[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む63 (1002ﾚｽ)

現代数学の系譜 工学物理雑談 古典ガロア理論も読む63 http://rio2016.5ch.net/test/read.cgi/math/1553946643/

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461: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2019/04/13(土) 00:27:13.56 ID:TPdnRaQt >>460 https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper Grothendieck's Tohoku paper (抜粋) The article "Sur quelques points d'algebre homologique" by Alexander Grothendieck,[1] now often referred to as the Tohoku paper,[2] was published in 1957 in the Tohoku Mathematical Journal. It has 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] Contents 1 Background 2 Later developments Background Material in the paper dates from Grothendieck's year at the University of Kansas in 1955?6. Research there allowed him to put homological algebra on an axiomatic basis, by introducing the abelian category concept.[5][6] つづく http://rio2016.5ch.net/test/read.cgi/math/1553946643/461

462: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2019/04/13(土) 00:28:15.73 ID:TPdnRaQt >>461 つづき A textbook treatment of homological algebra, "Cartan?Eilenberg" after the authors Henri Cartan and Samuel Eilenberg, appeared in 1956. Grothendieck's work was largely independent of it. 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 Tohoku 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?Eilenberg, 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 Tohoku 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, A. (1957), "Sur quelques points d'algebre homologique", Tohoku Mathematical Journal, (2), 9: 119?221, doi:10.2748/tmj/1178244839, MR 0102537. English translation. http://www.math.mcgill.ca/barr/papers/gk.pdf (引用終り) http://rio2016.5ch.net/test/read.cgi/math/1553946643/462

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