Geodesic
Projectivity and flatness over the graded ring of normalizing elements
Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 172-192.

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Let $k$ be a field, $H$ a cocommutative bialgebra, $A$ a commutative left $H$-module algebra, $\operatorname{Hom}(H,A)$ the $k$-algebra of the $k$-linear maps from $H$ to $A$ under the convolution product, $Z(H,A)$ the submonoid of $\operatorname{Hom}(H,A)$ 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 $A$.
Keywords: projective module, flat module, bialgebra, smash product, graded ring, normalizing element, weakly semi-invariant element. 
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     title = {Projectivity and flatness over the graded ring of normalizing elements},
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T. Guédénon. Projectivity and flatness over the graded ring of normalizing elements. Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 172-192. http://geodesic.mathdoc.fr/item/ADM_2015_19_2_a2/

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