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In a short article appearing on the arxiv today Cuntz and Deninger seem to give a new simple construction of the ring of Witt vectors. Seems like good news for people like me who use them on a daily basis but would probably need to go looking for a copy of Serre’s “Local Fields” if asked for the details on how to actually construct them. This post represents my processing the definition appearing in their paper.

Firstly, recall that for any perfect $\mathbb{F}_p$-algebra $R$, a strict $p$-ring for $R$ is a ring $A$ in which $p$ is not a zero-divisor, Hausdorff and complete for the $p$-adic topology, and with $A/p \cong R$. The key fact about these rings is the following.

Proposition: For every perfect $\mathbb{F}_p$-algebra $R$, a strict $p$-ring $W(R)$ exists, and is unique up to unique isomorphism as a ring over $R$ (in fact $R \mapsto W(R)$ is a functor).

This is of course well-known. The new construction goes as follows. View $R$ as a monoid (remember multiplication and the identity element), and form the free ring $\mathbb{Z}R$ thereon. This has elements $\sum_i n_i[r_i]$ with addition as a “free $\mathbb{Z}$-module” and multiplication coming from the monoid structure. For example, this means that weirdly $[0]\not= 0$ yet (but $[1]=1$). There is also (induced from the identity on $R$ as a monoid) a natural ring map $\mathbb{Z}R \rightarrow R$. Let $I$ be its kernel, and form the $I$-adic completion $A$. It turns out (and the proof in their paper is pretty short) that this is a strict $p$-ring and so canonically isomorphic to $W(R)$.

One interesting feature of this construction is that of course you can run it for $R$ not perfect, where the Witt construction also gives you something but which is non-canonical (a `Cohen ring’). Apparently this construction in general gives a different one from the Witt construction (so it is genuinely different: not a clever repackaging).