Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X=Spec(R)$ be an irreducible noetherian scheme and $\eta$ the unique minimal prime ideal of $R$. Let $U$ and $V$ be open sets in $X$ and $p$ a point with $p\in V\subseteq U\subseteq X$.

If $X$ is additionally integer (i.e. additionally reuced, i.e. $R$ is a domain and $\eta=(0)$) then one has a diagram

$$ \begin{array}{ccccccl} \mathcal{O}_X(U)& \hookrightarrow &\mathcal{O}_X(V)& \hookrightarrow&{\mathcal{O}}_{X,p}&\twoheadrightarrow& k(p)=Frac(R/p)&\\ &&&\searrow&\downarrow&&&\\ &&&&{\mathcal{O}}_{X,\eta}&\xrightarrow{=}& k(\eta)=Frac(R/\eta)&(=Frac(R)) \end{array} $$

where the indicated maps ($\hookrightarrow$), the diagonal map ($\searrow$) and ${\mathcal{O}}_{X,p}\hookrightarrow {\mathcal{O}}_{X,\eta}$ are inclusions. I have some questions relating to this situation.

$\mbox{1.}$ Is the map $k(p)\to k(\eta)$ an inclusion?

One may identify all the images of the inclusions in ${\mathcal{O}}_{X,\eta}=Frac(R)$ with their domains and has $$\mathcal{O}_X(U)=\bigcap_{p\in U}{\mathcal{O}}_{X,p}.~~(*)$$

This identification is very helpful for me since one can really "work" then inside the big ring ${\mathcal{O}}_{X,\eta}$.

$\mbox{2.}$ I would like to understand function fields and stalks in the non-reduced case (but $X$ still irreducible). Then one can write down the same diagram as above (instead of the equality $Frac(R/\eta)=Frac(R)$). Which of the arrows remain inclusions ? Can I write down something like $(*)$ in this case, too?

share|cite|improve this question
up vote 4 down vote accepted

Concerning 1.: Why do you think that there should be a map $k(p)\to k(\eta)$?

Consider for example the case $R=\mathbb{Z}$. Then for any two different points $p$ and $q$ of $\mathrm{Spec}\,\mathbb{Z}$ one has $\mathrm{Hom}(k(p),k(q))=\emptyset$ because these are two fields of different characteristic.

Concerning 2., the ring $R=k[x,y]/(x^2,xy)$ is enlightening. Topologically it is identical to $k[x,y]/(x)=k[y]$, so its spectrum is just a line, but with the origin infinitesimally thickened in the $x$-direction. The ideal $(x)$ is its unique minimal prime ideal. Whenever $U$ is an open subset not containing the point $(x,y)$ the restriction map $\mathcal{O}_X(X)\to \mathcal{O}_X(U)$ is not injective, because $x$ maps to zero in the latter ring. In particular $\mathcal{O}_X(X)\to \mathcal{O}_{X,\eta}$ is not injective.

share|cite|improve this answer
Well said, Philipp, very lucid (+1, of course). – Georges Elencwajg Jun 27 '11 at 20:24
Yes, thank you, Philipp! Can I still write $\mathcal{O}_X(U)$ as something having to do with the stalks $\mathcal{O}_{X,p}$ in the non-reduced case, like in (*)? This would help me much to understand what sections on $U$ should be. – geometrystudent Jun 28 '11 at 12:33

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.