現代数学の系譜工学物理雑談古典ガロア理論も読む63

[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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455(2): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2019/04/12(金) 23:52:03.38 ID:aUo1NtT0(9/12) AAS
>>448 関連

https://en.wikipedia.org/wiki/Leray_spectral_sequence
Leray spectral sequence
(抜粋)
In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946[1][2] by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.

Contents
1 Definition
2 Classical definition
3 Examples
4 Degeneration Theorem
4.1 Example with Monodromy
5 History and connection to other spectral sequences

Definition
Let f:X→Y be a continuous map of topological spaces, which in particular gives a functor f* from sheaves on X to sheaves on Y. Composing this with the functor Γ of taking sections on Sh(Y) is the same as taking sections on Sh(X), by the definition of the direct image functor f*:

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.

つづく

456: 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2019/04/12(金) 23:53:48.27 ID:aUo1NtT0(10/12) AAS
>>455

つづき

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.
This treatment, however, applied to Alexander?Spanier cohomology with compact supports, as applied to proper maps of locally compact Hausdorff spaces, as the derivation of the spectral sequence required a fine sheaf of real differential graded algebras on the total space, which was obtained by pulling back the de Rham complex along an embedding into a sphere.
Serre, who needed a spectral sequence in homology that applied to path space fibrations, whose total spaces are almost never locally compact, thus was unable to use the original Leray spectral sequence and so derived a related spectral sequence whose cohomological variant agrees, for a compact fiber bundle on a well-behaved space with the sequence above.

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.

References
2^ Miller, H. "Leray in Oflag XVIIA : the origins of sheaf theory, sheaf cohomology, and spectral sequences, Jean Leray (1906-1998)" (PDF). Gaz. Math. 84 (2000): 17?34.
http://www-math.mit.edu/~hrm/papers/ss.pdf
(引用終り)

460(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) 00:18:58.88 ID:TPdnRaQt(2/13) AAS
>>455 関連

https://en.wikipedia.org/wiki/Grothendieck_spectral_sequence
Grothendieck spectral sequence
(抜粋)
In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced in Tohoku paper, is a spectral sequence that computes the derived functors of the composition of two functors G◯F, from knowledge of the derived functors of F and G.

Contents
1 Examples
1.1 The Leray spectral sequence
1.2 Local-to-global Ext spectral sequence
2 Derivation

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