Grothendieck's Tôhoku paper

The article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck,[1] now often referred to as the Tôhoku paper,[2] was published in 1957 in the 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 Henri Cartan and Samuel Eilenberg, appeared in 1956. Grothendieck's work was largely independent of it. His abelian category concept had at least partially been anticipated by others.[7] David Buchsbaum in his doctoral thesis written under Eilenberg had introduced a notion of "exact category" close to the abelian category concept (needing only direct sums to be identical); and had formulated the idea of "enough injectives".[8] The Tôhoku paper contains an argument to prove that a Grothendieck category (a particular type of abelian category, the name coming later) has enough injectives; the author indicated that the proof was of a standard type.[9] In showing by this means that categories of sheaves of abelian groups admitted injective resolutions, Grothendieck went beyond the theory available in Cartan–Eilenberg, to prove the existence of a cohomology theory in generality.[10]

Later developments

After the Gabriel–Popescu theorem of 1964, it was known that every Grothendieck category is a quotient category of a 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. Grothendieck, A. (1957), "Sur quelques points d'algèbre homologique", Tôhoku Mathematical Journal, (2), 9 (2): 119–221, doi:10.2748/tmj/1178244839, MR 0102537. English translation.
  2. Schlager, Neil; Lauer, Josh (2000), Science and Its Times: 1950-present. Volume 7 of Science and Its Times: Understanding the Social Significance of Scientific Discovery, Gale Group, p. 251, ISBN 9780787639396.
  3. Sooyoung Chang (2011). Academic Genealogy of Mathematicians. World Scientific. p. 115. ISBN 978-981-4282-29-1.
  4. Jean-Paul Pier (1 January 2000). Development of Mathematics 1950-2000. Springer Science & Business Media. p. 715. ISBN 978-3-7643-6280-5.
  5. Pierre Cartier; Luc Illusie; Nicholas M. Katz; Gérard Laumon; Yuri I. Manin (22 December 2006). The Grothendieck Festschrift, Volume I: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck. Springer Science & Business Media. p. vii. ISBN 978-0-8176-4566-3.
  6. Piotr Pragacz (6 April 2005). Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes. Springer Science & Business Media. p. xiv–xv. ISBN 978-3-7643-7214-9.
  7. "Tohoku in nLab". Retrieved 2 December 2014.
  8. I.M. James (24 August 1999). History of Topology. Elsevier. p. 815. ISBN 978-0-08-053407-7.
  9. Amnon Neeman (January 2001). Triangulated Categories. Princeton University Press. p. 19. ISBN 0-691-08686-9.
  10. Giandomenico Sica (1 January 2006). What is Category Theory?. Polimetrica s.a.s. pp. 236–7. ISBN 978-88-7699-031-1.
  11. "Grothendieck category - Encyclopedia of Mathematics". Retrieved 2 December 2014.
  12. Charles A. Weibel (27 October 1995). An Introduction to Homological Algebra. Cambridge University Press. p. 150. ISBN 978-0-521-55987-4.
  13. Ravi Vakil (2005). Snowbird Lectures in Algebraic Geometry: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Algebraic Geometry : Presentations by Young Researchers, July 4-8, 2004. American Mathematical Soc. pp. 44–5. ISBN 978-0-8218-5720-5.
  14. Amnon Neeman, "Derived Categories and Grothendieck Duality", at p. 7
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