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It is a standard fact in the theory of coalgebras and comodules that, given a coalgebra $C$ and a comodule $M$ over $C$, the coradical filtration

$$M_n := \Delta^{-1}(M\otimes C_0 + M_{n-1}\otimes C)$$

is exhaustive: $M=\bigcup_{n\geq 0}M_n$. Despite this, I cannot find an article where it is shown, and would appreciate being pointed in the right direction.

In a similar vein, would anyone have a suggestion for a good introductory text on coalgebras I could recommend to a student?

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  • $\begingroup$ I know a reference for exhaustiveness of the coradical filtration on $C$ itself, but not for a general $C$-comodule $M$. Would that reference be useful to you, or is it only the comodule version that you need? $\endgroup$
    – user509184
    Commented 2 days ago
  • $\begingroup$ @user509184 That would be a great start! If I remember correctly, the proof is pretty similar (up to throwing some assumptions on M) and would probably suffice $\endgroup$
    – Aidan
    Commented 2 days ago
  • $\begingroup$ OK, it's posted. Maybe someone else will know where to look up the comodule version. $\endgroup$
    – user509184
    Commented 2 days ago
  • $\begingroup$ P.S. About a good introductory text on coalgebras: the way I learned coalgebras was to learn Hopf algebras from Milnor--Moore "On the structure of Hopf algebras," comodule theory from the first appendix to Ravenel's book "Complex cobordism and stable homotopy groups of spheres," then to read the Brzezinski--Wisbauer book "Corings and comodules" to see what stuff about Hopf algebras and Hopf algebroids gets more complicated if you work with coalgebras instead. (Mostly it's all the same story.) I liked following that path, and it keeps you close to the applications of the theory. $\endgroup$
    – user509184
    Commented yesterday

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Exhaustiveness of the coradical filtration of a coalgebra is proven in Theorem 5.2.2 of Montgomery's book "Hopf algebras and their actions on rings." This is not as general as what you asked for, since it's only the coradical filtration of a coalgebra, not the coradical filtration of an arbitrary comodule over a coalgebra.

Theorem 5.2.2 in Montgomery's book has several claims, only one of which is the exhaustiveness claim. Montgomery writes "Proof (Sketch)" when about to prove the theorem, but in the part where the exhaustiveness of the filtration is treated, the proof appears to be complete, and not only a sketch. It is the final paragraph of Montgomery's proof.

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