Witt-vector vectors

It is impossible if you want reasonable functoriality, and for finite-dimensional vector spaces to be carried to finite free modules for $A$ a field.

Presumably for $A = k$ a finite field of characteristic $p$, you want the composition of such a functor with "reduction modulo $p$" to be (canonically isomorphic to) the identity functor. (If not, then ignore the rest of what follows.) Then functoriality is impossible to achieve. Indeed, assume it is possible, so we get a homomorphic section $s$ to the reduction map ${\rm{GL}}_n(W(k)) \rightarrow {\rm{GL}}_n(k)$ for every $n>0$.

Let us see that there is no such section when $n \ge 2$ and $q := |k|>3$. (There must be a zillion ways to prove this; I just give the first argument that came to mind.) Let $T \subset {\rm{GL}}_n$ be a split maximal $k$-torus, so $s(T(k))$ consists of pairwise commuting elements of finite order dividing $q-1$. Since $W(k)[x]/(x^{q-1}-1)$ as a $W(k)$-algebra is a direct product of $q-1$ copies of $W(k)$, it follows that each element of $s(T(k))$ can be diagonalized over $W(k)$ (not just over $W(k)[1/p]$!). Since elements of $s(T(k))$ pairwise commute, we can achieve this diagonalization over $W(k)$ simultaneously, which is to say that we can conjugate to make $s(T(k))$ diagonal. In other words, for some $g \in {\rm{GL}}_n(W(k))$ (with reduction $g_0 \in {\rm{GL}}_n(k)$) the composition $c_g \circ s \circ c_{g_0}^{-1}$ is a section that identifies $c_{g_0}(T(k))$ with the $(q-1)$-torsion in the diagonal over $W(k)$. This forces $c_{g_0}(T(k))$ to be the diagonal of ${\rm{GL}}_n(k)$.

By replacing $s$ with $c_g \circ s \circ c_{g_0}^{-1}$ we may assume $s$ carries the diagonal into the diagonal. Now using that $q>3$, so the roots of ${\rm{GL}}_n$ are pairwise distinct on the group of $k$-points of the diagonal, it follows that $s$ must preserve "root groups"; i.e., for each $i \ne j$ with $1 \le i, j \le n$ (such $i, j$ exist since $n \ge 2$) and $U_{ij}$ the corresponding copy of $\mathbf{G}_a$ as a $W(k)$-subgroup of ${\rm{GL}}_n$ via the $ij$-entry of matrices, we see that $s(U_{ij}(k)) \subset U_{ij}(W(k))$. In particular, we get a homomorphic section $s_{ij}$ to the reduction map $W(k) \rightarrow k$. But that is absurd since $W(k)$ is torsion-free and $k$ is killed by $p$.

[This argument is inspired by the proof that for any semisimple Chevalley group $G$, $G(W_2(k))$ viewed as a $k$-group in the natural way has no Levi $k$-subgroup for $k$ algebraically closed of characteristic $p>0$.]

Let me restrict my attention to the only case I understand, which is $A = \mathbb{F}_p, W(A) = \mathbb{Z}_p$ (so here I mean $p$-typical Witt vectors). You don't provide any conditions you want your functor to satisfy, but I presume you don't just want e.g. the zero functor, so here is a very simple no-go for getting extra properties: no such functor can both

• be additive, and
• send finite free modules to finite free modules.

This is for the very simple reason that $\mathbb{F}_p$ (the endomorphism ring of a finite free $\mathbb{F}_p$-module of rank $1$) admits no ring homomorphism to $M_n(\mathbb{Z}_p)$ for any $n$, since the latter is torsion-free.

More generally this argument shows that the image of any additive functor from $\mathbb{F}_p$-modules to $\mathbb{Z}_p$-modules necessarily lands in modules whose underlying abelian groups are $p$-torsion. Of course there is a natural additive functor given by restriction of scalars along the quotient map $\mathbb{Z}_p \to \mathbb{F}_p$ with this property.