group scheme - Weblio 英和・和英辞典 (original) (raw)

単語を追加

× この辞書を今後表示しない

※辞書の非表示は、設定画面から変更可能

group scheme

名詞

group scheme (複数形 group schemes)

(category theory, scheme theory) A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group.
- 1973, Jean A. Dieudonné, Introduction to the Theory of Formal Groups, Marcel Dekker, page 6,
  It is instructive to give a few explicit examples of such[affine] group schemes:
- 2013, Philippe Gille, Arturo Pianzola, Torsors, Reductive Group Schemes and Extended Affine Lie Algebras, American Mathematical Society, page 11,
  We will tend to use boldface characters, such as G {\displaystyle \mathbf {G} } $\mathbf {G}$ , for algebraic groups over k {\displaystyle k} $k$ , as also for group schemes over X {\displaystyle {\mathfrak {X}}} ${\mathfrak {X}}$ that are obtained from groups over k {\displaystyle k} $k$ . A quintessential example is G X = G × k X {\displaystyle \mathbf {G} _{\mathfrak {X}}=\mathbf {G} \times _{k}{\mathfrak {X}}} $\mathbf {G} _{\mathfrak {X}}=\mathbf {G} \times _{k}{\mathfrak {X}}$ . For arbitrary group schemes, or more generally group functors, over X {\displaystyle {\mathfrak {X}}} ${\mathfrak {X}}$ we shall tend to use german characters, such as G {\displaystyle {\mathfrak {G}}} ${\mathfrak {G}}$ .
- 2014, J. S. Milne, The Work of John Tate, Helge Holden, Ragni Piene (editors), The Abel Prize 2008-2012, Springer, page 303,
  Every finite flat group scheme of order p {\displaystyle p} $p$ over S {\displaystyle S} $S$ is of the form G a , b L {\displaystyle G_{a,b}^{L}} $G_{a,b}^{L}$ for some triple ( L , a , b ) {\displaystyle (L,a,b)} $(L,a,b)$ , and G a , b L {\displaystyle G_{a,b}^{L}} $G_{a,b}^{L}$ is isomorphic to G a 1 , b 1 L 1 {\displaystyle G_{a_{1},b_{1}}^{L_{1}}} $G_{a_{1},b_{1}}^{L_{1}}$ if and only if there exists an isomorphism from L {\displaystyle L} $L$ to L 1 {\displaystyle L_{1}} $L_{1}$ carrying a {\displaystyle a} $a$ to a 1 {\displaystyle a_{1}} $a_{1}$ and b {\displaystyle b} $b$ to b 1 {\displaystyle b_{1}} $b_{1}$ .