[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)
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385(1): 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 ID:ZLEqKHqI(16/18) AAS
>>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.
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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 ID:ZLEqKHqI(17/18) AAS
>>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
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