Group scheme

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 of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have s, 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. A group scheme is a group object in a that has fiber products and some final object S. That is, it is an S-scheme G equipped with one of the equivalent sets of data a triple of morphisms μ: G ×S G → G, e: S → G, and ι: G → G, satisfying the usual compatibilities of groups (namely associativity of μ, identity, and inverse axioms) a functor from schemes over S to the , such that composition with the forgetful functor to sets is equivalent to the presheaf corresponding to G under the Yoneda embedding. (See also: group functor.) A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrased either by saying that a map f satisfies the equation fμ = μ(f × f), or by saying that f is a natural transformation of functors from schemes to groups (rather than just sets). A left action of a group scheme G on a scheme X is a morphism G ×S X→ X that induces a left action of the group G(T) on the set X(T) for any S-scheme T. Right actions are defined similarly.