6
$\begingroup$

For a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue class field $K$, consider the invariant $$\operatorname{def}(R) := \operatorname{dim}_K(\mathfrak{m}/\mathfrak{m}^2) - \operatorname{dim}(R).$$ Here are some questions:

  1. If $P \subseteq R$ is prime ideal, is it always true that $\operatorname{def}(R_P) \le \operatorname{def}(R)$?

  2. For which Noetherian rings is the function $\operatorname{Spec(}R) \to \mathbb{N}$, $P \mapsto \operatorname{def}(R_P)$ upper semicontinuous?

  3. Is there a name (and/or notation) that is commonly used for this invariant?

By a variant of the Jacobian criterion one can see that question 2 has an affirmaitve answer if $R$ is a finitely generated algebra over a perfect field. But are there more general results?

$\endgroup$

1 Answer 1

2
$\begingroup$

Edit. Finally the proof was not so long, so I include it complete:

Question 3. Embedding codimension (sometimes simply codimension).

Question 1. I don't have access here to "Lech, Inequalities related to certain couples of local rings, Acta math. 112 (1964), 69-89", but maybe that paper will answer it better, I can't remember. However, here is a proof when $A$ is quasi-excellent:

Notation: dim = Krull dimension, edim = embedding dimension, codim = edim - dim = embedding codimension.

Lemma 1. Let $(A,m) \to (B,n)$ be a flat homomorphism of noetherian local rings. Then dim $A$ + dim $B/mB$ = dim $B$.

Lemma 2. Let $(A,m) \to (B,n)$ be a flat homomorphism of noetherian local rings with regular closed fiber $B/mB$. Then edim $A$ + edim $B/mB$ = edim $B$.

Deduction of Lemma 2 from Andre-Quillen homology (references (x.y) are to Result y from Chapter x in Andre, Homologie des algebres commutatives, Springer, 1974): let $f:(A,m,k) \to (B,n,l)$ be a flat homomorphism of noetherian local rings with regular closed fiber. Let $p=n/mB$ be the maximal ideal of $B/mB$. It is sufficient to prove that we have an exact sequence of $l$-vector spaces $$0 \to m/m^2 \otimes_k l \to n/n^2 \to p/p^2 \to 0.$$ But this exact sequence is a part of the Jacobi -Zariski exact sequence in Andre-Quillen homology (5.1) associated to $B \to B/mB \to l$: $$H_2(B/mB,l,l) \to H_1(B,B/mB,l) \to H_1(B,l,l) \to H_1(B/mB,l,l) \to H_0(B,B/mB,l).$$

$H_2(B/mB,l,l)=0$ since $B/mB$ is regular by (6.26).

$H_1(B,B/mB,l)=H_1(A,k,l)= m/m^2 \otimes_k l$ by (4.54) (since $f$ is flat) and (6.1) respectively.

$H_1(B,l,l)= n/n^2$, and $H_1(B/mB,l,l)= p/p^2$ by (6.1).

$H_0(B,B/mB,l)=0$ by (4.60).

Another proof without Andre-Quillen homology can be found in arXiv:1205.2119v3, Lemma 3.1.

Corollary 3. Let $(A,m) \to (B,n)$ be a flat homomorphism of noetherian local rings with regular closed fiber. Then codim $A$ + codim $B/mB$ = codim $B$. In particular, codim $A \leq $ codim $B$. Another (trivial) particular case is codim $A$ = codim $\hat{A}$.

Definition. We say that a noetherian local ring is a G-ring if the completion homomorphism $A \to \hat{A}$ is regular. By Matsumura, Commutative Algebra, (33.C) Theorem 75 page 251 this is equivalent to the usual definition of G-ring, and by Theorems, 73, 76 and 77 this is also equivalent to be quasi-excellent (since it is local).

Corollary 4. Let $A$ be a local G-ring. Then for any prime ideal $p$ of $A$ we have codim $A_p \leq $ codim $A$.

Proof. The result is valid for a complete ring $A$ (choose a regular local ring $S$ with the same embedding dimension of $A$ such that $A=S/I$ and localize), so codim $\hat{A}_q \leq $ codim $\hat{A}$ = codim $A$. Since $A \to \hat{A}$ is faithfully flat, there exists a prime ideal $q$ of $\hat{A}$ contracting to $p$. The local homomorphism $A_p \to \hat{A}_q$ is regular and so by Corollary 3 codim $A_p \leq $ codim $\hat{A}_q$.

Question 2.

For excellent rings. An indirect proof, but short enough to be written here, is as follows. Embedding dimension is the diference between the invariant $\delta_2$ in "Ragusa, On Opennes ..." which is upper semicontinuous by Proposition 3.6 in that paper, and the complete intersection defect which is upper semicontinuous by the main theorem in "Alonso-Rodicio, On the upper ...". So for each $n$ the set of primes $p$ such that codim $A_p > n$ is constructibe. By Corollary 4 it is also stable under specialization. So by Matsumura, Commutative Algebra, (6.G) Lemma, page 46, it is closed.

$\endgroup$
4
  • $\begingroup$ Thanks a lot! The answer to Q1 should be enough since in the case I'm interested in, R is a quotient of a regular local ring. I think your argument is very nice, Many thanks for it. Lech's article has the required result, but with the hypothesis that ht(p) + dim(R/p) = dim(R). That is too restricitve. Is there any way how I can send you a copy of the article? About Q2, writing ecodim as the difference of two upper semicontinuous functions only seems to show that the locus where ecodim > d is constructible, not necessarily closed. Am I missing something? $\endgroup$ Nov 12, 2015 at 16:25
  • $\begingroup$ I'm sorry, my answer to Q2 depended on Q1 and so it becomes incomplete when I edited Q1. Nevertheless it is true. For an excellent ring (in fact for a local G-ring, i.e., a noetherian local ring $R$ with $R \to \hat{R}$ a regular homomorphism, in particular not necessarily catenary) the answer to your question 1 is affirmative (so the locus where ecodim > d is constructible and stable under specialization, and so closed). If this is interesting to you, I can email you next week with a proof. $\endgroup$
    – Vinteuil
    Nov 12, 2015 at 19:48
  • $\begingroup$ I have a copy of Lech's paper at my office, but I won't be there till next week. Thank you anyway. $\endgroup$
    – Vinteuil
    Nov 12, 2015 at 19:52
  • $\begingroup$ Thanks again! Yes, I'd be very interested in seeing a complete proof. My email address is [email protected]. $\endgroup$ Nov 13, 2015 at 15:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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