在代数几何中，一个概形上的群概形是范畴中的群对象。借由米田信夫引理，我们可以给出两种刻划：

• 以乘法、单比特与逆元定义：存在中的态射
  • 乘法：
  • 单比特：
  • 逆元：

并满足结合律等等群的性质。

• 以函子性定义：点函子透过遗忘函子分解。。

换言之：对于任意的-概形，构成一个群；而且对任意-态射，诱导映射都是群同态。

• 代数群：设为域，上的连通、光滑群概形称作上的代数群。

• 李代数：群概形自然地作用在它的全体矢量场上。的全体左不变矢量场称作的李代数，记为；它是上的层。

In mathematics, a group scheme is a type of algebro-geometric object 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 mo**** problems. The initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Raynaud and Michel Demazure in the early 1960s.