Automorphism Groups
For an object \(X\) of an algebraic-geometric category (a module, a scheme, a group scheme, a quadratic form, …), the automorphism group scheme \(\operatorname{Aut}(X)\) is the group scheme representing the functor
\[ B \;\mapsto\; \operatorname{Aut}_{B}(X_B), \]
where \(X_B = X \otimes B\) (or the appropriate base change) and the right-hand side is the abstract automorphism group in the category of \(B\)-objects.
Examples
- \(\operatorname{Aut}(\mathbb{G}_{a,R}) \cong \mathbb{G}_{m,R}\) (the automorphisms of the additive group are multiplications by units — for the underlying scheme, this is the polynomial automorphism \(T \mapsto u T\) with \(u \in R^\times\)).
- \(\operatorname{Aut}(\mathbb{G}_{m,R}) \cong \mathbb{Z}/2\) (the only automorphisms of \(\mathbb{G}_m\) are \(T \mapsto T^{\pm 1}\), recording the involution \(T \mapsto T^{-1}\)).
- \(\operatorname{Aut}(V) = \operatorname{GL}(V)\) for \(V\) a free module — see GL_n.
\(\operatorname{Aut}\) of a finite-rank free module
For \(V \cong R^n\) finite free, the automorphism scheme \(\operatorname{Aut}(V)\) IS the general linear group scheme:
\[ \operatorname{Aut}(V) \;=\; \operatorname{GL}(V) \;\cong\; \operatorname{GL}_n. \]
Its coordinate ring is \(R[X_{ij}, \det^{-1}]\) with matrix-multiplication Hopf structure. The isomorphism with \(\operatorname{GL}_n\) depends on a basis of \(V\); a basis-free presentation works via the functor of points.
Inner automorphisms
For a group \(G\), inner automorphisms form a homomorphism \(G \to \operatorname{Aut}(G)\) by conjugation \(g \mapsto (h \mapsto g h g^{-1})\). The image is denoted \(\operatorname{Inn}(G)\) and is a closed normal subgroup of \(\operatorname{Aut}(G)\). The quotient is the outer automorphism group \(\operatorname{Out}(G)\).
Outer automorphism groups of reductive groups
For a split connected reductive group \(G\) over a field, the outer automorphism group is canonically isomorphic to the automorphism group of the (based) root datum. This is one of the key inputs into the descent and forms theory: forms of \(G\) are classified by \(H^1(\operatorname{Gal}, \operatorname{Aut}(G))\), and the outer component carries inner-form vs outer-form information.
Automorphism schemes vs. abstract automorphism groups
The scheme-theoretic \(\operatorname{Aut}(X)\) is strictly stronger than the abstract group \(\operatorname{Aut}_R(X)\) for a fixed base ring \(R\): the former records automorphisms over all \(R\)-algebras functorially, the latter just over \(R\) itself. Mathematically the distinction matters when \(R\) is not algebraically closed (e.g., a field with non-trivial Galois group).