https://en.wikipedia.org/wiki/Generic_point Generic point (抜粋) In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.
In scheme theory, the spectrum of an integral domain has a unique generic point, which is the minimal prime ideal.
Contents 1 Definition and motivation 2 Examples 3 History
History
In the foundational approach of Andre Weil, developed in his Foundations of Algebraic Geometry, generic points played an important role, but were handled in a different manner. For an algebraic variety V over a field K, generic points of V were a whole class of points of V taking values in a universal domain Ω, an algebraically closed field containing K but also an infinite supply of fresh indeterminates. This approach worked, without any need to deal directly with the topology of V (K-Zariski topology, that is), because the specializations could all be discussed at the field level (as in the valuation theory approach to algebraic geometry, popular in the 1930s).
This was at a cost of there being a huge collection of equally generic points. Oscar Zariski, a colleague of Weil's at Sao Paulo just after World War II, always insisted that generic points should be unique. (This can be put back into topologists' terms: Weil's idea fails to give a Kolmogorov space and Zariski thinks in terms of the Kolmogorov quotient.)
In the rapid foundational changes of the 1950s Weil's approach became obsolete. In scheme theory, though, from 1957, generic points returned: this time a la Zariski. For example for R a discrete valuation ring, Spec(R) consists of two points, a generic point (coming from the prime ideal {0}) and a closed point or special point coming from the unique maximal ideal. For morphisms to Spec(R), the fiber above the special point is the special fiber, an important concept for example in reduction modulo p, monodromy theory and other theories about degeneration. The generic fiber, equally, is the fiber above the generic point. Geometry of degeneration is largely then about the passage from generic to special fibers, or in other words how specialization of parameters affects matters. (For a discrete valuation ring the topological space in question is the Sierpinski space of topologists. Other local rings have unique generic and special points, but a more complicated spectrum, since they represent general dimensions. The discrete valuation case is much like the complex unit disk, for these purposes.) (引用終り) 以上
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.
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 composi
501 名前：tion 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) []
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.
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]
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 Verdi
508 名前：er 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) []
https://en.wikipedia.org/wiki/J%C3%BCrgen_Neukirch Jurgen Neukirch (抜粋) Jurgen Neukirch (24 July 1937 ? 5 February 1997[1]) was a German mathematician known for his work on algebraic number theory.
Contributions He is known for his work on the embedding problem in algebraic number theory, the Bayer?Neukirch theorem on special values of L-functions, arithmetic Riemann existence theorems and the Neukirch?Uchida theorem in birational anabelian geometry. He gave a simple description of the reciprocity maps in local and global class field theory.
1960年代に撮影された仏ビュール・シュリベットでの講義に臨むアレクサンドル・グロタンディーク氏（2014年11月18日提供）。(c)AFP/Institut des Hautes Etudes Scientifiques
【5月11日 AFP】ドイツ生まれのフランス人で変わり者の天才数学者、アレクサンドル・グロタンディーク（Alexandre Grothendieck）氏による1万8000ページ近くに上る手書きメモがこのほど、母校の仏モンペリエ大学（University of Montpellier）によってインターネットで公開された。
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. 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.
大川 新之介先生ね それ、コンパクトだと思ったけど、章立て下記だけど、抜けている（未稿の）章があるね （下記）4章、10～14章も 米田の補題が練習問題か 3章終りに ”One can refer to [Pop73] for basics of additive and abelian categories. An English translation of Grothendieck's Tohoku paper is available at www.math.mcgill.ca/barr/papers/gk.pdf”とか（＾＾
www4.math.sci.osaka-u.ac.jp/~okawa/topics_in_algebra.pdf (引用開始) "Abstract. This is a note for the course `Topics in Algebra', which has been given during the winter semester of 2016-2017 at Osaka university." Date: January 13, 2017.
Contents 0. Plan of the lecture 1 1. Introduction and Overview 1 2. Category and functor 3 3. Additive category and abelian category 9 4. Recap on algebraic geometry 13 5. (Some) homological algebra 13 6. Derived category and the theorem of Bondal 17 7. Artin-Schelter algebras and AS-regular Z-algebras 17 8. Classication of 3-dimensional AS-regular algebras 21 9. Deformation theory of abelian categories 23 10. Blowup of noncommutative surfaces and SOD 26 11. Compact moduli of marked noncommutative del Pezzo surfaces 26 12. Noncommutative Hirzebruch surfaces 26 13. Relation to generalized complex geometry 26 14. Noncommutative P3{4-dimensional Sklyanin algebras and central extensions of 3-dimensional Sklyanin algebras 26 15. Comments on references 26 Acknowledgements 27 References 27 (引用終り)
538 名前：papers/okawa.pdf 非可換代数曲面 大川 新之介 Date: January 2, 2017
それと、下記に、関連トピックスがあるから、それを見た方が良いでしょうね 7章”Mori dream space” http://www4.math.sci.osaka-u.ac.jp/~okawa/talks.html Past talks and where I will be Schedule (temporary, with titles of the talks)
兆候はあるよね https://www.maths.nottingham.ac.uk/plp/pmzibf/guidestoiut.html Guides on IUT theory of Shinichi Mochizuki: reports, surveys, workshops materials, talks, lectures, videos (抜粋) https://www.maths.nottingham.ac.uk/plp/pmzibf/files/May2020.html Foundations and Perspectives of Anabelian Geometry, RIMS workshop, May 18-22 2020
Organisers: Ivan Fesenko (Univ. Nottingham), Arata Minamide (RIMS), Fucheng Tan (RIMS) This workshop is one of four workshops of special RIMS year "Expanding Horizons of Inter-universal Teichmuller Theory". The workshop will review fundamental developments in several branches of anabelian geometry, as well as report on recent developments. Anabelian geometry, together with higher class field theory and the Langlands correspondences, is one of three fundamental generalisations of class field theory.
https://en.wikipedia.org/wiki/Grothendieck_topology#CITEREFArtin1962 Grothendieck topology (抜粋) The term "Grothendieck topology" has changed in meaning. In Artin (1962) it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning. Giraud (1964) modified the definition to use sieves rather than covers. Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.
Contents 1 Overview 2 Definition 2.1 Motivation 2.2 Sieves 2.3 Grothendieck topology 2.3.1 Axioms 2.3.2 Grothendieck pretopologies 3 Sites and sheaves 4 Examples of sites 4.1 The discrete and indiscrete topologies 4.2 The canonical topology 4.3 Small site associated to a topological space 4.4 Big site associated to a topological space 4.5 The big and small sites of a manifold 4.6 Topologies on the category of schemes 5 Continuous and cocontinuous functors 5.1 Continuous functors 5.2 Cocontinuous functors 5.3 Morphisms of sites
追加 https://en.wikipedia.org/wiki/Mori_dream_space Mori dream space From Wikipedia, the free encyclopedia
In algebraic geometry, a Mori Dream Space is a projective variety whose cone of effective divisors has a well-behaved decomposition into certain convex sets called "Mori chambers". Hu & Keel (2000) showed that Mori dream spaces are quotients of affine varieties by torus actions. The notion is named so because it behaves nicely from the point of view of Mori's minimal model program.
See also spherical variety https://en.wikipedia.org/wiki/Spherical_variety References Hu, Yi; Keel, Sean (2000). "Mori dream spaces and GIT". The Michigan Mathematical Journal. 48 (1): 331?348. arXiv:math/0004017. doi:10.1307/mmj/1030132722. ISSN 0026-2285. MR 1786494. https://arxiv.org/pdf/math/0004017.pdf
History Etale cohomology was suggested by Grothendieck (1960), using some suggestions by J.-P. Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as Artin (Artin 1962) and SGA 4. Grothendieck used etale cohomology to prove some of the Weil conjectures (Dwork had already managed to prove the rationality part of the conjectures in 1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ?-adic cohomology.
Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincare duality and the Lefschetz fixed point theorem in this context.
