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Homological algebra

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Branch of mathematics

A diagram used in thesnake lemma, a basic result in homological algebra.

Homological algebra is the branch ofmathematics that studieshomology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations incombinatorial topology (a precursor toalgebraic topology) andabstract algebra (theory ofmodules andsyzygies) at the end of the 19th century, chiefly byHenri Poincaré andDavid Hilbert.

Homological algebra is the study of homologicalfunctors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence ofcategory theory. A central concept is that ofchain complexes, which can be studied through their homology andcohomology.

Homological algebra affords the means to extract information contained in these complexes and present it in the form of homologicalinvariants ofrings, modules,topological spaces, and other "tangible" mathematical objects. Aspectral sequence is a powerful tool for this.

It has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includescommutative algebra,algebraic geometry,algebraic number theory,representation theory,mathematical physics,operator algebras,complex analysis, and the theory ofpartial differential equations.K-theory is an independent discipline which draws upon methods of homological algebra, as does thenoncommutative geometry ofAlain Connes.

History

Homological algebra began to be studied in its most basic form in the late 19th century as a branch of topology and in the 1940s became an independent subject with the study of objects such as theext functor and thetor functor, among others.^[1]

Chain complexes and homology

Main article:Chain complex

The notion ofchain complex is central in homological algebra. An abstractchain complex is a sequence $(C_{\bullet },d_{\bullet })$ ofabelian groups andgroup homomorphisms, with the property that the composition of any two consecutivemaps is zero:

C_{\bullet }:\cdots \longrightarrow C_{n+1}{\stackrel {d_{n+1}}{\longrightarrow }}C_{n}{\stackrel {d_{n}}{\longrightarrow }}C_{n-1}{\stackrel {d_{n-1}}{\longrightarrow }}\cdots ,\quad d_{n}\circ d_{n+1}=0.

The elements ofC_n are calledn-chains and the homomorphismsd_n are called theboundary maps ordifferentials. Thechain groupsC_n may be endowed with extra structure; for example, they may bevector spaces ormodules over a fixedringR. The differentials must preserve the extra structure if it exists; for example, they must belinear maps or homomorphisms ofR-modules. For notational convenience, restrict attention to abelian groups (more correctly, to thecategoryAb of abelian groups); a celebratedtheorem by Barry Mitchell implies the results will generalize to anyabelian category. Every chain complex defines two further sequences of abelian groups, thecyclesZ_n = Kerd_n and theboundariesB_n = Imd_n+1, where Ker d and Im d denote thekernel and theimage ofd. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as

B_{n}\subseteq Z_{n}\subseteq C_{n}.

Subgroups of abelian groups are automaticallynormal; therefore we can define thenthhomology groupH_n(C) as thefactor group of then-cycles by then-boundaries,

H_{n}(C)=Z_{n}/B_{n}=\operatorname {Ker} \,d_{n}/\operatorname {Im} \,d_{n+1}.

A chain complex is calledacyclic or anexact sequence if all its homology groups are zero.

Chain complexes arise in abundance inalgebra andalgebraic topology. For example, ifX is atopological space then thesingular chainsC_n(X) are formallinear combinations ofcontinuous maps from the standardn-simplex intoX; ifK is asimplicial complex then thesimplicial chainsC_n(K) are formal linear combinations of then-simplices ofK; ifA = F/R is a presentation of an abelian groupA bygenerators and relations, whereF is afree abelian group spanned by the generators andR is the subgroup of relations, then lettingC₁(A) = R,C₀(A) = F, andC_n(A) = 0 for all othern defines a sequence of abelian groups. In all these cases, there are natural differentialsd_n makingC_n into a chain complex, whose homology reflects the structure of the topological spaceX, the simplicial complexK, or the abelian groupA. In the case of topological spaces, we arrive at the notion ofsingular homology, which plays a fundamental role in investigating the properties of such spaces, for example,manifolds.

On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes,R-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations.

Two objectsX andY are connected by a mapf between them. Homological algebra studies the relation, induced by the mapf, between chain complexes associated withX andY and their homology. This is generalized to the case of several objects and maps connecting them. Phrased in the language ofcategory theory, homological algebra studies thefunctorial properties of various constructions of chain complexes and of the homology of these complexes.
An objectX admits multiple descriptions (for example, as a topological space and as a simplicial complex) or the complex $C_{\bullet }(X)$ is constructed using some 'presentation' ofX, which involves non-canonical choices. It is important to know the effect of change in the description ofX on chain complexes associated withX. Typically, the complex and its homology $H_{\bullet }(C)$ are functorial with respect to the presentation; and the homology (although not the complex itself) is actually independent of the presentation chosen, thus it is aninvariant ofX.

Foundational aspects

Cohomology theories have been defined for many different objects such astopological spaces,sheaves,groups,rings,Lie algebras, andC*-algebras. The study of modernalgebraic geometry would be almost unthinkable withoutsheaf cohomology.

Central to homological algebra is the notion ofexact sequence; these can be used to perform actual calculations. A classical tool of homological algebra is that ofderived functor; the most basic examples are functorsExt andTor.

With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts before the subject settled down. An approximate history can be stated as follows:

Cartan–Eilenberg: In their 1956 book "Homological Algebra", these authors used projective and injectivemodule resolutions.
'Tohoku': The approach in acelebrated paper byAlexander Grothendieck which appeared in the Second Series of theTohoku Mathematical Journal in 1957, using theabelian category concept (to includesheaves of abelian groups).
Thederived category ofGrothendieck andVerdier. Derived categories date back to Verdier's 1967 thesis. They are examples oftriangulated categories used in a number of modern theories.

These move from computability to generality.

The computational sledgehammerpar excellence is thespectral sequence; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary.

There have been attempts at 'non-commutative' theories which extend first cohomology astorsors (important inGalois cohomology).

Standard tools

Exact sequences

Main article:Exact sequence

In the context ofgroup theory, a sequence

G_{0}\;{\xrightarrow {f_{1}}}\;G_{1}\;{\xrightarrow {f_{2}}}\;G_{2}\;{\xrightarrow {f_{3}}}\;\cdots \;{\xrightarrow {f_{n}}}\;G_{n}

ofgroups andgroup homomorphisms is calledexact if theimage of each homomorphism is equal to thekernel of the next:

\mathrm {im} (f_{k})=\mathrm {ker} (f_{k+1}).\!

Note that the sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain otheralgebraic structures. For example, one could have an exact sequence ofvector spaces andlinear maps, or ofmodules andmodule homomorphisms. More generally, the notion of an exact sequence makes sense in anycategory withkernels andcokernels.

Short

The most common type of exact sequence is theshort exact sequence. This is an exact sequence of the form

A\;{\overset {f}{\hookrightarrow }}\;B\;{\overset {g}{\twoheadrightarrow }}\;C

where ƒ is amonomorphism andg is anepimorphism. In this case,A is asubobject ofB, and the correspondingquotient isisomorphic toC:

C\cong B/f(A).

(wheref(A) = im(f)).

A short exact sequence of abelian groups may also be written as an exact sequence with five terms:

0\;{\xrightarrow {}}\;A\;{\xrightarrow {f}}\;B\;{\xrightarrow {g}}\;C\;{\xrightarrow {}}\;0

where 0 represents thezero object, such as thetrivial group or a zero-dimensional vector space. The placement of the 0's forces ƒ to be a monomorphism andg to be an epimorphism (see below).

Long

A long exact sequence is an exact sequence indexed by thenatural numbers.

Five lemma

Main article:Five lemma

Consider the followingcommutative diagram in anyabelian category (such as the category ofabelian groups or the category ofvector spaces over a givenfield) or in the category ofgroups.

The five lemma states that, if the rows areexact,m andp areisomorphisms,l is anepimorphism, andq is amonomorphism, thenn is also an isomorphism.

Snake lemma

Main article:Snake lemma

In anabelian category (such as the category ofabelian groups or the category ofvector spaces over a givenfield), consider acommutative diagram:

where the rows areexact sequences and 0 is thezero object.Then there is an exact sequence relating thekernels andcokernels ofa,b, andc:

\ker a\to \ker b\to \ker c{\overset {d}{\to }}\operatorname {coker} a\to \operatorname {coker} b\to \operatorname {coker} c

Furthermore, if the morphismf is amonomorphism, then so is the morphism ker a → ker b, and ifg' is anepimorphism, then so is coker b → coker c.

Abelian categories

Main article:Abelian category

Inmathematics, anabelian category is acategory in whichmorphisms and objects can be added and in whichkernels andcokernels exist and have desirable properties. The motivating prototype example of an abelian category is thecategory of abelian groups,Ab. The theory originated in a tentative attempt to unify severalcohomology theories byAlexander Grothendieck. Abelian categories are verystable categories, for example they areregular and they satisfy thesnake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category ofchain complexes of an Abelian category, or the category offunctors from asmall 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 inalgebraic geometry,cohomology and purecategory theory. Abelian categories are named afterNiels Henrik Abel.

More concretely, a category isabelian if

it has azero object,
it has all binaryproducts and binarycoproducts, and
it has allkernels andcokernels.
allmonomorphisms andepimorphisms arenormal.

Derived functors

Main article:Derived functor

Suppose we are given a covariantleft exact functorF :A →B between twoabelian categoriesA andB. If 0 →A →B →C → 0 is a short exact sequence inA, then applyingF yields the exact sequence 0 →F(A) →F(B) →F(C) and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (ifA is "nice" enough) there is onecanonical way of doing so, given by the right derived functors ofF. For everyi≥1, there is a functorRⁱF:A →B, and the above sequence continues like so: 0 →F(A) →F(B) →F(C) →R¹F(A) →R¹F(B) →R¹F(C) →R²F(A) →R²F(B) → ... . From this we see thatF is an exact functor if and only ifR¹F = 0; so in a sense the right derived functors ofF measure "how far"F is from being exact.

Ext functor

Main article:Ext functor

LetR be aring and let Mod_R be thecategory ofmodules overR. LetB be in Mod_R and setT(B) = Hom_R(A,B), for fixedA in Mod_R. This is aleft exact functor and thus has rightderived functorsRⁿT. The Ext functor is defined by

\operatorname {Ext} _{R}^{n}(A,B)=(R^{n}T)(B).

This can be calculated by taking anyinjective resolution

0\rightarrow B\rightarrow I^{0}\rightarrow I^{1}\rightarrow \cdots ,

and computing

0\rightarrow \operatorname {Hom} _{R}(A,I^{0})\rightarrow \operatorname {Hom} _{R}(A,I^{1})\rightarrow \cdots .

Then (RⁿT)(B) is thecohomology of this complex. Note that Hom_R(A,B) is excluded from the complex.

An alternative definition is given using the functorG(A)=Hom_R(A,B). For a fixed moduleB, this is acontravariant left exact functor, and thus we also have rightderived functorsRⁿG, and can define

\operatorname {Ext} _{R}^{n}(A,B)=(R^{n}G)(A).

This can be calculated by choosing anyprojective resolution

\dots \rightarrow P^{1}\rightarrow P^{0}\rightarrow A\rightarrow 0,

and proceeding dually by computing

0\rightarrow \operatorname {Hom} _{R}(P^{0},B)\rightarrow \operatorname {Hom} _{R}(P^{1},B)\rightarrow \cdots .

Then (RⁿG)(A) is the cohomology of this complex. Again note that Hom_R(A,B) is excluded.

These two constructions turn out to yieldisomorphic results, and so both may be used to calculate the Ext functor.

Tor functor

Main article:Tor functor

SupposeR is aring, and denoted byR-Mod thecategory ofleftR-modules and byMod-R the category of rightR-modules (ifR iscommutative, the two categories coincide). Fix a moduleB inR-Mod. ForA inMod-R, setT(A) =A⊗_RB. ThenT is aright exact functor fromMod-R to thecategory of abelian groupsAb (in the case whenR is commutative, it is a right exact functor fromMod-R toMod-R) and itsleft derived functorsL_nT are defined. We set

\mathrm {Tor} _{n}^{R}(A,B)=(L_{n}T)(A)

i.e., we take aprojective resolution

\cdots \rightarrow P_{2}\rightarrow P_{1}\rightarrow P_{0}\rightarrow A\rightarrow 0

then remove theA term and tensor the projective resolution withB to get the complex

\cdots \rightarrow P_{2}\otimes _{R}B\rightarrow P_{1}\otimes _{R}B\rightarrow P_{0}\otimes _{R}B\rightarrow 0

(note thatA⊗_RB does not appear and the last arrow is just the zero map) and take thehomology of this complex.

Spectral sequences

Main article:Spectral sequence

Fix anabelian category, such as a category of modules over a ring. Aspectral sequence is a choice of a nonnegative integerr₀ and a collection of three sequences:

For all integersr ≥r₀, an objectE_r, called asheet (as in a sheet ofpaper), or sometimes apage or aterm,
Endomorphismsd_r :E_r →E_r satisfyingd_rod_r = 0, calledboundary maps ordifferentials,
Isomorphisms ofE_r+1 withH(E_r), the homology ofE_r with respect tod_r.

The E₂ sheet of a cohomological spectral sequence

A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer. We have three indices,r,p, andq. For eachr, imagine that we have a sheet of graph paper. On this sheet, we will takep to be the horizontal direction andq to be the vertical direction. At each lattice point we have the object $E_{r}^{p,q}$ .

It is very common forn =p +q to be another natural index in the spectral sequence.n runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree (−r, r − 1), so they decreasen by one. In the cohomological case,n is increased by one. Whenr is zero, the differential moves objects one space down or up. This is similar to the differential on a chain complex. Whenr is one, the differential moves objects one space to the left or right. Whenr is two, the differential moves objects just like aknight's move inchess. For higherr, the differential acts like a generalized knight's move.

Functoriality

Acontinuous map of topological spaces gives rise to a homomorphism between theirnthhomology groups for alln. This basic fact ofalgebraic topology finds a natural explanation through certain properties of chain complexes. Since it is very common to studyseveral topological spaces simultaneously, in homological algebra one is led to simultaneous consideration of multiple chain complexes.

Amorphism between two chain complexes, $F:C_{\bullet }\to D_{\bullet },$ is a family of homomorphisms of abelian groups $F_{n}:C_{n}\to D_{n}$ that commute with the differentials, in the sense that $F_{n-1}\circ d_{n}^{C}=d_{n}^{D}\circ F_{n}$ for alln. A morphism of chain complexes induces a morphism $H_{\bullet }(F)$ of their homology groups, consisting of the homomorphisms $H_{n}(F):H_{n}(C)\to H_{n}(D)$ for alln. A morphismF is called aquasi-isomorphism if it induces an isomorphism on thenth homology for alln.

Many constructions of chain complexes arising in algebra and geometry, includingsingular homology, have the followingfunctoriality property: if two objectsX andY are connected by a mapf, then the associated chain complexes are connected by a morphism $F=C(f):C_{\bullet }(X)\to C_{\bullet }(Y),$ and moreover, the composition $g\circ f$ of mapsf: X → Y andg: Y → Z induces the morphism $C(g\circ f):C_{\bullet }(X)\to C_{\bullet }(Z)$ that coincides with the composition $C(g)\circ C(f).$ It follows that the homology groups $H_{\bullet }(C)$ are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.

The following definition arises from a typical situation in algebra and topology. A triple consisting of three chain complexes $L_{\bullet },M_{\bullet },N_{\bullet }$ and two morphisms between them, $f:L_{\bullet }\to M_{\bullet },g:M_{\bullet }\to N_{\bullet },$ is called anexact triple, or ashort exact sequence of complexes, and written as

0\longrightarrow L_{\bullet }{\overset {f}{\longrightarrow }}M_{\bullet }{\overset {g}{\longrightarrow }}N_{\bullet }\longrightarrow 0,

if for anyn, the sequence

0\longrightarrow L_{n}{\overset {f_{n}}{\longrightarrow }}M_{n}{\overset {g_{n}}{\longrightarrow }}N_{n}\longrightarrow 0

is ashort exact sequence of abelian groups. By definition, this means thatf_n is aninjection,g_n is asurjection, and Imf_n = Kerg_n. One of the most basic theorems of homological algebra, sometimes known as thezig-zag lemma, states that, in this case, there is along exact sequence in homology

\cdots \longrightarrow H_{n}(L){\overset {H_{n}(f)}{\longrightarrow }}H_{n}(M){\overset {H_{n}(g)}{\longrightarrow }}H_{n}(N){\overset {\delta _{n}}{\longrightarrow }}H_{n-1}(L){\overset {H_{n-1}(f)}{\longrightarrow }}H_{n-1}(M)\longrightarrow \cdots ,

where the homology groups ofL,M, andN cyclically follow each other, andδ_n are certain homomorphisms determined byf andg, called theconnecting homomorphisms. Topological manifestations of this theorem include theMayer–Vietoris sequence and the long exact sequence forrelative homology.

See also

Wikiquote has quotations related toHomological algebra.

Abstract nonsense, a term for homological algebra andcategory theory
Derivator
Homotopical algebra
List of homological algebra topics

References

^Weibel, Charles A. (1999). "History of homological algebra".History of Topology. pp. 797–836.doi:10.1016/b978-044482375-5/50029-8.ISBN 9780444823755.

Henri Cartan,Samuel Eilenberg,Homological Algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp.ISBN 0-691-04991-2
Grothendieck, Alexander (1957)."Sur quelques points d'algèbre homologique, I".Tohoku Mathematical Journal.9 (2):119–221.doi:10.2748/tmj/1178244839.
Saunders Mac Lane,Homology. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp.ISBN 3-540-58662-8
Peter Hilton; Stammbach, U.A Course in Homological Algebra. Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp.ISBN 0-387-94823-6
Gelfand, Sergei I.;Yuri Manin,Methods of Homological Algebra. Translated from Russian 1988 edition. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp.ISBN 3-540-43583-2
Gelfand, Sergei I.; Yuri Manin,Homological Algebra. Translated from the 1989 Russian original by the authors. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences (Algebra, V, Encyclopaedia Math. Sci., 38, Springer, Berlin, 1994). Springer-Verlag, Berlin, 1999. iv+222 pp.ISBN 3-540-65378-3
Weibel, Charles A. (1994).An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.ISBN 978-0-521-55987-4.MR 1269324.OCLC 36131259.

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