nLab
Tohoku

Alexander Grothendieck wrote in 1955 a revolutionary article on homological algebra, which was, after almost 3 years in redaction, published in 1957 in Tohoku Math. J.:

• A. GROTHENDIECK, Sur quelques points d’algèbre homologique, Tohoku Math. J. vol 9,n.2, 3, 1957, The Tohoku university, Sendai, Japan; MR21♯1328. Project Euclid open access pdf scans of French original: part 1, part 2. Russian translation as a separate booklet (Izdat. inostrannoj literatury, Moskva 1961): free djvu scan.

In Tohoku, Grothendieck observes that modules over rings, and sheaves of abelian groups have similar behaviour and that one can develop their homological algebra in a unique way; this includes the axiomatics of what is for the first time called abelian categories. Essentially, they were defined in an earlier paper by Buchsbaum as “exact categories”, with different motivation

• D. A. BUCHSBAUM, Exact categories and duality, Trans. Amer. Math. Soc.8 (1955), 1–34 (MR74407)

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”).

Grothendieck defined universal (co)homological functors and studied special properties of resolutions, including showing that Godement resolution of sheaves is really an injective resolution. There is also a section on sheaf cohomology of spaces with group action. In sheaf theory part of Tohoku, Grothendieck partly continues in spirit of his work from Kansas

• A. GROTHENDIECK, A general theory of fibre spaces with structure sheaf, University of Kansas 1955.

One of the most important discoveries in Tohoku is the spectral sequence for the derived functor of the composition of two functors (Grothendieck spectral sequence, which is now more naturally treated in terms of triangulated categories which Grothendieck invented later with Verdier).

Chapters:

1. Généralités sur les catégories abéliennes
2. Algebre homologique dans les catégories abéliennes
3. Cohomologie à coefficients dans un faisceau
4. Les Ext de faisceaux de modules
5. Étude cohomologique des espaces a opérateurs