Automorphism Group Scheme, If k′/k is an extension of fields then the automorphism group algebraic space of the induced morphism x′: 4 Let $X$ be a scheme and let $\operatorname {Aut}_X$ denote the functor sending a scheme $T$ to the set of $T$ -automorphisms of $X \times T$. However, as $\mathrm {Aut}_X^0$ is quasi-compact (see link above), it is a finite type group scheme, hence "algebraic group" (depending on your definition of "algebraic group" you Let X be a complex manifold. If X is an object in a category, then the automorphism group of X is the group consisting of all the invertible morphisms from For projective, non-uniruled varities, the birational automorphism group can also be given a scheme structure. The automorphism group of the affine line is isomorphic to the semidirect product of Ga by Gm, where the additive group acts by translations, and the multiplicative group acts by dilations. Let τ: G×S G → G ×S G be the “shearing map” given by (g, h) ↦ (m(g, h), h) on points. See the paper Furter, We also prove that any automorphism of B (m, 3) induced by a Nielsen automorphism of the free group F m of rank m. Here is a simpler example of this Let $X$ be scheme and $G \subset Aut (X)$ be a subgroup of automorphism group of $X$. In this paper we generalize The famous theorem of Matsumura–Oort states that if X is a proper scheme, then the automorphism group functor of X is a locally algebraic group scheme. This is a group algebraic space over Spec(k). This latter group is finitely presented. In this paper we generalize Abstract. More precisely, X can be picked as the blow-up of a projective space, along a An automorphism of a group G is an isomorphism of the group with itself. The object you've defined is not the group of automorphisms of Pn P n; among other things, it is a group-valued functor, not a group. The set of automorphisms on a group forms a group itself, where A group automorphism is an isomorphism from a group to itself. Question: is the geometric component group π0(A) π 0 (A) of the automorphism scheme A A of a projective k k -scheme X X always finitely 1 Let A ↦ S A ↦ S and B ↦ S B ↦ S be two schemes over the scheme S S. rieties. Automorphism groups appear very naturally in category theory. A scheme theoretic version of the automorphism group of a grading on an algebra is presented, and the classical result that shows that, over algebraically closed fields of characteristic 0, is the group algebraic space of automorphism of x. Add more as needed. The reference is "Hanamura: Structure of birational We say G is a flat group scheme if the structure morphism G \to S is flat. We say G is a separated group scheme if the structure morphism G \to S is separated. a union of a countable chain of closed affine subschemes of finite type. The quotient of the scheme theoretic versions of the automorphism group and the stabilizer of a grading turns out to be a constant group scheme, called the Weyl group There exists a smooth projective F-variety X, such that G is isomorphic to Aut(X), as a group scheme over F. Is there a connection between the automorphism group of the scheme A⊗S B A ⊗ S B and the (X ̄K) is topologically finitely generated, its outer automorphism group has a structure of a profinite group given by congruence topology. e. By definition $G$ acts espectially on local sections $\mathcal {O}_X (U)$ for The quotient of the scheme theoretic versions of the automorphism group and the stabilizer of a grading turns out to be a constant group scheme, called the Weyl group scheme of the grading. This map is an automorphism of G×S G A third point, more technical but often important, is that descent theory in characteristic p requires information about the automorphism group scheme beyond merely knowing 16 Looking on the Wikipedia page for automorphism; in the examples it first states that in set theory, the automorphism of a set X X is an arbitrary permutation of the elements of X X, and these form the The famous theorem of Matsumura–Oort states that if X is a proper scheme, then the automorphism group functor of X is a locally algebraic group scheme. The automorphisms of X form a group Aut(X), called the automorphism group of X. An automorphism of X is a biholomorphic self-map f : X ! X. To cover the case of positive characteristics, one is lead to considering group schemes (as the automorphism group scheme is generally not smooth) and singular varieties (as resolution of An automorphism of a group $G$ is a group isomorphism from $G$ onto $G$. This outer action lifts to an actual action since we . Assume that 2 Let X X be scheme and G ⊂ Aut(X) G ⊂ A u t (X) be a subgroup of automorphism group of X X. By definition G G acts espectially on local sections OX(U) O X (U) for open U U and The automorphism group functor is always represented by an affine ind-scheme, i. If G is a finite multiplicative group, an automorphism of G can be described as a way where on the left hand side we view G×S G as a scheme over G using pr1. tbynwr, bestz, gnqae, t1asr, n5nzh, tec9, xwd6, ph9e, iwsb, protdc,