Grothendieck's Tôhoku paper

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.{{cite book|author1=Pierre Cartier|author2=Luc Illusie|author3=Nicholas M. Katz |author4=Gérard Laumon |author5=Yuri I. Manin|title=The Grothendieck Festschrift, Volume I: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck|url=https://books.google.com/books?id=_BnSoQSKnNUC&pg=PR7|date=22 December 2006|publisher=Springer Science & Business Media|isbn=978-0-8176-4566-3|page=vii}}{{cite book|author=Piotr Pragacz|title=Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes|url=https://books.google.com/books?id=AqCXg0WtnrEC&pg=PR14|date=6 April 2005|publisher=Springer Science & Business Media|isbn=978-3-7643-7214-9|page=xiv–xv}}

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.{{cite web|url=http://ncatlab.org/nlab/show/Tohoku|title=Tohoku in nLab|accessdate=2 December 2014}} 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".{{cite book|author=I.M. James|title=History of Topology|url=https://books.google.com/books?id=7iRijkz0rrUC&pg=PA815|date=24 August 1999|publisher=Elsevier|isbn=978-0-08-053407-7|page=815}} 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.{{cite book|author=Amnon Neeman|title=Triangulated Categories|url=https://books.google.com/books?id=WptvM91FrbQC&pg=PA19|date=January 2001|publisher=Princeton University Press|isbn=0-691-08686-9|page=19}} 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.{{cite book|author=Giandomenico Sica|title=What is Category Theory?|url=https://books.google.com/books?id=tVOuvxqhBxwC&pg=PA236|date=1 January 2006|publisher=Polimetrica s.a.s.|isbn=978-88-7699-031-1|pages=236–7}}

After the Gabriel–Popescu theorem of 1964, it was known that every Grothendieck category is a quotient category of a module category.{{cite web|url=http://www.encyclopediaofmath.org/index.php/Grothendieck_category|title=Grothendieck category - Encyclopedia of Mathematics|accessdate=2 December 2014}}

The Tôhoku paper also introduced the Grothendieck spectral sequence associated to the composition of derived functors.{{cite book|author=Charles A. Weibel|title=An Introduction to Homological Algebra|url=https://books.google.com/books?id=flm-dBXfZ_gC&pg=PA150|date=27 October 1995|publisher=Cambridge University Press|isbn=978-0-521-55987-4|page=150}} In further reconsideration of the foundations of homological algebra, Grothendieck introduced and developed with Jean-Louis Verdier the derived category concept.{{cite book|author=Ravi Vakil|title=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|url=https://books.google.com/books?id=yWoSJyir6gcC&pg=PA43|year=2005|publisher=American Mathematical Soc.|isbn=978-0-8218-5720-5|pages=44–5}} 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".[http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=BB646E964DD6ADFC6527D0110FB4D95C?doi=10.1.1.611.7150&rep=rep1&type=pdf Amnon Neeman, "Derived Categories and Grothendieck Duality"], at p. 7

{{reflist}}

• {{citation|first=A.|last=Grothendieck|authorlink=Alexander Grothendieck|title=Sur quelques points d'algèbre homologique|journal=Tôhoku Mathematical Journal|volume=9|series=(2)|pages=119–221|year=1957|url=http://projecteuclid.org/euclid.tmj/1178244839}}. [http://www.math.mcgill.ca/barr/papers/gk.pdf English translation].
• [https://mathoverflow.net/q/18898 Grothendieck's Tohoku Paper and Combinatorial Topology]

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Category:Mathematics papers

Category:Homological algebra