[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)
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462(1): 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(4/13) AAS
>>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)
484: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/04/13(土) 20:17:46.11 ID:TPdnRaQt(13/13) AAS
>>462 まとめ
https://en.wikipedia.org/wiki/Leray_spectral_sequence
Leray spectral sequence
(抜粋)
History and connection to other spectral sequences
At the time of Leray's work, neither of the two concepts involved (spectral sequence, sheaf cohomology) had reached anything like a definitive state. Therefore it is rarely the case that Leray's result is quoted in its original form.
After much work, in the seminar of Henri Cartan in particular, the modern statement was obtained, though not the general Grothendieck spectral sequence.
Earlier (1948/9) the implications for fiber bundles were extracted in a form formally identical to that of the Serre spectral sequence, which makes no use of sheaves.
In the formulation achieved by Alexander Grothendieck by about 1957, the Leray spectral sequence is the Grothendieck spectral sequence for the composition of two derived functors.
https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper
Grothendieck's Tohoku paper
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]
Notes
1^ Grothendieck, A. (1957), "Sur quelques points d'algebre homologique", Tohoku Mathematical Journal, (2),
English translation. http://www.math.mcgill.ca/barr/papers/gk.pdf
https://en.wikipedia.org/wiki/Fiber_bundle
Fiber bundle
https://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Roundhairbrush.JPG/220px-Roundhairbrush.JPG
A cylindrical hairbrush showing the intuition behind the term "fiber bundle". This hairbrush is like a fiber bundle in which the base space is a cylinder and the fibers (bristles) are line segments.
The mapping π：E → B would take a point on any bristle and map it to its root on the cylinder.
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