In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence.
More precisely, let
be a first quadrant spectral sequence, meaning that vanishes except when p and q are both non-negative. Then there is an exact sequence
Here, the map is the differential of the -term of the spectral sequence.
The sequence is a consequence of the definition of convergence of a spectral sequence. The second page differential with codomain E21,0 originates from E2−1,1, which is zero by assumption. The differential with domain E21,0 has codomain E23,−1, which is also zero by assumption. Similarly, the incoming and outgoing differentials of Er1,0 are zero for all r ≥ 2. Therefore the (1,0) term of the spectral sequence has converged, meaning that it is isomorphic to the degree one graded piece of the abutment H 1(A). Because the spectral sequence lies in the first quadrant, the degree one graded piece is equal to the first subgroup in the filtration defining the graded pieces. The inclusion of this subgroup yields the injection E21,0 → H 1(A) which begins the five-term exact sequence. This injection is called an edge map.
The E20,1 term of the spectral sequence has not converged. It has a potentially non-trivial differential leading to E22,0. However, the differential landing at E20,1 begins at E2−2,2, which is zero, and therefore E30,1 is the kernel of the differential E20,1 → E22,0. At the third page, the (0, 1) term of the spectral sequence has converged, because all the differentials into and out of Er0,1 either begin or end outside the first quadrant when r ≥ 3. Consequently E30,1 is the degree zero graded piece of H 1(A). This graded piece is the quotient of H 1(A) by the first subgroup in the filtration, and hence it is the cokernel of the edge map from E21,0. This yields a short exact sequence
Because E30,1 is the kernel of the differential E20,1 → E22,0, the last term in the short exact sequence can be replaced with the differential. This produces a four-term exact sequence. The map H 1(A) → E20,1 is also called an edge map.
The outgoing differential of E22,0 is zero, so E32,0 is the cokernel of the differential E20,1 → E22,0. The incoming and outgoing differentials of Er2,0 are zero if r ≥ 3, again because the spectral sequence lies in the first quadrant, and hence the spectral sequence has converged. Consequently E32,0 is isomorphic to the degree two graded piece of H 2(A). In particular, it is a subgroup of H 2(A). The composite E22,0 → E32,0 → H2(A), which is another edge map, therefore has kernel equal to the differential landing at E22,0. This completes the construction of the sequence.
The five-term exact sequence can be extended at the cost of making one of the terms less explicit. The seven-term exact sequence is
This sequence does not immediately extend with a map to H3(A). While there is an edge map E23,0 → H3(A), its kernel is not the previous term in the seven-term exact sequence.
For spectral sequences whose first interesting page is E1, there is a three-term exact sequence analogous to the five-term exact sequence:
Similarly for a homological spectral sequence we get an exact sequence:
In both homological and cohomological case there are also low degree exact sequences for spectral sequences in the third quadrant. When additional terms of the spectral sequence are known to vanish, the exact sequences can sometimes be extended further. For example, the long exact sequence associated to a short exact sequence of complexes can be derived in this manner.
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 named after Niels Henrik Abel.
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
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.
An exact sequence is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.
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Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.
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In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.
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In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray, they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.
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In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.
In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors , from knowledge of the derived functors of and . Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
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In mathematics, an exact couple, due to William S. Massey (1952), is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple.
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi-isomorphism; if we pass to the derived category of complexes, this means that f is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism. If we are working in a t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.
In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's original paper addresses this for singular homology.
In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence.
In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G. The spectral sequence is named after Roger Lyndon, Gerhard Hochschild, and Jean-Pierre Serre.
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