Monomorphisms of Coalgebras

Details Monomorphisms of Coalgebras Monomorphisms of Coalgebras

2009-08-20

arxiv.org/abs/0908.2959v3

Mathematics

Rings and Algebras

6 pages. to appear in Colloquium Mathematicum

Scientific paper

We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, $\phi: C \to D$ is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide if and only if $\sum_{i \in I}\epsilon(a^{i})b^{i} = \sum_{i \in I} a^{i} \epsilon(b^{i})$, for all $\sum_{i \in I}a^{i} \otimes b^{i} \in C \square_{D} C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

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