# Cofree coalgebra

In algebra, the **cofree coalgebra** of a vector space or module is a coalgebra analog of the free algebra of a vector space. The cofree coalgebra of any vector space over a field exists, though it is more complicated than one might expect by analogy with the free algebra.

## Definition

If *V* is a vector space over a field **F**, then the cofree coalgebra *C* (*V*), of *V*, is a coalgebra together with a linear map *C* (*V*)→*V*, such that any linear map from a coalgebra *X* to *V* factors through a coalgebra homomorphism from *X* to *C* (*V*). In other words, the functor *C* is right adjoint to the forgetful functor from coalgebras to vector spaces.

The cofree coalgebra of a vector space always exists, and is unique up to canonical isomorphism.

Cofree cocommutative coalgebras are defined in a similar way, and can be constructed as the largest cocommutative coalgebra in the cofree coalgebra.

## Construction

*C* (*V*) may be constructed as a *completion* of the tensor coalgebra *T*(*V*) of *V*. For *k* ∈ **N** = {0, 1, 2, ...}, let *T*^{k}*V* denote the *k*-fold tensor power of *V*:

with *T*^{0}*V* = **F**, and *T*^{1}*V* = *V*. Then *T*(*V*) is the direct sum of all *T*^{k}*V*:

In addition to the graded algebra structure given by the tensor product isomorphisms *T*^{j}*V* ⊗ *T*^{k}*V* → *T*^{j+k}*V* for *j*, *k* ∈ **N**, *T*(*V*) has a graded coalgebra structure Δ : *T*(*V*) → *T*(*V*) ⊗ *T*(*V*) defined by extending

by linearity to all of *T*(*V*). This coproduct does not make *T*(*V*) into a bialgebra, but is instead dual to the algebra structure on *T*(*V*^{∗}), where *V*^{∗} denotes the dual vector space of linear maps *V* → **F**. Here an element of *T*(*V*) defines a linear form on *T*(*V*^{∗}) using the nondegenerate pairings

induced by evaluation, and the duality between the coproduct on *T*(*V*) and the product on *T*(*V*^{∗}) means that

This duality extends to a nondegenerate pairing

where

is the direct product of the tensor powers of *V*. (The direct sum *T*(*V*) is the subspace of the direct product for which only finitely many components are nonzero.) However, the coproduct Δ on *T*(*V*) only extends to a linear map

with values in the *completed tensor product*, which in this case is

and contains the tensor product as a proper subspace:

The completed tensor coalgebra *C* (*V*) is the largest subspace *C* satisfying

which exists because if *C*_{1} and *C*_{2} satisfiy these conditions, then so does their sum *C*_{1} + *C*_{2}.

It turns out^{[1]} that *C* (*V*) is the subspace of all *representative elements*:

Furthermore, by the finiteness principle for coalgebras, any *f* ∈ *C* (*V*) must belong to a finite dimensional subcoalgebra of *C* (*V*). Using the duality pairing with *T*(*V*^{∗}), it follows that *f* ∈ *C* (*V*) if and only if the kernel of *f* on *T*(*V*^{∗}) contains a two-sided ideal of finite codimension. Equivalently,

is the union of annihilators *I* ^{0} of finite codimension ideals *I* in *T*(*V*^{∗}), which are isomorphic to the duals of the finite dimensional algebra quotients *T*(*V*^{∗})/*I*.

### Example

When *V* = **F**, *T*(*V*^{∗}) is the polynomial algebra **F**[*t*] in one variable *t*, and the direct product

may be identified with the vector space **F**[[*τ*]] of formal power series

in an indeterminate *τ*. The coproduct Δ on the subspace **F**[*τ*] is determined by

and *C* (*V*) is the largest subspace of **F**[[*τ*]] on which this extends to a coalgebra structure.

The duality **F**[[*τ*]] × **F**[*t*] → **F** is determined by *τ*^{j}(*t*^{k}) = *δ*_{jk} so that

Putting *t*=*τ*^{-1}, this is the constant term in the product of two formal Laurent series. Thus, given a polynomial *p*(*t*) with leading term *t*^{N}, the formal Laurent series

is a formal power series for any *j* ∈ **N**, and annihilates the ideal *I*(*p*) generated by *p* for *j* < *N*. Since **F**[*t*]/*I*(*p*) has dimension *N*, these formal power series span the annihilator of *I*(*p*). Furthermore, they all belong to the localization of **F**[*τ*] at the ideal generated by *τ*, i.e., they have the form *f*(*τ*)/*g*(*τ*) where *f* and *g* are polynomials, and *g* has nonzero constant term. This is the space of rational functions in *τ* which are regular at zero. Conversely, any proper rational function annihilates an ideal of the form *I*(*p*).

Any nonzero ideal of **F**[*t*] is principal, with finite dimensional quotient. Thus *C* (*V*) is the sum of the annihilators of the principal ideals *I*(*p*), i.e., the space of rational functions regular at zero.

## References

- ↑ Hazewinkel 2003

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