# How does normalization behave on closed subschemes?

I'm trying to understand Harbater's "Mock Covers and Galois Extensions".

There, a "mock cover" of a domain $S$ is a finite map $p :$ Spec $T\rightarrow$ Spec $S$, where $T$ is a torsion-free $S$-algebra, and such that the restriction of $p$ to any irreducible component of Spec $T$ is an isomorphism onto Spec $S$.

Now let $A$ be a Dedekind domain. Suppose $Z$ is an integral scheme with a finite Galois torsion-free generically separable map $p : Z\rightarrow$ Spec $A[[t]]$, such that the preimage $Z|_{t=0}$ above $t = 0$ is a mock cover. Let $\tilde{Z}$ be the normalization of $Z$.

Is there anything we can say about the relation between $\tilde{Z}|_{t=0}$ and $Z|_{t=0}$? Should I expect them to be isomorphic? When can they fail to be isomorphic?

I have very little intuition on how normalization behaves on closed subschemes.

In general, you can say very little about the fibers of a normalization. It is possible that the total space is very nice yet the fibers have horrible singularities. (OK, not extremely horrible, but definitely not normal is not only possible, but the expected behavior for a special fiber).

In your case the situation is a little nicer, but based on only this much information one cannot tell what's happening. If $Z|_{t=0}$ is mock cover of ${\rm Spec} \left( A[[t]]/(t)\right)\simeq {\rm Spec}\, A$, then its irreducible components are isomorphic to ${\rm Spec}\, A$, so then your question depends on how these irreducible components come together.

The easiest case is if $Z|_{t=0}$ is a disjoint union of its irreducible components, in which case it is regular itself and since $Z|_{t=0}$ is a Cartier divisor in $Z$, it follows that then $Z$ is regular along $Z|_{t=0}$ and hence $\widetilde Z|_{t=0}=Z|_{t=0}$. However, this is a very special case.

It is more likely that the irreducible components of $Z|_{t=0}$ intersect, so it is itself not normal. Of course, this in itself is not necessarily a deal breaker, it just means that it is nearly impossible to tell what is happening.

The problem is that the normalization might separate some (or all) of these components, but which ones get separated and which don't depends on how the nearby fibers behave. I don't think you can give a simple criterion on what to expect.

So, the short answer is that based on the information given, what you hope for may or may not happen.

Example

For simplicity, let us assume that $A=k[x]$ for some field, although I am sure that the same construction can be done essentially for any $A$. Also, let us work over $A[t]$. Whatever we have can be specialized over $A[[t]]$ afterwards.

So then $A[t]=k[x,t]$ and hence ${\rm Spec}\, A[t]\simeq \mathbb A^2_k$. Next, let $X$ be a non-singular surface with a flat morphism $f:X\to \mathbb A^1_k\simeq {\rm Spec}\, k[t]$. Assume that $X_\lambda=f^{-1}((t-\lambda))$ is regular for $\lambda\neq 0$ and is a union of finitely many copies of $\mathbb A^1_k$ for $\lambda=0$. Assume further that $X$ admits two sections with images $C_1,C_2\subseteq X$ and let $Z$ be the surface obtained from $X$ by gluing $C_1$ to $C_2$ according to the identification of $C_1$ with $C_2$ given by $f$. It follows that $f$ factors through $Z$ and we still have a flat morphism $g:Z\to \mathbb A^1_k$. Now by Noether normalization $Z$ admits a finite morphism to $\mathbb A^2_k$ (I think one should be able to do this, so it is also Galois, but I haven't given this much thought, perhaps you can work this out). One can definitely make this so that it respects the morphism $g$: First take a finite map of $Z$ to $\mathbb A^2_k$, then project $\mathbb A^2_k$ to $\mathbb A^1_k$ in a direction that no irreducible component of any fiber of $g$ is mapped to a point. Call the composition $h:Z\to \mathbb A^1_k$ and take $p=g\times h: Z\to \mathbb A^2_k$. This $p$ should satisfy the original conditions. The normalization of $Z$ is obviously $X$ and hence the normalization separates some components of $Z|_{t=0}$ while leaving some together. Varying this construction you can get all kinds of examples.

• Dear Professor Kovacs, In your second to last paragraph, what do you mean by "nearby fibers"? Are you referring to the preimage of Z above primes other than $(t)$? Also, is there a simple example where normalization separates some components of the preimage above $t = 0$ but leaves others intersecting? Sep 5, 2016 at 19:56
• Dear @rtz, I don't think on this site one needs to address others as "Professor". For instance, I can't tell whether you are one, so I can't reciprocate.... :) Anyway, yes, by "nearby fibers" I meant fibers above other primes. I'll try to add an example. Sep 5, 2016 at 20:12
• I've just realized that I've been thinking about this for $A[t]$. I don't think that changes the outcome, but that certainly explains why you are puzzled by my using the words "nearby fibers". Sorry about that. In any case, it might be worth thinking about examples that come from examples above $A[t]$. Sep 5, 2016 at 20:18
• Thanks for your example! How do you deduce that $g : Z\rightarrow\mathbb{A}_k^1$ is flat? Sep 5, 2016 at 21:20
• I don't think you actually need that, but it is because $Z$ is irreducible, $\mathbb A^1_k$ is regular of dim 1 and $g$ is dominant. Sep 5, 2016 at 21:27