[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む24 [無断転載禁止]©2ch.net (730ﾚｽ)
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660(2): 現代数学の系譜11 ガロア理論を読む 2016/10/29(土)19:07 ID:vwUy6eEC(28/46) AAS
>>659 付録
層
外部ﾘﾝｸ:en.wikipedia.org
Sheaf (mathematics)
(抜粋)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one.
For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical.
They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.
There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category.
省7
16: 現代数学の系譜11 ガロア理論を読む 2016/10/07(金)16:08 ID:++KBxzq2(14/40) AAS
前スレ
660 自分返信：現代数学の系譜11 ガロア理論を読む[] 投稿日：2016/10/07(金) 11:07:07.89 ID:++KBxzq2 [6/17]
>>620-621 補足
１．時枝も手放しで、あの解法>>2-4が成り立つとは思っていなかったろう
２．「箱入り無数目」という半分ふざけて逃げた非数学的題にその気持ちが現れていると思う
３．記事の前半>>2-4は解法の数学的解説だが、記事の後半>>5-7は数学的逃げの言い訳を二つ書いている>>574
４．一つは、”非可測集合を経由した”から>>5。一つは、"(1)無限を直接扱う，(2)有限の極限として間接に扱う，二つの方針が可能である.が、この二つは区別されるべき"と
５．そして、>>574-581に示したが、この二つの言い訳は数学的に不成立だ
６．だから、数学的な議論は、これで記事の前半に絞られたわけだ
661 自分返信：現代数学の系譜11 ガロア理論を読む[] 投稿日：2016/10/07(金) 11:07:50.22 ID:++KBxzq2 [7/17]
省5
661(2): 現代数学の系譜11 ガロア理論を読む 2016/10/29(土)19:10 ID:vwUy6eEC(29/46) AAS
>>660 つづき
1945 Jean Leray publishes work carried out as a prisoner of war, motivated by proving fixed point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences.
1947 Henri Cartan reproves the de Rham theorem by sheaf methods, in correspondence with Andre Weil (see De Rham-Weil theorem). Leray gives a sheaf definition in his courses via closed sets (the later carapaces).
1948 The Cartan seminar writes up sheaf theory for the first time.
1950 The "second edition" sheaf theory from the Cartan seminar: the sheaf space (espace etale) definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports.
Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, in several complex variables.
1951 The Cartan seminar proves the Theorems A and B based on Oka's work.
1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Jean-Pierre Serre, as is Serre duality.
1954 Serre's paper Faisceaux algebriques coherents (published in 1955) introduces sheaves into algebraic geometry. These ideas are immediately exploited by Hirzebruch, who writes a major 1956 book on topological methods.
1955 Alexander Grothendieck in lectures in Kansas defines abelian category and presheaf, and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
省3
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