[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78 (1002ﾚｽ)

現代数学の系譜 工学物理雑談 古典ガロア理論も読む78 http://rio2016.5ch.net/test/read.cgi/math/1571400076/

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385: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/02(土) 11:27:29.67 ID:ZLEqKHqI >>384 つづき Finite projective spaces and planes Further information on finite projective planes: Projective plane § Finite projective planes For finite projective spaces of dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF(q), whose order (that is, number of elements) is q (a prime power). A finite projective space defined over such a finite field has q + 1 points on a line, so the two concepts of order coincide. Notationally, PG(n, GF(q)) is usually written as PG(n, q). All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. The smallest projective plane is the Fano plane, PG(2, 2) with 7 points and 7 lines. The smallest 3-dimensional projective spaces is PG(3,2), with 15 points, 35 lines and 15 planes. Algebraic geometry An important property of projective spaces and projective varieties is that the image of a projective variety under a morphism of algebraic varieties is closed for Zariski topology (that is, it is an algebraic set). This is a generalization to every ground field of the compactness of the real and complex projective space. A projective space is itself a projective variety, being the set of zeros of the zero polynomial. つづく http://rio2016.5ch.net/test/read.cgi/math/1571400076/385

386: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/02(土) 11:28:01.09 ID:ZLEqKHqI >>385 つづき Scheme theory Scheme theory, introduced by Alexander Grothendieck during the second half of 20th century, allows defining a generalization of algebraic varieties, called schemes, by gluing together smaller pieces called affine schemes, similarly as manifolds can be built by gluing together open sets of {\displaystyle \mathbb {R} ^{n}.}{\displaystyle \mathbb {R} ^{n}.} The Proj construction is the construction of the scheme of a projective space, and, more generally of any projective variety, by gluing together affine schemes. In the case of projective spaces, one can take for these affine schemes the affine schemes associated to the charts (affine spaces) of the above description of a projective space as a manifold. See also: Algebraic geometry of projective spaces https://ja.wikipedia.org/wiki/%E5%B0%84%E5%BD%B1%E7%A9%BA%E9%96%93 射影空間 http://rio2016.5ch.net/test/read.cgi/math/1571400076/386

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