Dergiler / Turkish Journal of Mathematics / 2013 / Cilt: 37 - Sayı: 3
Structure theorems for rings under certain coactions of a Hopf algebra
- Sayfa
- 427–436
- DOI
- —
Özet
Let \{D1,..., Dn\} be a system of derivations of a k-algebra A, k a field of characteristic p > 0, defined by a coaction d of the Hopf algebra Hc = k[X1,..., Xn]/(X1p,..., Xnp), c \in \{0,1\}, the Lie Hopf algebra of the additive group and the multiplicative group on A, respectively. If there exist x1, \dots, xn \in A, with the Jacobian matrix (Di(xj)) invertible, [Di,Dj] = 0, Dip = cDi, c \in \{0, 1\}, 1 \leq i, j \leq n, we obtain elements y1,..., yn \in A, such that Di(yj) = dij(1 + cyi), using properties of Hc-Galois extensions. A concrete structure theorem for a commutative k-algebra A, as a free module on the subring Ad of A consisting of the coinvariant elements with respect to d, is proved in the additive case.
Abstract
Let \{D1,..., Dn\} be a system of derivations of a k-algebra A, k a field of characteristic p > 0, defined by a coaction d of the Hopf algebra Hc = k[X1,..., Xn]/(X1p,..., Xnp), c \in \{0,1\}, the Lie Hopf algebra of the additive group and the multiplicative group on A, respectively. If there exist x1, \dots, xn \in A, with the Jacobian matrix (Di(xj)) invertible, [Di,Dj] = 0, Dip = cDi, c \in \{0, 1\}, 1 \leq i, j \leq n, we obtain elements y1,..., yn \in A, such that Di(yj) = dij(1 + cyi), using properties of Hc-Galois extensions. A concrete structure theorem for a commutative k-algebra A, as a free module on the subring Ad of A consisting of the coinvariant elements with respect to d, is proved in the additive case.