Geodesic
Projectivity and flatness over the graded ring of semi-coinvariants
Algebra and discrete mathematics, Tome 10 (2010) no. 1, pp. 42-56

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Let $k$ be a field, $C$ a bialgebra with bijective antipode, $A$ a right $C$-comodule algebra, $G$ any subgroup of the monoid of grouplike elements of $C$. We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of semi-coinvariants of $A$. When $A$ and $C$ are commutative and $G$ is any subgroup of the monoid of grouplike elements of the coring $A\otimes C$, we prove similar results for the graded ring of conormalizing elements of $A$. 
@article{ADM_2010_10_1_a4,
     author = {T. Gu\'ed\'enon},
     title = {Projectivity and flatness over the graded ring of semi-coinvariants},
     journal = {Algebra and discrete mathematics},
     pages = {42--56},
     publisher = {mathdoc},
     volume = {10},
     number = {1},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2010_10_1_a4/}
}
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T. Guédénon. Projectivity and flatness over the graded ring of semi-coinvariants. Algebra and discrete mathematics, Tome 10 (2010) no. 1, pp. 42-56. http://geodesic.mathdoc.fr/item/ADM_2010_10_1_a4/