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I gather that rings of germs of functions at a point $p$ on a manifold/variety/etc. are local with the maximal ideal containing exactly the germs of functions which vanish at $p$. So in some sense, these rings, which happen to be local, describe the local behavior of functions. But what about other local rings? $\mathbb Z/p^n\mathbb Z$ is local for primes $p$ and $n\geq1$. Can we interpret it as a ring of germs of functions on some space?

I found a way to do so for fields $F$, at least. They can be seen as the ring of functions (or germs thereof, makes no difference in this case) on a one-element topological space $\{p\}$, where $x(p):=x$ for all $x\in F$. Which seems like it would make sense in a context where local rings actually are rings of germs: local rings with trivial maximal ideal (fields) are germs of functions on a trivial space. But how to generalize this?

Vercassivelaunos
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1 Answers1

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Depends on your definition of "function". The following answer is based on the notion of schemes in algebraic geometry.

A function on (a scheme) $X$ is a morphism $X\to\mathbb{G}_a$, where $\mathbb{G}_a=\mathrm{Spec}(\mathbb{Z}[t])$ (where $t$ is a "canonically" chosen variable). Added: Here canonically means that one cannot replace $t$ by another generator of $\mathbb{Z}[t]$ as a ring.

For the case $X=\mathrm{Spec}(R)$, this is amounts to a map $\mathbb{Z}[t]\to R$.

The points of the space $\mathrm{Spec}(R)$ are prime ideals in $R$. Added: The basic open sets (used to define germs of functions) in $\mathrm{Spec}(R)$ are of the form $D(a)=\{\mathfrak{p} : a\not\in\mathfrak{p}\}=\mathrm{Spec}(R_a)$ for various $a$ in $R$.

The "ring of germs of functions at a point" of the scheme $\mathrm{Spec}(R)$ is $R_{\mathfrak{p}}$ where $\mathfrak{p}$ is a point of the scheme $\mathrm{Spec}(R)$.

Note that if $R$ is a local ring and $\mathfrak{m}$ is its maximal ideal, the n $\mathfrak{m}$ is also a prime ideal of $R$ and $R=R_{\mathfrak{m}}$.

Kapil
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  • Thank you for your answer! I have a few clarifying questions left, though. By "canonically chosen variable", do you mean a free variable? So just the usual polynomial ring? And how is $R_{\mathfrak p}$ a ring of germs at $\mathfrak p$? What functions would the germs consist of? – Vercassivelaunos Jan 02 '22 at 10:45
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    @Vercassivelaunos, a simple way to think of $\operatorname{Spec} R$ is a space whose (global) ring of functions is precisely $R$. In the Zariski topology, we can look at the stalk of a function $r\in R$ at a point $\mathfrak{p}$ (which is a topological procedure defined for all sheaves), and show that it is isomorphic to $R_{\mathfrak p}$. I recommend the book by Eisenbud-Harris on schemes. – Aurelio Jan 02 '22 at 19:14