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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].

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 14L15 Group schemes 16W20 Automorphisms and endomorphisms 16W25 Derivations, actions of Lie algebras