Smooth automorphism group schemes.

*(English)*Zbl 0996.16025
Coelho, Flávio Ulhoa (ed.) et al., Representations of algebras. Proceedings of the conference, São Paulo, Brazil, July 19-24, 1999. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 224, 71-89 (2002).

Let \(A\) be a finite-dimensional algebra over a field \(k\) of positive characteristic. The issue here is to determine when the automorphism group scheme \(\operatorname{Aut}_A\) is smooth. The authors do this by considering the algebra \(H\) that represents \(\operatorname{Aut}_A\). It is shown that \(\operatorname{Aut}_A\) is smooth if and only if every \(k\)-derivation of \(A\) is integrable, that is every \(D\in\text{Der}(A)\) gives rise to a sequence of \(k\)-endomorphisms on \(A\), namely \(D^{(0)}=\text{I}\), \(D^{(1)}=D\), \(D^{(2)},D^{(3)},\dots\), satisfying certain conditions. This result is proved in the first section using Hopf algebra-theoretic techniques.

Attention is then turned to monomial algebras, from which numerous examples are obtained. A condition is given for when a derivation \(D\) on \(R=k[X,Y_1,\dots,Y_n]/I\) is integrable (where \(I\) is a monomial ideal). This is used to show that if no minimal monomial in \(I\) has positive degree in any \(X_j\) which is divisible by \(p=\text{char }k\) then \(R=k[X_1,\dots,X_n]/I\) has a smooth automorphism group scheme.

One problem that remains unsolved is to use the minimal monomials generating \(I\) to quickly determine if the corresponding monomial algebra has a smooth automorphism scheme.

Finally, it is shown that having a smooth automorphism scheme is a Morita invariant. Specifically, if \(\int HH^1(A)\) is the subspace of \(A\) consisting of integrable derivations of \(A\) modulo the inner derivations of \(A\), then if \(A\) and \(B\) are Morita equivalent then \(\int HH^1(A)\) is isomorphic to \(HH^1(B)\).

For the entire collection see [Zbl 0974.00038].

Attention is then turned to monomial algebras, from which numerous examples are obtained. A condition is given for when a derivation \(D\) on \(R=k[X,Y_1,\dots,Y_n]/I\) is integrable (where \(I\) is a monomial ideal). This is used to show that if no minimal monomial in \(I\) has positive degree in any \(X_j\) which is divisible by \(p=\text{char }k\) then \(R=k[X_1,\dots,X_n]/I\) has a smooth automorphism group scheme.

One problem that remains unsolved is to use the minimal monomials generating \(I\) to quickly determine if the corresponding monomial algebra has a smooth automorphism scheme.

Finally, it is shown that having a smooth automorphism scheme is a Morita invariant. Specifically, if \(\int HH^1(A)\) is the subspace of \(A\) consisting of integrable derivations of \(A\) modulo the inner derivations of \(A\), then if \(A\) and \(B\) are Morita equivalent then \(\int HH^1(A)\) is isomorphic to \(HH^1(B)\).

For the entire collection see [Zbl 0974.00038].

Reviewer: Alan Koch (Decatur)

##### MSC:

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

14L15 | Group schemes |

16W20 | Automorphisms and endomorphisms |

16W25 | Derivations, actions of Lie algebras |