Real Spectrum of a Commutative Ring

Question

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance.

$\text{Spec}_{r}(A)$

Answer 1

Let start with orderings on fields. Let $F$ be a field and $P\subseteq F$. We say that $P$ is an ordering if $P+P\subseteq P$, $P\cdot P\subseteq P$, $P\cup(-P)=F$ and $P\cap(-P)=\{0\}$. Here we consider $P+P=\{a+b:a,b\in P\}$, $P\cdot P=\{ab:a,b\in P\}$ and $-P=\{-a:a\in P\}$.

Thus, the real spectrum of $F$ is the set $$\mbox{Spec}_r(F)=\mbox{Sper}(F)=\{P\subseteq F:P\mbox{ is an ordering}\}.$$

For a commutative ring, the Definition is the following. Let $A$ be a commutative ring and $P\subseteq A$. We say that $P$ is an ordering if $P+P\subseteq P$, $P\cdot P\subseteq P$, $P\cup(-P)=A$ and $P\cap(-P)$ is a prime ideal of $A$, called the support of the ordering $P$. The real spectrum of $A$ is the set $$\mbox{Spec}_r(A)=\mbox{Sper}(A)=\{P\subseteq A:P\mbox{ is an ordering}\}.$$

Calculate $\mbox{Sper}(A)$ is a hard task in general even for fields. There are here some initial examples:

1. Let $F=\mathbb Q(\alpha)$ where $\alpha^2=2$. We can define an ordering $P$ on $F$ by using the embedding $\varphi:F\rightarrow\mathbb R$ with $\varphi(\alpha)=\sqrt2$. Similarly, we can define another ordering $P'\ne P$ on $F$ by using the $\mathbb Q$-automorphism $\varphi':\mathbb Q(\sqrt2)\rightarrow\mathbb Q(-\sqrt2)$ with $\varphi'(\alpha)=-\sqrt2$.
2. Let $F=K(x)$, where $K$ is a field given with an ordering $P_0$. We can extend this ordering on $F$ in several ways. First, we declare a polynomial $$f(x)=a_0+a_1x+...+a_nx^n\in K[x],\,a_n\ne0,$$ positive if $a_n\in\dot P_0$. Then we declare a rational function $g(x)/f(x)$ positive if the polynomial $f(x)g(x)$ is positive. The set of positive elements in $F$ defined in this way, together with $0$, gives an ordering $P_1$ on $F$. Note that in this ordering, we have $$0<...<x^{-2}<x^{-1}<a<x<x^2<...$$ for any $a\in\dot P_0$, as we can readily check. We can also get a second extension of $P_0$ as follows. Declare a polynomial $$f(x)=a_rx^r+a_{r+1}x^{r+1}+...+a_nx^n\in K[x],\,r\le n,\,a_n,a_r\ne0$$ positive if $a_r\in\dot P_0$, and extend this positivity notion to $F=K(x)$ as before. This results in a second ordering $P_2$ extending $P_0$. With respect to this ordering $P_2$, we have instead $$0<...<x^2<x<a<x^{-1}<x^{-2}<...$$ for any $a\in\dot P_0$. These orderings are examples of \textbf{nonarchimedean} orderings on $F$: these are orderings with respect to which there are elements that are larger than all integers (and hence all rational numbers) in $F$.
3. Consider $P_0$ the usual ordering on $\mathbb R$. Let $C$ be any subset of $\mathbb R$ with the property \begin{align*} \mbox{For any pair }a<b\in\mathbb R: b\in C\Rightarrow a\in C \end{align*} (for example, take $C$ as an open interval $(-\infty,b)$). We can define an ordering $P_C$ on $F=\mathbb R(x)$ as follows. For any nonzero polynomial $f(x)\in\mathbb R[x]$, write down the factorization of $f$ into irreducible factors $$f(x)=r(x-a_1)...(x-a_n)q_1(x)...q_m(x),$$ where $r,a_1,...,a_n\in\mathbb R$ and the $q_i$'s are monic irreducible quadratic polynomials. We shall take $f(x)\in\dot P_C$ iff $r\in\dot P_0$ and the number of $a_i\notin C$ is even, or $r\notin\dot P_0$ and the number of $a_i\notin C$ is odd. For nonzero rational functions $g(x)/f(x)$, we take (as before) $g/f\in\dot P_C$ iff $gf\in\dot P_C$. It can be shown that the $P_C$ obtained in this manner is an ordering on $\mathbb R(x)$, and is, in fact, the unique ordering $P$ on $\mathbb R(x)$ with respect to which $C=\{b\in\mathbb R:b<_Px\}$.
4. Take $A$ to be the polynomial ring $\mathbb R[t]$ again, and $$P_{\infty^+}=\{a_0+a_1t+...+a_kt^k:k\ge0,\,a_k>0\}\cup\{0\}.$$ Then $P_{\infty^+}$ is an ordering in $A$.
5. We have constructed three orderings $P_0,P_{0^+},P_{\infty^+}$ on $\mathbb R[t]$. Applying the automorphism $t\mapsto-t$ to $P_{0^+},P_{\infty^+}$ yields two additional orderings $P_{0^-},P_{\infty^-}$. Applying the automorphism $t\mapsto t-a$ ($a\in\mathbb R$) to $P_0,P_{0^+},P_{0^-}$ yields orderings $P_a,P_{a^+},P_{a^-}$ with $P_{a^+},P_{a^-}\subsetneq P_a$. The orderings $P_{a^+},P_{a^-}$ for $a\in\mathbb R$ together with $P_{\infty^+},P_{\infty^-}$ have support $\{0\}$. The ordering $P_a$ has as support $(t-a)$. We have $$\mbox{Sper}(\mathbb R[t])=P_{\infty^+}\cup P_{\infty^-}\cup\{P_a,P_{a^+},P_{a^-}\}_{a\in\mathbb R[t]}.$$ The support $\{0\}$ orderings on $\mathbb R[t]$ are just the orderings on the field $\mathbb R(t)$.

These are the first examples. I was inspired mostly in the following references:

• Spaces of orderings and abstract real spectra from Murray Marshall;
• Introduction to quadratic forms over fields from T. Y. Lam;
• Valuations, orderings, and Milnor K-theory from Ido Efrat;
• Quadratic forms with Applications to Algebraic Geometry and Topology from A. Pfister;
• Real Algebraic Geometry from Jacek Bochnak, Michel Coste and Marie-Françoise Roy.

I do not know your interest in this subject but I think these references are good enough for learning and further lectures in quadratic forms, real algebraic geometry and so on.