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 1955–56.[5] Research there allowed him to put homological algebra on an axiomatic basis, by introducing the abelian category concept.[6]

That same year, a textbook treatment of homological algebra, Homological Algebra by Henri Cartan and Samuel Eilenberg, appeared. Grothendieck's work was largely independent of it; however, 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 and Eilenberg's book 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 (1957).
  2. ↑ Schlager & Lauer (2000), p. 251, "commonly referred to as the Tohoku Paper".
  3. ↑ Chang (2011), p. 115, "He [...] revolutionized homological algebra in his "Tohoku paper"".
  4. ↑ Lawvere (2000), p. 715, "Grothendieck's Tohoku paper [...] showed that this conceptual basis for homological algebra over a ring also applies to linear objects varying as sheaves over a space".
  5. ↑ Cartier et al. (2006), p. vii, "his Tôhoku article dates from his visit to Kansas".
  6. ↑ Pragacz (2005), pp. xiv–xv, "Grothendieck constructs his axiomatic theory of Abelian categories. His main result asserts that the sheaves of modules form an Abelian category".
  7. ↑ nLab, "Essentially, they were defined in an earlier paper by Buchsbaum as 'exact categories', with different motivation [...] Saunders MacLane had rudiments of the definition of abelian category, around 1950, but it was a bit different and less invariant notion (and under a different name 'bicategory')".
  8. ↑ Weibel (1999), p. 815.
  9. ↑ Neeman (2001), p. 19, "Grothendieck said that it had been well-known, and he was merely sketching it".
  10. ↑ Marquis (2006), pp. 236–7, "Grothendieck solved the problem not by giving an explicit construction, but rather by finding the appropriate categorical framework for homological algebra in general and providing a categorical property from which the construction required could be performed. Grothendieck's work was therefore astonishing not only because it showed how to subsume the cohomological theory of sheaves within homological algebra [...] but also in the way he used categories to obtain this result".
  11. ↑ Encyclopedia of Mathematics, "A full subcategory 𝔖 of the category R𝔐 of left R-modules [...] An Abelian category is a Grothendieck category if and only if it is equivalent to some quotient category R𝔐/𝔖".
  12. ↑ Weibel (1994), p. 150, "In his classic paper [Tohoku], Grothendieck introduced a spectral sequence associated to the composition of two functors. Today it is one of the organizational principles of Homological Algebra".
  13. ↑ Căldăraru (2005), p. 43, "The derived category is a rather complicated object, having its roots in efforts by Grothendieck and Verdier to put firmer foundations on homological algebra than those originally laid out by Grothendieck in his Tôhoku paper".
  14. ↑ Neeman (2010), p. 297, "In his 1958 talk at the Edinburgh ICM Grothendieck announced that he had a solution, but also that the homological algebra language necessary to state it did not yet exist. Verdier’s thesis project was to develop the framework. Derived categories were born in that thesis".

References

  • Căldăraru, Andrei (2005). "Derived categories of sheaves: a skimming". In Vakil, Ravi (ed.). Snowbird Lectures in Algebraic Geometry. Contemporary Mathematics. Vol. 388. Providence, RI: American Mathematical Society. pp. 43–75. ISBN 0-8218-3719-2. MR 2182889.
  • Marquis, Jean-Pierre (2006). "What is category theory?". In Sica, Giandomenico (ed.). What Is Category Theory?. Monza: Polimetrica. pp. 221–255. ISBN 978-88-7699-031-1.
  • Neeman, Amnon (2010). "Derived categories and Grothendieck duality". In Holm, Thorsten; Jørgensen, Peter; Rouquier, Raphaël (eds.). Triangulated Categories. London Mathematical Society Lecture Note Series. Vol. 375. Cambridge University Press. pp. 290–350. ISBN 978-0-521-74431-7.
  • Pragacz, Piotr (2005). Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes. Basel: Birkhäuser. ISBN 978-3-7643-7214-9.

Further reading

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