4
$\begingroup$

Given a specific ring $R$ (eg, $R=k[x_{1}, \cdots, x_{n}]/I)$ is there a (simple) way to determine whether or not $R$ satisfies Serre's condition $S_{n}$? In particular, is there a way to do this in Macaulay2?

($R$ satisfies $S_{n}$ if $depth R_{p} \geq \min(height(p), n)$ for all $p \in Spec R$.)

$\endgroup$
  • $\begingroup$ So wait, can we assume that R is finitely presented (over a base ring with known properties) or no? (If we can't assume that, I am very doubtful that you'll be able to compute all that much with software). $\endgroup$ – Harry Gindi Aug 27 '10 at 21:40
  • $\begingroup$ sure, we can assume that R is finitely presented. Even if R is a quotient of a polynomial ring, I'm not sure how to go about determining if it's $S_{n}$. $\endgroup$ – LAM Aug 28 '10 at 6:10
5
$\begingroup$

I think there is a neat answer to this question.

Lemma: Let $S$ be regular local of dimension $d$, $M$ a f.g $S$-module. Then: $$\text{depth}(M)\geq n \Longleftrightarrow \text{Ext}^i(M,S)=0\ \text{for} \ i>d-n$$

Proof: LHS is equivalent to $e= \text{pd}_SM \leq d-n$. By using a minimal free resolution of $M$ to compute Ext, one sees that $\text{Ext}^i(M,S)$ is not $0$ for $i=e$ (Nakayama's lemma) and $0$ for $i>e$.

Now let $A=k[x_1,\cdots,x_d]$, $R=A/I$. Here is the main:

Claim: $R\ \text{is}\ (S_n) \Longleftrightarrow \text{dim}(\text{Ext}_A^i(R,A))\leq d-n-i \ \forall i>d- \text{dim}(R)$

(of course, we only need to check for values of $i$ up to $d$, as $A$ has finite global dimension $d$).

Proof: By Lemma one needs to check that for all $p\in \text{Spec} \ A$: $$\text{Ext}_{A_p}^i(M_p,A_p)=0\ \text{for} \ i>\text{dim}(A_p) -\min\{n,\text{\dim}(R_p)\}=\max\{\text{dim}(A_p)-n,\text{dim}(A_p)-\text{dim}(R_p)\}$$

This condition is equivalent to the fact that for all $i>0$ and each $p$ in the support of $\text{Ext}_A^i(R,A)$ we must have $i\leq \max\{\text{dim}(A_p)-n,d-\text{dim}(R)\}$. Note if $i < d-\text{dim}(R)$, $\text{Ext}^i(R,A)=0$, so the claim follows.

You can compute both Ext and dimension with Macaulay 2.

$\endgroup$
  • $\begingroup$ I am unclear about several things: (1) What are $M$ and $S$ in the second part? (2) How do you pass between the Ext's of $A$-modules and $R$-modules? (3) Where does the dimension in the claim come from? $\endgroup$ – Victor Protsak Aug 28 '10 at 16:13
  • $\begingroup$ @Victor: In the second part, applying Lemma for $M=M_p$,$S=A_p$ and $n=\min\{n,ht(p)\}$. For (3), see my next to last sentence. $\endgroup$ – Hailong Dao Aug 28 '10 at 17:49
  • $\begingroup$ Oh, I see, I may messed up something about the dimensions. $\endgroup$ – Hailong Dao Aug 28 '10 at 18:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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