# When does the regularity of $A$ implies the regularity of $A[w]$?

Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular.

Recall that a commutative noetherian ring is called regular if all its localizations at maximal ideals are regular local rings.

Let $w$ be algebraic over $A$ (I do not mind to assume that $w$ is integral over $A$. I do NOT want to assume that $w$ is in the field of fractions of $A$), and assume that $A[w]$ is noetherian.

My question: When is $A[w]$ regular? (I suspect there exists a counterexample, even under the additional conditions: $A$ is a UFD, $w$ is integral over $A$).

Notice the following question, which deals with $w$ transcendental over $A$: Does the regularity of $A$ imply the regularity of $A[X]$?

• Some simplifications that don't really answer your question: I think $A$ will automatically be a UFD (Auslander-Buchsbaum) and in particular normal. And certainly $A[w]$ will be Noetherian by the Hilbert basis theorem.
– Hoot
Jun 11, 2015 at 1:35
• Thanks for trying to help. A clarification: In the specific example I have in mind, $A$ is not local but a UFD. (If I am not wrong, generally: A regular ring is normal-- this can be found in Matsumura's book. A regular local ring is a UFD). I thought the Hilbert basis theorem works when $w$ is transcendental over $A$, not algebraic? Jun 11, 2015 at 1:51
• Thanks! Actually, I have once asked a similar question (in which $A$ is a polynomial ring in two indeterminates): math.stackexchange.com/questions/1287044/… Jun 11, 2015 at 2:29
• I don't understand your emphasis on $A[w]$ noetherian: AFAIK a regular ring is assumed noetherian (that is, $A$ is noetherian by definition), and I don't think there are many books in CA which miss this condition. Jun 11, 2015 at 9:02

The nonregular ring $R=k[x,y]/\langle x^2 - y^3\rangle$, which is the coordinate ring of the cusp (not regular at the maximal ideal $\langle x,y\rangle$), is of this form. Indeed, you can set $A:=k[t^2]\cong k[t]$ which is a polynomial ring and then, $R=A[t^3]$. Here, $t^3$ is integral over $A$, but it is not in the field of fractions for reasons of degree.

Geometry: You are basically taking a smooth affine scheme $X$ and then you consider a closed subscheme $Z\subseteq \mathbb A^1_X$ whose projection to $X$ is finite. That closed subscheme may well be singular. However, this is not the case if that finite projection is actually étale, i.e. smooth of relative dimension zero - in this case $Z$ must be smooth. There are several equivalent definitions of being étale, one would be flat and unramified. These are now just properties of the ring extension $A\subseteq A[w]$.

Edit. When you can guarantee flatness and you want to use the above criterion, you have to check the following: For any prime ideal $\mathfrak q\subseteq A[w]$, the prime ideal $\mathfrak p := \mathfrak q\cap A$ (of $A$) generates the maximal ideal of the local ring $A[w]_{\mathfrak q}$. This is the condition for $A\hookrightarrow A[w]$ to be unramified.

• very nice answer, especially the geometric interpretation. One can note that in this case the extension (or the morphism) is not flat. Jun 11, 2015 at 7:40
• @Jesko Huttenhain: Thanks! What if I know that the extension $A \subset A[w]$ is flat? (in the special example I have in mind, $A[w]$ is actually a free $A$-module). Jun 11, 2015 at 16:03
• @user237522 In spite of the first comment above, the given example is such $A\subset A[w]$ is flat, in fact even free. (I've thought this is pretty clear from the way I've constructed this example in my answer.) Jun 11, 2015 at 16:24
• @user26857: That's what I thought...next time I will count on myself. Jun 11, 2015 at 16:53
• @user237522: I added an explanation of "unramified". I am not sure how easy that is to check, though. Jun 11, 2015 at 17:20

If $A$ is regular, then $A[X]$ is regular. Now let $f$ be a monic polynomial in the square of a maximal ideal of $A[X]$. Then $A[X]/(f)=A[w]$ is not regular. (It's worthwhile to mention that $A[w]$ is however CM; see Bruns and Herzog, Proposition 2.2.11.)

Concrete example: $A=k[t]$, and $f(X)=X^2-t^3$. The maximal ideal $\mathfrak m=(t,X)$ has the property $f\in\mathfrak m^2$. (I've chosen this example on purpose, to show that the example given by the other answer is a particular case of the construction I proposed: just replace here $t$ by $t^2$.)

• Thanks! In the special example I have in mind (in which $A$ is a UFD and $A[w]$ is a domain), $A[w]$ is even a complete intersection ring (regular $\subset$ complete intersection $\subset$ Gorenstein $\subset$ Cohen-Macaulay). However, I am not able to show it is regular. Actually it is good enough for me to show that $A[w]$ is integrally closed. Jun 11, 2015 at 16:18
• @user237522 Please rephrase your comment in such a way I can understand exactly what's the hypothesis and what you want to show. Jun 11, 2015 at 16:21
• Thank you very much! Now, $A$ is a regular UFD, $w$ is not in the field of fractions of $A$, $w$ is integral over $A$, so $A[w]$ is a finitely generated $A$-module (then by a corollary on page 58 of one of Bourbaki's books, $A[w]$ is a projective $A$-module. But my $A$ is $K[x,y]$, so $A[w]$ is a f.g. free $A$-module, by Quillen-Suslin). I wish to show that $A[w]$ is integrally closed (if I am not wrong, for a domain being normal and integrally closed is the same. If one shows $A[w]$ is regular, then it is normal; however, showing normality is good enough for me). Jun 11, 2015 at 16:43
• @user237522 "Now, A is a regular UFD, w is not in the field of fractions of A, w is integral over A, so A[w] is a finitely generated A-module. I wish to show that A[w] is integrally closed." As far as I can see you wanted to generalize this to A a regular UFD. I suggest you to ask the question for A = K[X,Y] as a new question, as now we are far from the original one. (I don't understand why you ask a general question if have in mind a clear one.) Jun 11, 2015 at 18:49
• I admit it was not clear if my question was about a regular UFD $A$, or $A=K[x,y]$. The reason for me asking a more general question than the specific example I have in mind, is that only recently I have started to be interested in regularity, separability, normality etc., so I thought that maybe I am missing some known facts that can be applied to my case. After asking some relevant questions and getting counterexamples, it seems that my specific case needs special attention. Jun 11, 2015 at 20:02