Comodule exercises desired

Question

This Question is inspired by a Quote of Moore's

"There are two ‘evil’ influences at work here: 1. we are toilet trained with algebras not coalgebras 2. some of us are addicted to manifolds and so think of differential forms as given by God and all the rest are the works of man."

I understand what a coalgebra is and what a comodule over a coalgebra is. I am curious if anyone knows of some lecture notes, a text or fun little toy examples that i could work on. Ideally I would like to better understand what a cotensor product is and maybe feel comfortable computing some Cotor groups. I am currently looking at such objects in the category of chain complexes, so more concrete suggestions will be appreciated. So if you have: 1. the name of a text that has lots of examples that it works through, essentially something that tries to correct for Moore's first point, or 2. a list of examples to work through, like "hey, look at this algebra and this module etc...", or 3. illuminating thoughts or mantras that I can recite while working with such gadgets I would really appreciate hearing from you.

Thanks, Sean

Answer 1

My favorite exercise is: prove that a comodule (over a coalgebra over a field) is coflat if and only if it is injective. This presumes that you already know that any coalgebra is the union of its finite-dimensional subcoalgebras, and any comodule is the union of its finite-dimensional subcomodules (if you don't know that, prove that first).

Also: construct a counterexample showing that infinite products in the category of comodules are not exact in general. Describe as explicitly as possible the cofree coalgebra with one cogenerator, i.e. the coalgebra $F$ such that coalgebra morphisms from any coalgebra $C$ into $F$ correspond bijectively to linear functions on $C$. Prove existence of the cofree coalgebra with any vector space of cogenerators.

Define a cosimple coalgebra as a coalgebra having no nonzero proper subcoalgebras, and a cosemisimple coalgebra as a direct sum of cosimple coalgebras. Prove that a coalgebra is cosemisimple if and only if the abelian category of comodules over it is semisimple. Prove that any coalgebra $C$ contains a maximal cosemisimple subcoalgebra which contains any other cosemisimple subcoalgebra of this coalgebra. Consider the quotient coalgebra of $C$ by this maximal cosemisimple subcoalgebra; it will be a coalgebra without counit. Prove that any element of this quotient coalgebra $D$ is annihilated by the iterated comultiplication map $D\to D^{\otimes n}$ for a large enough $n$ (depending on the element).

Define Cotor between an unbounded complex of right comodules and an unbounded complex of left comodules as the cohomology of the total complex of the cobar complex of the coalgebra with coefficients in these two complexes of comodules, the total complex being constructed by taking infinite direct sums along the diagonal planes. Construct a counterexample showing that the Cotor between two acyclic complexes can be nonzero.