Algebra Seminar talk

2018-01-19 (12:00)
Sophie Frisch (TU Graz)
On the Spectrum of Rings of Functions

For $D$ a domain and $E\subseteq D$, we investigate the prime spectrum of rings of functions from $E$ to $D$, that is, of rings contained in $\prod_{e\in E} D$ and containing $D$, where we identify elements of $D$ with the corresponding constant functions.

Among other things, we characterize, when $M$ is a maximal ideal of finite index in $D$, those prime ideals lying above $M$ which contain the kernel of the canonical map to $\prod_{e\in E} (D/M)$ as being precisely the prime ideals corresponding to ultrafilters on $E$.

We give a sufficient condition for when all primes above $M$ are of this form and thus establish a correspondence to the prime spectra of ultraproducts of residue class rings of $D$. As a corollary, we obtain a description using ultrafilters, differing from Chabert's original one which uses elements of the $M$-adic completion, of the prime ideals in the ring of integer-valued polynomials $\mathrm{Int}(D)$ lying above a maximal ideal of finite index.