Ring of "Sequences up to finitely many entries"

Question

Given any ring $R$ we can look at the ring of $R$-sequences $\text{Sets}(\mathbb N,R) \cong R^\mathbb N$. Take the quotient $R^\mathbb N/\sim$ by the equivalence relation ~ given by setting two sequences equivalent if they agree up to finitely many terms. Does this ring have a specific name in the literature?

Also: The diagonal map $R \xrightarrow{\Delta} R^\mathbb N$ induces an $R$-algebra structure on $R^\mathbb N/\sim$. Is $\text{Spec}(R^\mathbb N/\sim)$ in any way interesting as a scheme over $R$?

Answer 1

I don't know a name for this ring, but let me say a little about its spectrum. Let $S=R^\mathbb{N}/\sim$. For any subset $A\subset\mathbb{N}$, there is an element $e_A\in S$ which is $1$ on each element of $A$ and $0$ on each element of $\mathbb{N}\setminus A$. These elements $e_A$ are idempotent, and $e_A=e_B$ iff the symmetric difference $A\mathbin{\Delta}B$ is a finite set. If $P\subset S$ is a prime ideal, then $f(P)=\{A\in\mathbb{N}:1-e_A\in P\}$ is an ultrafilter on $\mathbb{N}$. Since $e_F=0$ for $F$ finite, $f(P)$ cannot contain any finite sets, so $f(P)$ is a nonprincipal ultrafilters. This defines a continuous map $f:\operatorname{Spec} S\to \beta\mathbb{N}\setminus\mathbb{N}$, where $\beta\mathbb{N}\setminus\mathbb{N}$ is the space of all nonprincipal ultrafilters on $\mathbb{N}$ (also known as the Stone-Cech remainder of $\mathbb{N}$).

We can get a handle on the fibers of this map $f$. If $U$ is a nonprincipal ultrafilter on $\mathbb{N}$, then $f^{-1}(U)$ is just the closed subscheme of $\operatorname{Spec} S$ consisting of those prime ideals that contain $1-e_A$ for each $A\in U$. That is, it is the spectrum of the quotient of $S$ by the ideal generated by these elements $1-e_A$ for $A\in U$. Equivalently, it is the quotient of $R^{\mathbb{N}}$ by ideal generated by the elements $1-e_A$ in $R^{\mathbb{N}}$, since every cofinite set $A$ is in $U$ and the elements $1-e_A$ for $A$ cofinite exactly generate the kernel of the map $R^{\mathbb{N}}\to S$. This quotient of $R^{\mathbb{N}}$ is better-known as the ultrapower of $R$ by the ultrafilter $U$.

So to sum up, $\operatorname{Spec} S$ has a canonical continuous map to the space of nonprincipal ultrafilters on $\mathbb{N}$, and the fibers of this map are the spectra of the ultrapowers of $R$. If you were looking at $\operatorname{Spec} R^\mathbb{N}$ instead, the description would be exactly the same, except that you would also include the principal ultrafilters. For more related discussion, you may be interested in the answers to this question which focuses on the case $R=\mathbb{Z}$.