Valuations and seminorms on rings play a big role in number theory and analytic geometry, with seminorms being heavily used in Berkovich geometry and valuations featuring heavily in adic geometry.
Let's quickly recall their definitions:
A valuation on a ring $A$ is a pair $(\Gamma,\nu)$ consisting of
- The Value Group. A totally ordered abelian group $\Gamma$;
- The Valuation. A map of of sets $$\nu\colon A\to\Gamma\mathbin{\textstyle\coprod}\{0\},$$ where $\Gamma\mathbin{\textstyle\coprod}\{0\}$ is the totally ordered abelian group with zero consisting of the disjoint union of $\Gamma$ with $\{0\}$, with multiplication extending that of $\Gamma$ by declaring $0\cdot0=0$ and $0\cdot a=0$ for all $a\in\Gamma$, and where we have $0<a$ for all $a\in\Gamma$.
satisfying the following conditions:
- We have $\nu(0)=0$.
- We have $\nu(1)=1$.
- For each $a,b\in A$, we have $\nu(ab)=\nu(a)\nu(b)$.
- For each $a,b\in A$, we have $\nu(a+b)\leq\max(\nu(a),\nu(b))$.
A seminorm on a ring $A$ is a map $|-|\colon A\to\mathbb{R}_{\geq0}$ satisfying the following conditions:
- We have $|0|=0$.
- We have $|1|=1$.
- For each $a,b\in A$, we have $|a+b|\leq|a|+|b|$.
- For each $a,b\in A$, we have $|ab|\leq|a||b|$.
Moreover, $|-|$ is a norm if $|a|=0$ implies $a=0$.
Question. Are there notions of valuations, seminorms, and norms on ring spectra extending the above notions?
