# A closer look at Spec

Let’s think about what is going on in a different way. So now let’s think of $f \in R$ elements of the ring as functions with domain $Spec(R)$. We define the value of the function at a point in our space $f(P)$ to be the residue class in $R/P$. This looks weird at first, since the image space depends on the point that you are evaluating the function.

Before worrying about that too much, let’s see if we can get this notion to match up with what we did yesterday. We have the nice property that $f(P)=0$ if and only if $f \in P$. (Remember that even though we think of f as a function, it is really an element of the ring).

Define for any subset of the ring S the zero set: $Z(S)=\{P\in Spec(R): f(P)=0, \forall f \in S\}$. Now from what I just noted in the previous paragraph, we get that these are just precisely the elements of $Spec(R)$ that contain S, i.e. the closed sets of the Zariski topology. Thus we can define our basis for the Zariski topology to be the collection of $D(f)=Spec(R)\setminus Z(f)$.

We also will want what is “an inverse” to the zero set. We want the ideal that vanishes on a subset of Spec. So given $Y\subset Spec(R)$, define $I(Y)=\{f \in R : f(P)=0, \forall P\in Y\}$. Now this isn’t really an inverse, but we get close in the following sense:

If $J\subset R$ is an ideal, then $\displaystyle I(Z(J))=\sqrt{J}$. Taking the ideal of the zero set is the radical of the ideal. And the radical has two equivalent definitions: $\displaystyle \sqrt{J}=\cap_{P\in Spec(R), P\supset J} P=\{a\in R : \exists n\in \mathbb{N}, a^n\in J\}$.

If we take the ideal and zero set in the other order we get that $Z(I(Y))=\overline{Y}$ : the closure in the Zariski topology.

We can abstract one step further and put a sheaf on $D(f)$. Note that for any $f\in R$ we have that $\{1, f, f^2, \ldots\}$ is a multiplicative set, so we can localize at it. Since I haven’t talked at all about sheaves, I’m not sure if I want to go any further with this, so maybe I’ll do some more examples next time and possibly start to scratch this surface.