- Homological algebra
**Homological algebra**is the branch ofmathematics which studies homology 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 of modules and syzygies) at the end of the 19th century, chiefly byHenri Poincaré andDavid Hilbert .The development of homological algebra was closely intertwined with the emergence of

category theory . By and large, homological algebra is the study of homologicalfunctor s and the intricate algebraic structures that they entail. The hidden fabric of mathematics is woven of, which manifest themselves through their homology andchain complex escohomology . Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules,topological space s, and other 'tangible' mathematical objects. A powerful tool for doing this is provided byspectral sequence s.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes

commutative algebra ,algebraic geometry ,algebraic number theory ,representation theory ,mathematical physics ,operator algebra s,complex analysis , and the theory ofpartial differential equation s.K-theory is an independent discipline which draws upon methods of homological algebra, as doesnoncommutative geometry ofAlain Connes .**Chain complexes and homology**The

is the central notion of homological algebra. It is a sequence $(C\_ullet,\; d\_ullet)$ ofchain complex abelian group s andgroup homomorphism s, with the property that the composition of any two consecutive maps is zero:: $C\_ullet:\; cdots\; o\; C\_\{n+1\}\; egin\{matrix\}\; d\_\{n+1\}\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}C\_n\; egin\{matrix\}\; d\_n\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}C\_\{n-1\}\; egin\{matrix\}\; d\_\{n-1\}\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}cdots,\; quad\; d\_n\; circ\; d\_\{n+1\}=0.$ The elements of "C"_{"n"}are called "n"-**chains**and the homomorphisms "d"_{"n"}are called the**boundary maps**or**differentials**. The**chain groups**"C"_{"n"}may be endowed with extra structure; for example, they may bevector space s or modules over a fixed ring "R". The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of "R"-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the category**Ab**of abelian groups); a celebrated theorem by Barry Mitchell implies the results will generalize to anyabelian category . Every chain complex defines two further sequences of abelian groups, the**cycles**"Z"_{"n"}= Ker "d"_{"n"}and the**boundaries**"B"_{"n"}= Im "d"_{"n"+1}, where Ker "d" and Im "d" denote the kernel and the image of "d". 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 automatically normal; therefore we can define the "n"th

**homology group**"H"_{"n"}("C") as thefactor group of the "n"-cycles by the "n"-boundaries,: $H\_n(C)\; =\; Z\_n/B\_n\; =\; operatorname\{Ker\},\; d\_n/\; operatorname\{Im\},\; d\_\{n+1\}.$

A chain complex is called

**acyclic**or an**exact sequence**if all its homology groups are zero.Chain complexes arise in abundance in algebra and

algebraic topology . For example, if "X" is atopological space then thesingular chain s "C"_{"n"}("X") are formallinear combination s ofcontinuous map s from the standard "n"-simplex into "X"; if "K" is asimplicial complex then the simplicial chains "C"_{"n"}("K") are formal linear combinations of the "n"-simplices of "X"; if "A" = "F"/"R" is a presentation of an abelian group "A" bygenerators and relations , where "F" is afree abelian group spanned by the generators and "R" is the subgroup of relations, then letting "C"_{1}("A") = "R", "C"_{0}("A") = "F", and "C"_{"n"}("A") = 0 for all other "n" defines a sequence of abelian groups. In all these cases, there are natural differentials "d"_{"n"}making "C"_{"n"}into a chain complex, whose homology reflects the structure of the topological space "X", the simplicial complex "K", or the abelian group "A". 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,manifold s.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 objects "X" and "Y" are connected by a map "f" between them. Homological algebra studies the relation, induced by the map "f", between chain complexes associated to "X" and "Y" 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 the functorial properties of various constructions of chain complexes and of the homology of these complexes.

* An object "X" admits multiple descriptions (for example, as a topological space and as a simplicial complex) or the complex $C\_ullet(X)$ is constructed using some 'presentation' of "X", which involves non-canonical choices. It is important to know the effect of change in the description of "X" on chain complexes associated to "X". Typically, the complex and its homology $H\_ullet(C)$ 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 an invariant of "X".**Functoriality**A continuous map of topological spaces gives rise to a homomorphism between their "n"th

homology group s for all "n". 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.A

**morphism**between two chain complexes, $F:\; C\_ullet\; o\; D\_ullet$, is a family of homomorphisms of abelian groups "F"_{"n"}:"C"_{"n"}→ "D"_{"n"}that commute with the differentials, in the sense that "F"_{"n" -1}• "d"_{"n"}^{"C"}= "d"_{"n"}^{"D"}• "F"_{"n"}for all "n". A morphism of chain complexes induces a morphism $H\_ullet(F)$ of their homology groups, consisting of the homomorphisms "H"_{"n"}("F"): "H"_{"n"}("C") → "H"_{"n"}("D") for all "n". A morphism "F" is called a**quasi-isomorphism**if it induces an isomorphism on the "n"th homology for all "n".Many constructions of chain complexes arising in algebra and geometry, including

singular homology , have the following functoriality property: if two objects "X" and "Y" are connected by a map "f", then the associated chain complexes are connected by a morphism "F" = "C"("f") from $C\_ullet(X)$ to $C\_ullet(Y),$ and moreover, the composition "g" • "f" of maps "f": "X" → "Y" and "g": "Y" → "Z" induces the morphism "C"("g" • "f") from $C\_ullet(X)$ to $C\_ullet(Z)$ that coincides with the composition "C"("g") • "C"("f"). It follows that the homology groups $H\_ullet(C)$ 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 $L\_ullet,\; M\_ullet,\; N\_ullet$ and two morphisms between them, $f:L\_ullet\; o\; M\_ullet,\; g:\; M\_ullet\; o\; N\_ullet,$is called an

**exact triple**, or a**short exact sequence of complexes**, and written as: $0\; ightarrow\; L\_ullet\; egin\{matrix\}\; f\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}\; M\_ullet\; egin\{matrix\}\; g\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}\; N\_ullet\; ightarrow\; 0,$

if for any "n", the sequence

: $0\; ightarrow\; L\_n\; egin\{matrix\}\; f\_n\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}\; M\_n\; egin\{matrix\}\; g\_n\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}\; N\_n\; ightarrow\; 0$

is a

short exact sequence of abelian groups. By definition, this means that "f"_{"n"}is an injection, "g"_{"n"}is asurjection , and Im "f"_{"n"}= Ker "g"_{"n"}. One of the most basic theorems of homological algebra states that, in this case, there is a**long exact sequence in homology**,: $ldots\; ightarrow\; H\_n(L)\; egin\{matrix\}\; H\_n(f)\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}\; H\_n(M)\; egin\{matrix\}\; H\_n(g)\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}\; H\_n(N),\; egin\{matrix\}\; delta\_n\; \backslash \; ightarrow\; \backslash \; ,\; end\{matrix\}\; ,H\_\{n-1\}(L)\; egin\{matrix\}\; H\_\{n-1\}(f)\; \backslash \; longrightarrow\; \backslash \; ,\; end\{matrix\}H\_\{n-1\}(M)\; ightarrowldots,$

where the homology groups of "L", "M", and "N" cyclically follow each other, and "δ"

_{"n"}are certain homomorphisms determined by "f" and "g", called the. Topological manifestations of this theorem include theconnecting homomorphism sMayer-Vietoris sequence and the long exact sequence forrelative homology .**Foundational aspects**Cohomology theories have been defined for many different objects such as

topological space s, sheaves, groups, rings,Lie algebra s, andC*-algebra s. The study of modernalgebraic geometry would be almost unthinkable withoutsheaf cohomology .Central to homological algebra is the notion of

exact sequence ; these can be used to perform actual calculations. A classical tool of homological algebra is that ofderived functor ; the most basic examples are functors Ext and Tor.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 injective module resolutions.

* 'Tohoku': The approach in a celebrated paper byAlexander Grothendieck which appeared in the Second Series of theTohoku Mathematical Journal in 1957, using theabelian category concept (to include sheaves of abelian groups).

* Thederived category ofGrothendieck and Verdier. Derived categories date back to Verdier's 1967 thesis. They are examples of triangulated categories used in a number of modern theories.These move from computability to generality.

The computational sledgehammer "par excellence" is the

spectral 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 as "

torsor s" (important inGalois cohomology ).**References***

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

*Alexander Grothendieck , "Sur quelques points d'algèbre homologique". Tôhoku Math. J. (2) 9, 1957, 119--221

*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., "An introduction to homological algebra". Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. xiv+450 pp. ISBN 0-521-43500-5; 0-521-55987-1

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