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 moduli problems. The initial development of the theory of group schemes was due to Alexandre Grothendieck, Michel Raynaud, and Michel Demazure in the early 1960s.
Other articles related to "group scheme, group, scheme, group schemes":
... The group scheme of -th roots of unity is by definition the kernel of the -power map on the multiplicative group, considered as a group scheme ... That is, for any integer we can consider the morphism on the multiplicative group that takes -th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the ... The resulting group scheme is written ...
... Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear ... and Cartier constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length ... vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V Wn → Wn+1, and then ...
Famous quotes containing the words scheme and/or group:
“Your scheme must be the framework of the universe; all other schemes will soon be ruins.”
—Henry David Thoreau (18171862)
“He hung out of the window a long while looking up and down the street. The worlds second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (18961970)