If V is a vector space over a fieldF, then the cofree coalgebra C , of V, is a coalgebra together with a linear mapC → V, such that any linear map from a coalgebra X to V factors through a coalgebra homomorphism from X to C . In other words, the functorC 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 may be constructed as a completion of the tensor coalgebraT of V. For k ∈ N =, let TkV denote the k-fold tensor power of V: with T0V = F, and T1V = V. Then T is the direct sum of all TkV: In addition to the graded algebra structure given by the tensor product isomorphisms TjV ⊗ TkV → Tj+kV for j, k ∈ N, T has a graded coalgebra structure Δ : T → T ⊠ T defined by extending by linearity to all of T. Here, the tensor product symbol ⊠ is used to indicate the tensor product used to define a coalgebra; it must not be confused with the tensor product ⊗, which is used to define the bilinear multiplication operator of the tensor algebra. The two act in different spaces, on different objects. Additional discussion of this point can be found in the tensor algebra article. The sum above makes use of a short-hand trick, defining to be the unit in the field. For example, this short-hand trick gives, for the case of in the above sum, the result that for. Similarly, for and, one gets Note that there is no need to ever write as this is just plain-old scalar multiplication in the algebra; that is, one trivially has that With the usual product this coproduct does not make T into a bialgebra, but is instead dual to the algebra structure on T, where V∗ denotes the dual vectorspace of linear mapsV → F. It can be turned into a bialgebra with the product where denotes the binomial coefficient. This bialgebra is known as the divided power Hopf algebra. The product is dual to the coalgebra structure on T which makes the tensor algebra a bialgebra. Here an element ofT defines a linear form on T using the nondegenerate pairings induced by evaluation, and the duality between the coproduct on T and the product on T means that This duality extends to a nondegenerate pairing where is the direct product of the tensor powers of V. However, the coproduct Δ on T 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 is the largest subspace C satisfying which exists because if C1 and C2 satisfiy these conditions, then so does their sum C1 + C2. It turns out that C is the subspace of all representative elements: Furthermore, by the finiteness principle for coalgebras, any f ∈ C must belong to a finite-dimensional subcoalgebra of C . Using the duality pairing with T, it follows that f ∈ Cif and only if the kernel of f on T contains a two-sided ideal of finite codimension. Equivalently, is the union of annihilators I0 of finite codimension ideals I in T, which are isomorphic to the duals of the finite-dimensional algebra quotients T/I.
Example
When V = F, T is the polynomial algebraF in one variable t, and the direct product may be identified with the vector space F of formal power series in an indeterminate τ. The coproduct Δ on the subspace F is determined by and C is the largest subspace of F on which this extends to a coalgebra structure. The duality F × F → F is determined by τj = δjk so that Putting t=τ−1, this is the constant term in the product of two formal Laurent series. Thus, given a polynomial p with leading term tN, the formal Laurent series is a formal power series for any j ∈ N, and annihilates the ideal I generated by p for j < N. Since F/I has dimension N, these formal power series span the annihilator of I. Furthermore, they all belong to the localization of F at the ideal generated by τ, i.e., they have the form f/g 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. Any nonzero ideal of F is principal, with finite-dimensional quotient. Thus C is the sum of the annihilators of the principal ideals I, i.e., the space of rational functions regular at zero.
