下記 Algebraic geometryより ”Applications Algebraic geometry now finds applications in statistics,[9] control theory,[10][11] robotics,[12] error-correcting codes,[13] phylogenetics[14] and geometric modelling.[15] There are also connections to string theory,[16] game theory,[17] graph matchings,[18] solitons[19] and integer programming.[20]” いまどき、工学でも”Algebraic geometry”は、道具の一つですｗ（＾＾ (引用終り) https://en.wikipedia.org/wiki/Algebraic_geometry Algebraic geometry (抜粋) Contents 1 Basic notions 1.1 Zeros of simultaneous polynomials 1.2 Affine varieties 1.3 Regular functions 1.4 Morphism of affine varieties 1.5 Rational function and birational equivalence 1.6 Projective variety 2 Real algebraic geometry 3 Computational algebraic geometry 3.1 Grobner basis 3.2 Cylindrical algebraic decomposition (CAD) 3.3 Asymptotic complexity vs. practical efficiency 4 Abstract modern viewpoint 5 History 5.1 Before the 16th century 5.2 Renaissance 5.3 19th and early 20th century 5.4 20th century 6 Analytic geometry 7 Applications
層とスキーム ザリスキー位相を持った空間 X = Spec(R) が与えられると，その構造層 OX が開集合 Df 上 Γ(Df, OX) を R の f における局所化 Rf とすることで定義される．これは B 層（英語版）を定義し，したがって層を定義することを示すことができる．より詳しくは，開集合 Df たちはザリスキー位相の基底であるので，任意の開集合 U に対し，
関手として 圏論のことばを用いて Spec が関手であることを見ることは有用である．任意の環準同型 f: R → S は連続写像 Spec(f): Spec(S) → Spec(R) を誘導する（なぜなら S の任意の素イデアルの引き戻しは R の素イデアルなので）．このようにして，Spec は可換環の圏から位相空間の圏への反変関手と見ることができる．
環 R の極大左イデアル（きょくだいひだりいである、英: maximal left ideal）とは、R 以外の左イデアルの中で（集合の包含関係に関して）極大なもののことである。 すなわち、左イデアル I を真に含む左イデアルが
424 名前： R しかないときに I を R の極大左イデアルという。極大右イデアルおよび極大両側イデアルも同様に定義される。これらのイデアルは（環が 0 でなく単位元をもつとき）ツォルンの補題によって存在が保証される[1]。可換環においては、左・右・両側の区別はない。唯一の極大左イデアルをもつ環は局所環と呼ばれる。
https://en.wikipedia.org/wiki/Alexander_Grothendieck Alexander Grothendieck
Category theory Grothendieck's emphasis on the role of universal properties across varied mathematical structures brought category theory into the mainstream as an organizing principle for mathematics in general. Among its uses, category theory creates a common language for describing similar structures and techniques seen in many different mathematical systems.[71] His notion of abelian category is now the basic object of study in homological algebra.[72] The emergence of a separate mathematical discipline of category theory has been attributed to Grothendieck's influence, though unintentional.[73]
https://ja.wikipedia.org/wiki/%E5%B1%80%E6%89%80%E7%92%B0 局所環 抽象代数学における局所環（きょくしょかん、英: local ring[1]）は、1938年にヴォルフガンク・クルルによって導入された概念で[2]、比較的簡単な構造を持つ環であり、代数多様体や可微分多様体上で定義される関数の、あるいは代数体を座や素点上の関数として見るときの「局所的な振る舞い」を記述すると考えられるものである。局所環およびその上の加群について研究する可換環論の一分野を局所環論と呼ぶ。
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.
