Non-reduced automorphism schemes
Let \(A\) be a finite-dimensional associative algebra over an algebraically closed field \(k\). Denote by \(\mathfrak{Aut}(A)\) the affine group scheme of automorphisms of \(A\) and by \(\mathfrak{Inn}(A)\) the smooth characteristic subgroup of inner automorphisms. Then \(\mathfrak{Out}(A)=\mathfrak{Aut}(A)/\mathfrak{Inn}(A)\) is the affine group scheme of outer automorphisms. By the theorem of Cartier, \(\mathfrak{Aut}(A)\) is always reduced if \(\text{char}(k)=0\), but this is not the case if \(\text{char}(k)=p>0\). The authors prove that the group schemes \(\mathfrak{Aut}(A)\) and \(\mathfrak{Out}(A)\) are not reduced if \(A\) admits a \(\mathbb{Z}_p\)-grading which can not be lifted to a \(\mathbb{Z}\)-grading. The proof is based on the following result: an infinitesimal multiplicative subgroup \(\mathfrak{M}\) of a smooth algebraic group \(\mathfrak{G}\) over an algebraically closed field \(k\) of characteristic \(p>0\) is contained in a maximal torus of \(\mathfrak{G}\). This applies in particular when \(A\) is the group algebra of a \(p\)-group. Another example, where the above result shows that \(\mathfrak{Aut}(A)\) is not reduced, is illustrated by the direct calculation of the automorphism group.
scientific article; zbMATH DE number 2092908