| dbo:abstract |
Ein Gruppenschema ist in der algebraischen Geometrie die Verallgemeinerung einer algebraischen Gruppe.Typische Beispiele sind affine algebraische Gruppen oder abelsche Varietäten. Im Unterschied zur klassischen Sichtweise können Gruppenschemata über beliebigen Schemata definiert werden. Solche finden Anwendung in der Theorie von Modulräumen abelscher Varietäten. (de) In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Raynaud and Michel Demazure in the early 1960s. (en) 대수기하학에서 군 스킴(群scheme, 영어: group scheme, 프랑스어: schéma en groupe)은 군과 유사한 구조를 갖는 스킴이다. 즉, 대수군의 정의에서 대수다양체를 스킴으로 대체한 것이다. (ko) 在代數幾何中，一個概形上的群概形是範疇中的。藉由米田信夫引理，我們可以給出兩種刻劃： * 以乘法、單位元與逆元定義：存在中的態射 * 乘法： * 單位元： * 逆元： 並滿足結合律等等群的性質。 * 以函子性定義：點函子透過遺忘函子分解。。 換言之：對於任意的-概形，構成一個群；而且對任意-態射，誘導映射都是群同態。 * 代數群：設為域，上的連通、光滑群概形稱作上的代數群。 * 李代數：群概形自然地作用在它的全體向量場上。的全體左不變向量場稱作的李代數，記為；它是上的層。 (zh) |
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Ein Gruppenschema ist in der algebraischen Geometrie die Verallgemeinerung einer algebraischen Gruppe.Typische Beispiele sind affine algebraische Gruppen oder abelsche Varietäten. Im Unterschied zur klassischen Sichtweise können Gruppenschemata über beliebigen Schemata definiert werden. Solche finden Anwendung in der Theorie von Modulräumen abelscher Varietäten. (de) 대수기하학에서 군 스킴(群scheme, 영어: group scheme, 프랑스어: schéma en groupe)은 군과 유사한 구조를 갖는 스킴이다. 즉, 대수군의 정의에서 대수다양체를 스킴으로 대체한 것이다. (ko) 在代數幾何中，一個概形上的群概形是範疇中的。藉由米田信夫引理，我們可以給出兩種刻劃： * 以乘法、單位元與逆元定義：存在中的態射 * 乘法： * 單位元： * 逆元： 並滿足結合律等等群的性質。 * 以函子性定義：點函子透過遺忘函子分解。。 換言之：對於任意的-概形，構成一個群；而且對任意-態射，誘導映射都是群同態。 * 代數群：設為域，上的連通、光滑群概形稱作上的代數群。 * 李代數：群概形自然地作用在它的全體向量場上。的全體左不變向量場稱作的李代數，記為；它是上的層。 (zh) In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, (en) |
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