion of a direct limit and is characterized by the following mapping property. For a directed set and a family of -modules , let be an inverse system. is some -module with some homomorphisms , where for each , (1)such that if there exists some -module with homomorphisms , where for each , (2)then a unique homomorphism is induced and the above diagram commutes.The inverse limit can be constructed as follows. For a given inverse system, , write(3)

### Injective module

An injective module is the dual notion to the projective module. A module over a unit ring is called injective iff whenever is contained as a submodule in a module , there exists a submodule of such that the direct sum is isomorphic to (in other words, is a direct summand of ). The subset of is an example of a noninjective -module; it is a -submodule of , and it is isomorphic to ; , however, is not isomorphic to the direct sum . The field of rationals and its quotient module are examples of injective -modules.A direct product of injective modules is always injective. The corresponding property for direct sums does not hold in general, but it is true for modules over Noetherian rings.The notion of injective module can also be characterized by means of commutative diagrams, split exact sequences, or exact functors...

### Indecomposable module

A non-zero module which is not the direct sum of two of its proper submodules. The negation of indecomposable is, of course, decomposable. An abstract vector space is indecomposable iff it has dimension 1.As a consequence of Kronecker basis theorem, an Abelian group is indecomposable iff it is either isomorphic to or to , where is a prime power. This is not the case for , and in fact we have

### Indecomposable

A p-form is indecomposable if it cannot be written as the wedge product of one-formsA -form that can be written as such a product is called decomposable.

### Cofree module

A module having dual properties with respect to a freemodule, as enumerated below. 1. Every free module is projective; every cofree module is injective. 2. For every module , there is a surjective homomorphism from a free module to ; for every module , there is an injective homomorphism from to a cofree module. 3. A module is projective iff it can be completed by a direct sum to a free module; a module is injective iff it can be completed by a direct product to a cofree module. Every cofree module over a unit ring is isomorphic to a direct productindexed on some set .

### Ideal height

The notion of height is defined for proper ideals in a commutative Noetherian unit ring . The height of a proper prime ideal of is the maximum of the lengths of the chains of prime ideals contained in ,The height of any proper ideal is the minimum of the heights of the prime ideals containing .

### Coequalizer

A coequalizer of a pair of maps in a category is a map such that 1. , where denotes composition. 2. For any other map with the same property, there is exactly one map such that i.e., one has the above commutative diagram. It can be shown that the coequalizer is an epimorphismand that, moreover, it is unique up to isomorphism.In the category of sets, the coequalizer is given bythe quotient setand by the canonical map , where is the minimal equivalence relation on that identifies and for all .The same construction is valid in the categories of additive groups, modules, and vector spaces. In these cases, the cokernel of a morphism can be viewed, in a more abstract categorical setting, as the coequalizer of and the zero map.The dual notion of the coequalizer is the equalizer.

### Horseshoe lemma

Given a short exact sequence of modules(1)let(2)(3)be projective resolutions of and , respectively. Then there is a projective resolution of (4)such that the above diagrams are commutative. Here, is the injection of the first summand, whereas is the projection onto the second factor for .The name of this lemma derives from the shape of the diagram formed by the shortexact sequence and the given projective resolutions.

### Cocycle

In a cochain complex of modulesthe module of -cocycles is the kernel of , which is a submodule of .

### Split exact sequence

A short exact sequence of groups(1)is called split if it essentially presents as the direct sum of the groups and .More precisely, one can construct a commutative diagram as diagrammed above, where is the injection of the first summand and is the projection onto the second summand , and the vertical maps are isomorphisms.Not all short exact sequences of groups are split. For example the short exact sequence diagrammed above cannot be split, since and are non isomorphic finite groups. Note that this is also a short exact sequence of -modules: this shows that being split is a distinguished property of short exact sequences also in the category of modules. In fact, it is related to particular classes of modules.Given a module over a unit ring , all short exact sequences(2)are split iff is projective, and all short exact sequences(3)are split iff is injective.A short exact sequence of vectorspaces is always split...

### Homology cycle

In a chain complex of modulesthe module of -cycles is the kernel of , which is a submodule of .

### Spectral sequence

A spectral sequence is a tool of homological algebra that has many applications in algebra, algebraic geometry, and algebraic topology. Roughly speaking, a spectral sequence is a system for keeping track of collections of exact sequences that have maps between them.There are many definitions of spectral sequences and many slight variations that are useful for certain purposes. The most common type is a "first quadrant cohomological spectral sequence," which is a collection of Abelian groups where , , and are integers, with and nonnegative and for some positive integer , usually 2. The groups come equipped with maps(1)such that(2)There is the further restriction that(3)The maps are called boundary maps.A spectral sequence may be visualized as a sequence of grids, one for each value of . The s and s denote positions on the grid, where is the -coordinate and is the -coordinate. The diagram above shows this for .The entire collection..

### Homology boundary

In a chain complex of modulesthe module of -boundaries is the image of . It is a submodule of and is contained in the module of -cycles , which is the kernel of .The complex is called exact at if .In the chain complexwhere all boundary operators are the multiplication by 4, for all the module of -boundaries is , whereas the module of -cycles is .

### Snake lemma

A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequenceThis commutative diagram shows how the first commutative diagram (shown here in blue) can be modified to exhibit the long exact sequence (shown here in red) explicitly. The map is called a connecting homomorphism and describes a curve from the end of the upper row () to the beginning of the lower row (), which suggested the name given to this lemma.The snake lemma is explained in the first scene of Claudia Weill's film Itis My Turn (1980), starring Jill Clayburgh and Michael Douglas.

### Homology

Homology is a concept that is used in many branches of algebra and topology. Historically, the term "homology" was first used in a topological sense by Poincaré. To him, it meant pretty much what is now called a bordism, meaning that a homology was thought of as a relation between manifolds mapped into a manifold. Such manifolds form a homology when they form the boundary of a higher-dimensional manifold inside the manifold in question.To simplify the definition of homology, Poincaré simplified the spaces he dealt with. He assumed that all the spaces he dealt with had a triangulation (i.e., they were "simplicial complexes"). Then instead of talking about general "objects" in these spaces, he restricted himself to subcomplexes, i.e., objects in the space made up only on the simplices in the triangulation of the space. Eventually, Poincaré's version of homology was dispensed with and..

### Coboundary

In a cochain complex of modulesthe module of -coboundaries is the image of . It is a submodule of and is contained in the module of -cocycles .The cochain complex is called exact at if .In the right complex of -modulesfor all , the th module is , and the th coboundary operator maps every element of to the residue class of in . The module of -coboundaries is the set of the residue classes of 0 and in , and the module of -cocycles the set of the residues classes of all even numbers .

### Small submodule

A submodule of a module such that for any proper submodule of , the submodule generated by is not the entire module . is also called superfluous submodule.

### Chain homotopy

Suppose and are two chain homomorphisms. Then a chain homotopy is given by a sequence of mapssuch thatwhere denotes the boundary operator.

### Chain homomorphism

Also called a chain map. Given two chain complexes and , a chain homomorphism is given by homomorphisms such thatwhere and are the boundary operators.

### Serre's problem

Serre's problem, also called Serre's conjecture, asserts that the implication "free module projective module" can be reversed for every module over the polynomial ring , where is a field (Serre 1955).The hard part of the proof, the one concerning finitely generated modules, was given simultaneously, and independently, by D. Quillen in Cambridge, Massachusetts and A. A. Suslin in Leningrad (St. Petersburg) in 1976. As a result, the statement is often referred to as the "Quillen-Suslin theorem."The solution to this difficult problem is part of the work for which Quillen wasawarded the Fields Medal in 1978.Quillen and Suslin received, for other contributions in algebra, the ColePrize in 1975 and 2000 respectively.

### Hom

Given two modules and over a unit ring , denotes the set of all module homomorphisms from to . It is an -module with respect to the addition of maps,(1)and the product defined by(2)for all . denotes the covariant functor from the category of -modules to itself which maps every module to , and maps every module homomorphism(3)to the module homomorphism(4)such that, for every ,(5)A similar definition is given for the contravariant functor , which maps to and maps to(6)where, for every ,(7)

### Chain homology

For every , the kernel of is called the group of cycles,(1)The letter is short for the German word for cycle, "Zyklus." The image is contained in the group of cycles because , and is called the group of boundaries,(2)The quotients are the homology groups of the chain.Given a short exact sequence of chaincomplexes(3)there is a long exact sequence in homology.(4)In particular, a cycle in with , is mapped to a cycle in . Similarly, a boundary in gets mapped to a boundary in . Consequently, the map between homologies is well-defined. The only map which is not that obvious is , called the connecting homomorphism, which is well-defined by the snake lemma.Proofs of this nature are (with a modicum of humor) referred to as diagramchasing.

### Hilbert series

Given a finitely generated -graded module over a graded ring (finitely generated over , which is an Artinian local ring), define the Hilbert function of as the map such that, for all ,(1)where denotes the length. If is the dimension of , then there exists a polynomial of degree with rational coefficients (called the Hilbert polynomial of ) such that for all sufficiently large .The power series(2)is called the Hilbert series of . It is a rational function that can be written in a unique way in the form(3)where is a finite linear combination with integer coefficients of powers of and . If is positively graded, i.e., for all , then is an ordinary polynomial with integer coefficients in the variable . If moreover , then , i.e., the Hilbert series is a polynomial.

### Chain equivalence

Chain equivalences give an equivalence relation on the space of chain homomorphisms. Two chain complexes are chain equivalent if there are chain maps and such that is chain homotopic to the identity on and is chain homotopic to the identity on .

### Ring regular sequence

Given a commutative unit ring , and an -module , a sequence of elements of is called a regular sequence for (or an -sequence for short), if, for all , 1. The multiplication by is injective on . 2. . If only condition (1) is fulfilled, the sequence is called weakly regular. An -sequence is usually simply called a regular sequence.

### Chain contraction

Let be a commutative ring, let be an R-module for , 1, 2, ..., and define a chain complex of the formA chain contraction is a collection of R-modules morphisms such that, for all ,Here, is the boundary map of .

### Resolution

Resolution is a widely used word with many different meanings. It can refer to resolution of equations, resolution of singularities (in algebraic geometry), resolution of modules or more sophisticated structures, etc. In a block design, a partition of a BIBD's set of blocks into parallel classes, each of which in turn partitions the set , is called a resolution (Abel and Furino 1996).A resolution of the module over the ring is a complex of -modules and morphisms and a morphism such that(1)satisfying the following conditions: 1. The composition of any two consecutive morphisms is the zero map, 2. For all , , 3. , where ker is the kernel and im is the image. Here, the quotient(2)is the th homology group.If all modules are projective (free), then the resolution is called projective (free). There is a similar concept for resolutions "to the right" of , which are called injective resolutions.In mathematical logic, the rule(3)is known as resolution..

### Hilbert cube

The Cartesian product of a countable infinity of copies of the interval . It can be denoted or , where and are the first infinite cardinal and ordinal, respectively. It is homeomorphic to the product space of any countable infinity of closed bounded positive-length intervals.According to another interesting description (Cullen 1968, pp. 164-165), the Hilbert cube can be identified up to homeomorphisms with the metric space formed by all sequences of real numbers such that for all , where the metric is defined asIt is then a subspace of the metric space called a Hilbert space which is formed by all real sequences such that the series converges.The Hilbert cube can be used to characterize classes of topologicalspaces. 1. A topological space that is second countable and T4 is homeomorphic to a subspace of the Hilbert cube. 2. A topological space that is separable andmetrizable is homeomorphic to a subspace of the Hilbert cube. Other statements..

### Chain complex

A chain complex is a sequence of maps(1)where the spaces may be Abelian groups or modules. The maps must satisfy . Making the domain implicitly understood, the maps are denoted by , called the boundary operator or the differential. Chain complexes are an algebraic tool for computing or defining homology and have a variety of applications. A cochain complex is used in the case of cohomology.Elements of are called chains. For each , the kernel of is called the group of cycles,(2)The letter is short for the German word for cycle, "Zyklus." The image is contained in the group of cycles because . It is called the group of boundaries.(3)The quotients are the homology groups of the chain.For example, the sequence(4)where every space is and each map is given by multiplication by 4 is a chain complex. The cycles at each stage are and the boundaries are . So the homology at each stage is the group of two elements . A simpler example is given by a linear transformation..

### Rees module

Given a module over a commutative unit ring and a filtration(1)of ideals of , the Rees module of with respect to is(2)which is the set of all formal polynomials in the variable in which the coefficient of is of the form , where and . It is a graded module over the Rees ring .The subscript distinguishes it from the so-called extended Rees module, defined as(3)where for all . This module includes all polynomials containing negative powers of .If is a proper ideal of , the notation (or ) indicates the (extended) Rees module of with respect to the -adic filtration.

### Cartesian equation

An equation representing a locus in the -dimensional Euclidean space. It has the form(1)where the left-hand side is some expression of the Cartesian coordinates , ..., . The -tuples of numbers fulfilling the equation are the coordinates of the points of .For example, the locus of all points in the Euclidean plane lying at distance 1 from the origin is the circle that can be represented using the Cartesian equation(2)Similarly, the locus of all points of the three-dimensional Euclidean space lying at distance 1 from the origin is a sphere of radius 1 centered at the origin can be represented using the Cartesian equation(3)Often the letters , , are used instead of indexed coordinates , , .The intersection of two loci and is the set of points whose coordinates fulfil the system of equations(4)(5)For example, the system(6)(7)represents the intersection of the coordinate plane (the set of points for which ) with the coordinate plane (the set of points..

A decomposition of a module into a direct sum of submodules. The index set for the collection of submodules is then called the grading set.Graded modules arise naturally in homology. In particular, for every integer , there exists an th homology group of a space , and usually the "total homology" of the space is considered to be the direct sum of all the s. This makes the "total" homology of a module graded over the integers.

A minimal free resolution of a finitely generated graded module over a commutative Noetherian -graded ring in which all maps are homogeneous module homomorphisms, i.e., they map every homogeneous element to a homogeneous element of the same degree. It is usually written in the form(1)where indicates the ring with the shifted graduation such that, for all ,(2)For all nonnegative integers and all integers , is the number of copies of appearing in the th module of the resolution, and is called graded Betti number. The ordinary th Betti number is .For example, if is the polynomial ring over a field , with the usual graduation, the graded free resolution of is(3)In , the constant polynomials have degree 2. It follows that has degree 5. Similarly, has degree 5 in .The graded free resolution can be used to compute the Hilbertfunction...

### Quotient module

If is a submodule of the module over the ring , the quotient group has a natural structure of -module with the product defined byfor all and all .

### Free module

The free module of rank over a nonzero unit ring , usually denoted , is the set of all sequences that can be formed by picking (not necessarily distinct) elements , , ..., in . The set is a particular example of the algebraic structure called a module since is satisfies the following properties. 1. It is an additive Abelian group with respectto the componentwise sum of sequences,(1)2. One can multiply any sequence with any element of according to the rule(2)and this product fulfils both the associative and the distributive law. The term free module extends to all modules which are isomorphic to , i.e., which have essentially the same structure as . Note that not all modules are free. For example, the quotient ring , where is an integer greater than 1 is not free, since it is a -module having elements, and therefore it cannot be isomorphic to any of the modules , which are all infinite sets. Hence it is not free as a -module, while, of course, it is free as a module..

### Baer's criterion

Baer's criterion, also known as Baer's test, states that a module over a unit ring is injective iff every module homomorphism from an ideal of to can be extended to a homomorphism from to .

### Form envelope

Given a differential p-form in the exterior algebra , its envelope is the smallest subspace such that is in the subspace . Alternatively, is spanned by the vectors that can be written as the tensor contraction of with an element of .For example, the envelope of in is , and the envelope of in is all of .

### Proof without words

A proof that is only based on visual elements, without any comments.An arithmetic identity can be demonstrated by a picture showing a self-evident equality between numerical quantities. The above figure shows that the difference between the th pentagonal number and is equal to three times the th triangular number. Of course, the situation depicted is a particular case of the formula (here it corresponds to ), but it is presented in a way that can be immediately generalized.Another form of proof without words frequently used in elementary geometry is thedissection proof.

### Flat module

A module over a unit ring is called flat iff the tensor product functor (or, equivalently, the tensor product functor ) is an exact functor.For every -module, obeys the implicationwhich, in general, cannot be reversed.A -module is flat iff it is torsion-free: hence and the infinite direct product are flat -modules, but they are not projective. In fact, over a Noetherian ring or a local ring, flatness implies projectivity only for finitely generated modules. This property, together with Serre's problem, allows it to be concluded that the three above implications are equivalences if is a finitely generated module over a polynomial ring , where is a field.

Given a commutative unit ring and a filtration(1)of ideals of , the associated graded ring of with respect to is the graded ring(2)The addition is defined componentwise, and the product is defined as follows. If is the residue class of mod , and is the residue class of mod , then is the residue class of mod . is a quotient ring of the Rees ring of with respect to ,(3)If is a proper ideal of , then the notation indicates the associated graded ring of with respect to the -adic filtration of ,(4)If is Noetherian, then is as well. Moreover is finitely generated over . Finally, if is a local ring with maximal ideal , then(5)

### Projective module

A projective module generalizes the concept of the free module. A module over a nonzero unit ring is projective iff it is a direct summand of a free module, i.e., of some direct sum . This does not imply necessarily that itself is the direct sum of some copies of . A counterexample is provided by , which is a module over the ring with respect to the multiplication defined by . Hence, while a free module is obviously always projective, the converse does not hold in general. It is true, however, for particular classes of rings, e.g., if is a principal ideal domain, or a polynomial ring over a field (Quillen and Suslin 1976). This means that, for instance, is a nonprojective -module, since it is not free.A direct sum of projective modules is always projective, but this property does not apply to direct products. For example, the infinite direct product is not a projective -module.According to its formal definition, a module is projective if, whenever is a quotient..

### Filtration

A filtration of ideals of a commutative unit ring is a sequence of idealssuch that for all indices . An example is the -adic filtration associated with a proper ideal of ,A ring equipped with a filtration is called a filteredring.

Given a module over a commutative unit ring and a filtration(1)of ideals of , the associated graded module of with respect to is(2)which is a graded module over the associated graded ring with respect to the addition and the multiplication by scalars defined componentwise.If is a proper ideal of , then the notation indicates the associated graded module of with respect to the -adic filtration of ,(3)

### Noetherian module

A module is Noetherian if it obeys the ascending chain condition with respect to inclusion, i.e., if every set of increasing sequences of submodules eventually becomes constant.If a module is Noetherian, then the following are equivalent. 1. satisfies the ascending chain condition on submodules. 2. Every submodule of is finitely generated. 3. Every set of submodules of contains a maximal element.

### Ascending chain condition

The ascending chain condition, commonly abbreviated "A.C.C.," for a partially ordered set requires that all increasing sequences in become eventually constant.A module fulfils the ascending chain condition if its set of submodules obeys the condition with respect to inclusion. In this case, is called Noetherian.

### Faithfully flat module

A module over a unit ring is called faithfully flat if the tensor product functor is exact and faithful.A faithfully flat module is always flat and faithful, but the converse does not hold in general. For example, is a faithful and flat -module, but it is not faithfully flat: in fact reduces all the quotient modules (and the maps between them) to zero, since for all and all :

### Nine lemma

A diagram lemma also known as lemma. According to its most general statement, the commutative diagram illustrated above with exact rows and columns can be completed by two morphismswithout losing commutativity.Moreover, the short exact sequenceis exact.The lemma is also true if the roles of the first and the third row are interchanged.

### Faithful module

A module over a unit ring is called faithful if for all distinct elements , of , there exists such that . In other words, the multiplications by and by define two different endomorphisms of .This condition is equivalent to requiring that whenever , , one has that for some , i.e., , so that the annihilator of is reduced to . This shows, in particular, that any torsion-free module is faithful. Hence the field of rationals and the polynomial rings are faithful -modules.More generally, any ring containing as a subring is faithful as a module over , since 1 is annihilated only by 0.The -modules are not faithful, since they are annihilated by . In general, a finite module over an infinite ring cannot be faithful, since in this case the infinitely many elements of the ring have to give rise to only a finite number of module endomorphisms...

### Acyclic chain complex

Let be a commutative ring and let be an R-module for . A chain complex of the formis said to be acyclic if its th homology group is trivial for all values .A straightforward result in homological algebra states that a chain complex with each free is acyclic if and only if there exists a chain contraction .

### Module tensor product

The tensor product between modules and is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring . The familiar formulas hold, but now is any element of ,(1)(2)(3)This generalizes the definition of a tensor product for vector spaces since a vector space is a module over the scalar field. Also, vector bundles can be considered as projective modules over the ring of functions, and group representations of a group can be thought of as modules over CG. The generalization covers those kinds of tensor products as well.There are some interesting possibilities for the tensor product of modules that don't occur in the case of vector spaces. It is possible for to be identically zero. For example, the tensor product of and as modules over the integers, , has no nonzero elements. It is enough to see that . Notice that . Then(4)since in and in . In general, it is easier to show that elements are zero than to show..

### Exact functor

A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into short exact sequences.A covariant functor is called left exact ifit preserves the exactness of all sequencesand it is called right exact if it preserves the exactness of all sequences("Left" and "right" are interchanged in the corresponding definitionsfor contravariant functors.)A functor is exact iff it isboth left and right exact.Every tensor product functor is right exact. For every module over a unit ring , the covariant functor and the contravariant functor are left exact; the first is exact iff is projective and the second iff is injective.

### Module multiplicity

Module multiplicity is a number associated with every nonzero finitely generated graded module over a graded ring for which the Hilbert series is defined. If , the Hilbert series of can be written in the formand the multiplicity of is the integerIf is the polynomial ring over the field , the multiplicity of the quotient ring , where is a polynomial of degree , is equal to . This example shows the geometric origin of the notion. The number is in fact the so-called intersection multiplicity of the algebraic variety of defined by the equation , of which is the coordinate ring (i.e., a line of chosen in a sufficiently general way intersects in distinct points).The definition of multiplicity can be extended to nonzero finitely generated modules over a Noetherian local ring . If is the maximal ideal of , one can define the multiplicity of as the multiplicity of the associated graded module of with respect to ...

### Zero module

Every module over a ring contains a so-called "zero element" which fulfils the properties suggested by its name with respect to addition,and with respect to multiplication by any element of ,This shows that the set is closed under both module operations, and, therefore, it itself is a module, called the zero module. It also deserves the name trivial module, since it is the simplest module possible.

### Module length

The length of all composition series of a module . According to the Jordan-Hölder theorem for modules, if has any composition series, then all such series are equivalent. The length of a module without composition series is conventionally set equal to .A module has finite length iff it is both Artinian and Noetherian; this includes the case where is finite.An abstract vector space has finite length iff it is finite-dimensional, and in this case the length coincides with the dimension.

### Euler system

A mathematical structure first introduced by Kolyvagin (1990) and defined as follows. Let be a finite-dimensional -adic representation of the Galois group of a number field . Then an Euler system for is a collection of cohomology classes for a family of Abelian extensions of , with a relation between and whenever (Rubin 2000, p. 4).Wiles' proof of Fermat's last theorem via the Taniyama-Shimura conjecture made use of Euler systems.