Alexander Grothendieck wrote in 1955 a revolutionary article on homological algebra, which was, after almost 3 years in redaction, published in 1957 in Tôhoku Mathematical Journal:
In Tôhoku, as it is nowadays called, Grothendieck observes that modules over rings, and sheaves of abelian groups have similar behaviour and that one can develop their homological algebra in a unified 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
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”). The Tôhoku paper also introduces the new weaker notion of an additive category (in which he also postulates the existence of finite products), as well as some additional axioms (including AB5) to abelian categories ensuring existence of sufficiently many injective objects, what is now called a Grothendieck category. See additive and abelian categories for more.
The Tôhoku paper is the place where the notion of an equivalence of categories is introduced for the first time. In fact the definition in question is a definition of an adjoint equivalence (unit and counit isomorphisms and the corresponding triangle identities are a part of the definition). This was predating just a little bit Kan’s introduction of adjoint functors in general.
Grothendieck defined universal (co)homological functors and studied special properties of resolutions, including showing that the 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 Tôhoku, Grothendieck partly continues in spirit of his work from Kansas
During his work on the Tôhoku article in Kansas, Grothendieck did not have access to the manuscript of the 1956 book of Cartan–Eilenberg, about which he heard from his correspondence with Serre. Thus some of the constructions are overlapping with Cartan–Eilenberg, while being independent.
One of the most important discoveries in Tôhoku is the spectral sequence for the derived functor of the composition of two functors (the Grothendieck spectral sequence, which is now more naturally treated in terms of triangulated categories which Grothendieck invented later with Verdier).
Chapters:
The original paper is
English translation:
A Russian translation:
Last revised on October 30, 2023 at 06:47:09. See the history of this page for a list of all contributions to it.