TITLE: Geometric interpretation of the fundamental theorem for coalgebras?
QUESTION [10 upvotes]: Given an element $m$ in a coalgebra $C$, there always exists a finite-dimensional subcoalgebra $D \subset C$ containing $m$; this is the fundamental theorem for coalgebras. This obviously isn't the case for algebras. However, to make a proper analogue of this theorem for algebras, one should formulate this theorem using arrows and then dualize. An element of $C$ determines a linear map $F:\mathbb{k} \to C$ and vice versa. The fundamental theorem then becomes:
For any linear map $F: \mathbb{k} \to C$ there exists a map $G:\mathbb{k} \to D$, such that $i \circ G=F$, for $D$ some finite dimensional coalgebra, and $i$ an injective map of coalgebras.
Dualizing to algebras, one easily sees that the statement becomes equivalent to the question: is any linear map $f:A \to \mathbb{k}$ zero on an ideal of finite codimension, where $A$ is any algebra. Spelling this out for $k[x]$, one is led to the question: do there exist sequences $(f_{n})_{n}$ that do not satisfy any recursion equation. Does anyone know about a sequence like this, or an existence proof? 
A perhaps less sensical question: does the reformulation of the fundamental theorem in terms of algebras have some nice geometrical interpretation? Say in the case of affine algebras?

REPLY [2 votes]: EDIT: In this answer all coalgebras are cocommutative and all algebras commutative
As was already mentioned the statement you mention is equivalent to the statement that there's an equivalence $CoAlg \cong Ind(CoAlg_{fin})$ (everything is over a fixed field $k$). 
We can make this statement more geometric by noticing that 
$$Ind(Coalg_{fin}(k)) \cong Flat(CoAlg_{fin}^{op},Set) \cong Flat(Alg_{fin},Set) \cong Flat(Aff_{fin}^{op},Set)$$ 
Where "$Flat(-,-)$" means flat functors i.e. functors preserving finite limits. This last category is equivalent to the category of pro-representable functors on artin rings or in more geometric terms formal schemes (which are set-theoretically supported on a discrete set of points).
In summary the category of Coalgebras over $k$ is equivalent to the category of Formal Schemes (ind-finite over $k$).