The dual convolution algebra and rational modules

For a coalgebra $C$ over $k$, write $C^{\ast }:={\mathrm{Hom}}_k(C,k)$ for its full algebraic dual, using Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$. Define the convolution candidate product by $(f\ast g)(c):=(f\otimes g)(\mathrm{\Delta }(c))$ for $f,g\in C^{\ast }$ and $c\in C$, and take $\epsilon$ as its candidate unit. Thus, when $\mathrm{\Delta }(c)={\sum }_i{c}_i\otimes d_i$, $(f\ast g)(c)={\sum }_{i}f(c_i)g(d_i)$. Every such sum is finite because tensors are algebraic tensors under Scalars, tensor powers, the empty tensor, opposite algebras and finite sums.

The theorem Right comodules are rational modules over the dual algebra ↗ proves that this product is associative and unital, so it makes $C^{\ast }$ a $k$-algebra. A left $C^{\ast }$-module here is a $k$-vector space $M$ with a $k$-bilinear action $C^{\ast }\times M\to M$ satisfying $f\cdot (g\cdot m)=(f\ast g)\cdot m$ and $\epsilon \cdot m=m$. This is the usual unital module convention of Unital left and right modules over a ring; unqualified module means left module for the resulting $k$-algebra.

Such a module is rational if for every $m\in M$ there exists a finite tensor $t_m\in M\otimes C$ such that $f\cdot m=({\mathrm{id}}_M\otimes f)(t_m)\text{for every }f\in C^{\ast }\mathrm{.}$ This definition asserts existence only. The theorem proves uniqueness of $t_m$ under its stated choice hypothesis before naming it as a coaction value.

The convolution formula uses the order $f$ on the first tensor factor and $g$ on the second. Sources that identify $(C\otimes C{)}^{\ast }$ with $C^{\ast }\otimes C^{\ast }$ using the reversed tensor-factor convention require the corresponding switch; the formula above fixes the convention used throughout this page.

The draft supplier Coalgebra and coalgebra map over a field is present in batch 1 and states the coassociative comultiplication and two-sided counit used here. This definition introduces the convolution candidate only; Right comodules are rational modules over the dual algebra ↗ proves that it is associative and unital.