Grothendieck's Tôhoku paper

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The article "Sur quelques points d'algèbre homologique" by

Tôhoku Mathematical Journal. It revolutionized the subject of homological algebra, a purely algebraic aspect of algebraic topology.[3] It removed the need to distinguish the cases of modules over a ring and sheaves of abelian groups over a topological space.[4]

Background

Material in the paper dates from Grothendieck's year at the University of Kansas in 1955–6. Research there allowed him to put homological algebra on an axiomatic basis, by introducing the abelian category concept.[5][6]

A textbook treatment of homological algebra, "Cartan–Eilenberg" after the authors

cohomology theory in generality.[10]

Later developments

After the

module category.[11]

The Tôhoku paper also introduced the Grothendieck spectral sequence associated to the composition of derived functors.[12] In further reconsideration of the foundations of homological algebra, Grothendieck introduced and developed with Jean-Louis Verdier the derived category concept.[13] The initial motivation, as announced by Grothendieck at the 1958 International Congress of Mathematicians, was to formulate results on coherent duality, now going under the name "Grothendieck duality".[14]

Notes

  1. MR 0102537. English translation
    .
  2. ISBN 9780787639396
    .
  3. ISBN 978-981-4282-29-1
    .
  4. ISBN 978-3-7643-6280-5
    .
  5. ISBN 978-0-8176-4566-3
    .
  6. ISBN 978-3-7643-7214-9
    .
  7. ^ "Tohoku in nLab". Retrieved 2 December 2014.
  8. ISBN 978-0-08-053407-7
    .
  9. ISBN 0-691-08686-9
    .
  10. ISBN 978-88-7699-031-1
    .
  11. ^ "Grothendieck category - Encyclopedia of Mathematics". Retrieved 2 December 2014.
  12. ISBN 978-0-521-55987-4
    .
  13. ISBN 978-0-8218-5720-5
    .
  14. ^ Amnon Neeman, "Derived Categories and Grothendieck Duality", at p. 7

External links

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