Smooth automorphism group schemes (Q2782429)

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 \(\Aut_A\) is smooth. The authors do this by considering the algebra \(H\) that represents \(\Aut_A\). It is shown that \(\Aut_A\) 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.NEWLINENEWLINENEWLINEAttention 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.NEWLINENEWLINENEWLINEOne 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.NEWLINENEWLINENEWLINEFinally, 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 \(HH^1(B)\).NEWLINENEWLINEFor the entire collection see [Zbl 0974.00038].