Homološka algebra

Dijagram koji se koristi u lemi zmije, osnovnom rezultatu u homološkoj algebri.

Homološka algebra je grana matematike koja izučava homologiju u opštem algebarskom okruženju. To je relativno mlada disciplina, čije poreklo se može pratiti do istraživanja kombinatorne topologije (preteče algebarske topologije) i apstraktne algebre (teorije modula i linearnih relacija) s kraja 19. veka, uglavnom zaslugom Anrija Poenkarea i Dejvida Hilberta.

Razvoj homološke algebre bio je usko isprepleten s nastankom teorije kategorija. Uopšte, homološka algebra je proučavanje homoloških funktora i zamršenih algebričnih struktura koje oni uključuju. Jedan prilično koristan i sveprisutan koncept u matematici su lančani kompleksi, koji se mogu proučavati putem njihove homologije i kohomologije. Homološka algebra pruža sredstva za izdvajanje informacija sadržanih u ovim kompleksima i njihovo predstavljanje u obliku homoloških invarijanati prstenova, modula, topoloških prostora i drugih 'opipljivih' matematičkih objekata. Moćan alat za ovo pružaju spektralne sekvence.

Homološka algebra je od samog nastanka igrala ogromnu ulogu u algebarskoj topologiji. Njen uticaj se postepeno proširio i trenutno uključuje komutativnu algebru, algebarsku geometriju, teoriju algebarskih brojeva, teoriju reprezentacije, matematičku fiziku, operatorske algebre, kompleksnu analizu i teoriju parcijalnih diferencijalnih jednačina. K-teorija je nezavisna disciplina koja se zasniva na metodama homološke algebre, kao i nekomutativna geometrija Alena Kona.

Istorija homološke algebre

Proučavanje homološke algebre je započeto u njenom najosnovnijem obliku tokom 1800-ih kao grane topologije. Tek je tokom 1940-ih godina ona postala samostalni predmet proučavanja, sa izučavanjem tema kao što su: ext funktor i tor funktor, između ostalog.

Lančani kompleksi i homologija

The notion of chain complex is central in homological algebra. An abstract chain complex is a sequence $(C_{\bullet },d_{\bullet })$ of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps 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 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 be vector spaces 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 any abelian 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 nth homology group H_n(C) as the factor 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.

Literatura

• 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
• Serge Lang: Algebra. 3rd edition, Springer 2002, ISBN 978-0-387-95385-4, pp. 157–159 (online copy, стр. 157, на сајту Гугл књиге)
• M. F. Atiyah; I. G. Macdonald: Introduction to Commutative Algebra. Oxford 1969, Addison–Wesley Publishing Company, Inc. ISBN 0-201-00361-9.
• Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics, 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1 
• Hovey, Mark (1999), Model categories, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1359-1 
• Quillen, Daniel (1967), Homotopical Algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-03914-5 
• R.L. Taylor, Covering groups of non connected topological groups, Proceedings of the American Mathematical Society, vol. 5 (1954), 753–768.
• R. Brown and O. Mucuk, Covering groups of non-connected topological groups revisited, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 115 (1994), 97–110.
• R. Brown and T. Porter, On the Schreier theory of non-abelian extensions: generalisations and computations, Proceedings of the Royal Irish Academy, vol. 96A (1996), 213–227.
• G. Janelidze and G. M. Kelly, Central extensions in Malt'sev varieties, Theory and Applications of Categories, vol. 7 (2000), 219–226.
• P. J. Morandi, Group Extensions and H³. From his collection of short mathematical notes.
• Buchsbaum, David A. (1955), „Exact categories and duality”, Transactions of the American Mathematical Society, 80 (1): 1—34, ISSN 0002-9947, JSTOR 1993003, MR 0074407, doi:10.1090/S0002-9947-1955-0074407-6 
• Freyd, Peter (1964), Abelian Categories, New York: Harper and Row 
• Mitchell, Barry (1965), Theory of Categories, Boston, MA: Academic Press 
• Popescu, Nicolae (1973), Abelian categories with applications to rings and modules, Boston, MA: Academic Press 
• Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3 
• Hatcher, Allen (2002). Algebraic Topology. Cambridge: Cambridge University Press. ISBN 0-521-79540-0. 
• Cox, David (2004). Galois Theory. Wiley-Interscience. ISBN 9781118031339. MR 2119052. 
• Jacobson, Nathan (2009). Basic Algebra I (2nd изд.). Dover Publications. ISBN 978-0-486-47189-1. 
• Rose, John S. (2012). A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8.  Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York, Heidelberg: Springer-Verlag, ISBN 0-387-90244-9, MR 0463157 
• May, J. Peter (1999), A Concise Course in Algebraic Topology (PDF), University of Chicago Press, ISBN 0-226-51182-0, MR 1702278 
• Switzer, Robert (1975), Algebraic Topology — Homology and Homotopy, Springer-Verlag, ISBN 3-540-42750-3, MR 0385836 
• Thom, René (1954), „Quelques propriétés globales des variétés différentiables”, Commentarii Mathematici Helvetici, 28: 17—86, MR 0061823, S2CID 120243638, doi:10.1007/BF02566923 

Spoljašnje veze

• Weisstein, Eric W. „Snake Lemma”. MathWorld. 
• Snake Lemma Архивирано на сајту Wayback Machine (25. септембар 2012) at PlanetMath
• Homological conjectures, old and new, Melvin Hochster, Illinois Journal of Mathematics Volume 51, Number 1 (2007), 151-169.
• On the direct summand conjecture and its derived variant by Bhargav Bhatt.