The article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck, now often referred to as the ''Tôhoku'' paper, was published in 1957 in the ''

Tôhoku Mathematical Journal The ''Tohoku Mathematical Journal'' is a mathematical research journal published by Tohoku University in Japan. It was founded in August 1911 by Tsuruichi Hayashi. History Due to World War II the publication of the journal stopped in 1943 with ...

''. It has revolutionized the subject of homological algebra, a purely algebraic aspect of

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

. It removed the need to distinguish the cases of

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

s over a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

and sheaves of abelian groups over a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

Background

Material in the paper dates from Grothendieck's year at the

University of Kansas The University of Kansas (KU) is a public research university with its main campus in Lawrence, Kansas, United States, and several satellite campuses, research and educational centers, medical centers, and classes across the state of Kansas. T ...

in 1955–6. Research there allowed him to put homological algebra on an axiomatic basis, by introducing the

abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...

concept. A textbook treatment of homological algebra, "Cartan–Eilenberg" after the authors Henri Cartan and

Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...

, appeared in 1956. Grothendieck's work was largely independent of it. His abelian category concept had at least partially been anticipated by others.

David Buchsbaum David Alvin Buchsbaum (November 6, 1929 – January 8, 2021) was a mathematician at Brandeis University who worked on commutative algebra, homological algebra, and representation theory. He proved the Auslander–Buchsbaum formula and the Ausland ...

in his doctoral thesis written under Eilenberg had introduced a notion of "

exact category In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and ...

" close to the abelian category concept (needing only direct sums to be identical); and had formulated the idea of " enough injectives". The ''Tôhoku'' paper contains an argument to prove that a

Grothendieck category In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves ...

(a particular type of abelian category, the name coming later) has enough injectives; the author indicated that the proof was of a standard type. In showing by this means that categories of sheaves of abelian groups admitted

injective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to def ...

s, Grothendieck went beyond the theory available in Cartan–Eilenberg, to prove the existence of a

cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

in generality.

Later developments

After the Gabriel–Popescu theorem of 1964, it was known that every Grothendieck category is a

quotient category In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, bu ...

of a module category. The ''Tôhoku'' paper also introduced the Grothendieck spectral sequence associated to the composition of

derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...

s. In further reconsideration of the foundations of homological algebra, Grothendieck introduced and developed with Jean-Louis Verdier the

derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...

concept. The initial motivation, as announced by Grothendieck at the 1958 International Congress of Mathematicians, was to formulate results on

coherent duality In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' the ...

, now going under the name "Grothendieck duality".Amnon Neeman, "Derived Categories and Grothendieck Duality"
at p. 7

Notes

External links

*
English translation

Grothendieck's Tohoku Paper and Combinatorial Topology
{{DEFAULTSORT:Grothendieck's Tohoku paper Mathematics papers Homological algebra