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Homology de Saunders MacLane

Homology: Classics in Mathematics

Autor Saunders MacLane
en Limba Engleză Paperback – 15 feb 1995

Bazându-ne pe arhivele editoriale Springer Berlin, Heidelberg, notăm că lucrarea Homology de Saunders MacLane reprezintă un pilon istoric al matematicii moderne, fiind reeditată în prestigioasa serie Classics in Mathematics. Această ediție din 1995 păstrează rigoarea textului care a definit algebra omologică nu doar ca instrument pentru topologie, ci ca disciplină de sine stătătoare. Remarcăm modul în care autorul distilează decenii de cercetare susținută de Air Force Office of Scientific Research, transformând intuițiile din colaborările sale cu Eilenberg în teoreme fundamentale.

Structura volumului reflectă o progresie pedagogică precisă, pornind de la fundamentele de categorii și functori în primul capitol, spre aplicații complexe în omologia grupurilor. Notăm cu interes capitolul al II-lea, dedicat grupurilor diferențiale și complexelor, unde este introdusă secvența exactă de omologie, un instrument indispensabil în cercetarea matematică actuală. Cartea extinde cadrul propus de Homological Algebra (PMS–19), Volume 19 de Henry Cartan prin integrarea unei perspective mai accentuate pe teoria categoriilor, oferind o viziune unificată asupra extensiilor de module și rezoluțiilor.

Spre deosebire de abordările introductive, precum cea din An Elementary Approach to Homological Algebra, textul lui Saunders MacLane este dens și destinat nivelului de cercetare, punând accent pe demonstrații complete și pe eleganța diagramelor comutative. Este o resursă care documentează tranziția de la studiul seturilor de factori la teoria modernă a functorilor derivați, rămânând o referință critică pentru oricine studiază structura algebrică a spațiilor topologice.

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Specificații

ISBN-13: 9783540586623
ISBN-10: 3540586628
Pagini: 440
Ilustrații: X, 422 p.
Dimensiuni: 155 x 235 x 23 mm
Greutate: 0.61 kg
Ediția:1995
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Classics in Mathematics

Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

De ce să citești această carte

Recomandăm această lucrare cercetătorilor și studenților avansați care doresc să stăpânească fundamentele algebrei omologice direct de la unul dintre arhitecții disciplinei. Cititorul câștigă o înțelegere profundă a legăturii dintre algebră și topologie, beneficiind de un text care a stabilit standardele de notație și rigoare în domeniu. Este o investiție în biblioteca de bază a oricărui matematician orientat spre structuri abstracte.

Despre autor

Saunders MacLane (1909–2005) a fost un matematician american de excepție, profesor la Universitatea din Chicago și Harvard. Absolvent al Universității Yale și doctor în filozofie la Göttingen sub îndrumarea lui Hermann Weyl, MacLane a revoluționat matematica secolului XX prin descoperirea teoriei categoriilor alături de Samuel Eilenberg. Cercetările sale inițiale în logică și teoria numerelor algebrice au evoluat spre topologie algebrică, unde a pus bazele cohomologiei grupurilor. Activitatea sa academică și tratatele sale rămân fundamentale pentru înțelegerea modului în care structurile matematice interacționează.

Descriere scurtă

In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL­ LING. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL ElLENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. Both REINHOLD BAER and JOSEF SCHMID read and commented on my entire manuscript; their advice has led to many improvements. ANDERS KOCK and JACQUES RIGUET have read the entire galley proof and caught many slips and obscurities. Among the others whose sug­ gestions have served me well, I note FRANK ADAMS, LOUIS AUSLANDER, WILFRED COCKCROFT, ALBRECHT DOLD, GEOFFREY HORROCKS, FRIED­ RICH KASCH, JOHANN LEICHT, ARUNAS LIULEVICIUS, JOHN MOORE, DIE­ TER PUPPE, JOSEPH YAO, and a number of my current students at the University of Chicago - not to m~ntion the auditors of my lectures at Chicago, Heidelberg, Bonn, Frankfurt, and Aarhus. My wife, DOROTHY, has cheerfully typed more versions of more chapters than she would like to count. Messrs.

Cuprins

I. Modules, Diagrams, and Functors.- 1. The Arrow Notation.- 2. Modules.- 3. Diagrams.- 4. Direct Sums.- 5. Free and Projective Modules.- 6. The Functor Horn.- 7. Categories.- 8. Functors.- II. Homology of Complexes.- 1. Differential Groups.- 2. Complexes.- 3. Cohomology.- 4. The Exact Homology Sequence.- 5. Some Diagram Lemmas.- 6. Additive Relations.- 7. Singular Homology.- 8. Homotopy.- 9. Axioms for Homology.- III. Extensions and Resolutions.- 1. Extensions of Modules.- 2. Addition of Extensions.- 3. Obstructions to the Extension of Homomorphisms.- 4. The Universal Coefficient Theorem for Cohomology.- 5. Composition of Extensions.- 6. Resolutions.- 7. Injective Modules.- 8. Injective Resolutions.- 9. Two Exact Sequences for Extn.- 10. Axiomatic Description of Ext.- 11. The Injective Envelope.- IV. Cohomology of Groups.- 1. The Group Ring.- 2. Crossed Homomorphisms.- 3. Group Extensions.- 4. Factor Sets.- 5. The Bar Resolution.- 6. The Characteristic Class of a Group Extension.- 7. Cohomology of Cyclic and Free Groups.- 8. Obstructions to Extensions.- 9. Realization of Obstructions.- 10. SCHUR’S Theorem.- 11. Spaces with Operators.- V. Tensor and Torsion Products.- 1. Tensor Products.- 2. Modules over Commutative Rings.- 3. Bimodules.- 4. Dual Modules.- 5. Right Exactness of Tensor Products.- 6. Torsion Products of Groups.- 7. Torsion Products of Modules.- 8. Torsion Products by Resolutions.- 9. The Tensor Product of Complexes.- 10. The KÜNNETH Formula.- 11. Universal Coefficient Theorems.- VI. Types of Algebras.- 1. Algebras by Diagrams.- 2. Graded Modules.- 3. Graded Algebras.- 4. Tensor Products of Algebras.- 5. Modules over Algebras.- 6. Cohomology of free Abelian Groups.- 7. Differential Graded Algebras.- 8. Identities on Horn and ?.- 9. Coalgebras and HOPFAlgebras.- VII. Dimension.- 1. Homological Dimension.- 2. Dimensions in Polynomial Rings.- 3. Ext and Tor for Algebras.- 4. Global Dimensions of Polynomial Rings.- 5. Separable Algebras.- 6. Graded Syzygies.- 7. Local Rings.- VIII. Products.- 1. Homology Products.- 2. The Torsion Product of Algebras.- 3. A Diagram Lemma.- 4. External Products for Ext.- 5. Simplicial Objects.- 6. Normalization.- 7. Acyclic Models.- 8. The EILENBERG-ZILBER Theorem.- 9. Cup Products.- IX. Relative Homological Algebra.- 1. Additive Categories.- 2. Abelian Categories.- 3. Categories of Diagrams.- 4. Comparison of Allowable Resolutions.- 5. Relative Abelian Categories.- 6. Relative Resolutions.- 7. The Categorical Bar Resolution.- 8. Relative Torsion Products.- 9. Direct Products of Rings.- X. Cohomology of Algebraic Systems.- 1. Introduction.- 2. The Bar Resolution for Algebras.- 3. The Cohomology of an Algebra.- 4. The Homology of an Algebra.- 5. Homology of Groups and Monoids.- 6. Ground Ring Extensions and Direct Products.- 7. Homology of Tensor Products.- 8. The Case of Graded Algebras.- 9. Complexes of Complexes.- 10. Resolutions and Constructions.- 11. Two-stage Cohomology of DGA-Algebras.- 12. Cohomology of Commutative DGA-Algebras.- 13. Homology of Algebraic Systems.- XI. Spectral Sequences.- 1. Spectral Sequences.- 2. Fiber Spaces.- 3. Filtered Modules.- 4. Transgression.- 5. Exact Couples.- 6. Bicomplexes.- 7. The Spectral Sequence of a Covering.- 8. Cohomology Spectral Sequences.- 9. Restriction, Inflation, and Connection.- 10. The Lyndon Spectral Sequence.- 11. The Comparison Theorem.- XII. Derived Functors.- 1. Squares.- 2. Subobjects and Quotient Objects.- 3. Diagram Chasing.- 4. Proper Exact Sequences.- 5. Ext without Projectives.- 6. The Category of Short Exact Sequences.- 7. Connected Pairs of Additive Functors.- 8. Connected Sequences of Functors.- 9. Derived Functors.- 10. Products by Universality.- 11. Proper Projective Complexes.- 12. The Spectral KÜNNETH Formula.- List of Standard Symbols.

Notă biografică

Biography of Saunders Mac Lane
Saunders Mac Lane was born on August 4, 1909 in Connecticut. He studied at Yale University and then at the University of Chicago and at Göttingen, where he received the D.Phil. in 1934. He has tought at Harvard, Cornell and the University of Chicago.
Mac Lane's initial research was in logic and in algebraic number theory (valuation theory). With Samuel Eilenberg he published fifteen papers on algebraic topology. A number of them involved the initial steps in the cohomology of groups and in other aspects of homological algebra - as well as the discovery of category theory. His famous and undergraduate textbook Survey of modern algebra, written jointly with G. Birkhoff, has remained in print for over 50 years. Mac Lane is also the author of several other highly successful books.