Projectivity and flatness over the graded ring of normalizing elements

• Дата: 2015
• Автор: Guédénon, T.
• Формат: Стаття
• Мова: English
• Опубліковано: Lugansk National Taras Shevchenko University 2015
• Теми: projective module; flat module; bialgebra; smash product; graded ring; normalizing element; weakly semi-invariant element; 16D40; 16W50; 16W30
• Онлайн доступ: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/44
• Назва журналу: Algebra and Discrete Mathematics

Let \(k\) be a field, \(H\) a cocommutative bialgebra, \(A\) a commutative left \(H\)-module algebra, \(Hom(H,A)\) the $k$-algebra of the \(k\)-linear maps from \(H\) to \(A\) under the convolution product, \(Z(H,A)\) the submonoid of \(Hom(H,A)\) whose elements satisfy the cocycle condition and \(G\) any subgroup of the monoid \(Z(H,A)\). We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of normalizing elements of \(A\). When \(A\) is not necessarily commutative we obtain similar results over the graded ring of weakly semi-invariants of \(A\) replacing \(Z(H,A)\) by the set \(\chi(H,Z(A)^H)\) of all algebra maps from \(H\) to \(Z(A)^H\), where \(Z(A)\) is the center of \(A\).