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Let $(C, \Delta, \epsilon)$ be a coalgebra over a commutative ring $k$. Let $M$ be a right comodule over $C$, that is a $k$-module $M$ together with a $k$-linear map $\delta \colon M \to M \otimes C$ such that $(1 \otimes \epsilon)\delta = 1$ and $(1 \otimes \Delta)\delta = (\delta \otimes 1 )\delta$. Let $P$ and $N$ be subcomodules of $M$, i.e. $P$ is a $k$-module such that $\delta(P) \subseteq P \otimes C$, similarly for $N$.

Is it true that $N \cap P$ is a subcomodule of $M$?

To me it seems simple that $\delta(N \cap P) \subseteq \delta(N) \cap \delta(P) \subseteq (N \otimes C) \cap (P \otimes C) = (N \cap P) \otimes C$...

But there is something wrong with my reasoning, since my internet searches tell me that $C$ has to be flat. Can someone help?

Paul Slevin
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You write $N \otimes C \cap P \otimes C$ and implicitly assume that this intersection makes sense in $M \otimes C$. In other words, you assume that $N \otimes C \to M \otimes C$ is injective (the same for $P \otimes C$). This is not true in general. And even if these maps are injective, it is not clear at all why $N \otimes C \cap P \otimes C = (N \cap P) \otimes C$. But both are true when $C$ is flat / $k$.

Martin Brandenburg
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  • in what way does $\delta (N) \subseteq N \otimes C $ make sense? – Paul Slevin Oct 19 '12 at 15:45
  • Oh it doesn't make sense for general $C$. Your definition of a sub-comodule is wrong. It should be a comodule $N$ together with a monomorphism of comodules $N \to P$, which should be a injective linear map $i : N \to P$ such that the obvious diagrams commute. – Martin Brandenburg Oct 19 '12 at 18:45
  • Is it correct to say that $N$ is a subcomodule of $M$ if there exists a monomorphism $j \colon N \to M$ in $k$-$\mathsf{Mod}$ such that $N$ has a coaction $\kappa \colon N \to N \otimes C$ and $ \delta j = (j \otimes 1) \kappa$ ? – Paul Slevin Oct 22 '12 at 09:16
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    yes, this is what I meant – Martin Brandenburg Oct 24 '12 at 10:33