The article "Sur quelques points d'algèbre homologique" by

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

, 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 revolutionized the subject of

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

, 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. It removed the need to distinguish the cases of modules over a

ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

and sheaves of

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

s over a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

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. Two branch campuses are in the Kansas City metropolitan area on the Kansas side: the university's medical school and hospital ...

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 o ...

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

Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...

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, specifically in category theory, an exact category is a category equipped with short exact sequences. The concept is due to Daniel Quillen and is designed to encapsulate the properties of short exact sequences in abelian categories ...

" close to the abelian category concept (needing only

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...

s to be identical); and had formulated the idea of "

enough injectives In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. ...

". 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 in ...

(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) that is used to defi ...

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 In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian category, abelian categories, introduced by . It characterizes certain abelian categories (the Grothendieck category, Grothendieck categories) as Quotient of a ...

of 1964, it was known that every Grothendieck category is a

quotient category In mathematics, a quotient category is a category (mathematics), category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of small categories, category of (locally small) categories ...

of a

module category In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the rin ...

. The ''Tôhoku'' paper also introduced the

Grothendieck spectral sequence In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his ''Tôhoku'' paper, is a spectral sequence that computes the derived functors of the composition of two functors ...

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 Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Groth ...

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 The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

, 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' th ...

, 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