2
$\begingroup$

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.)

Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing down explicitly (I mean "analytically", not by computer). But on the other hand the ideal has a geometric/algebraic meaning. So that some naturally related ideals are simpler. Then, instead of computing $J$ we could at least bound/approximate it.

For example: $\sqrt{J}\supseteq\overline{J}\supseteq J$. (the integral closure in the middle) The radical $\sqrt{J}$ can sometimes be computed "set-theoretically", by going over the points of $Spec(R)$. While for $\overline{J}$ one can use the criterion of projections onto DVR's (initially by Teissier). And over DVR things are usually simpler. One can also try the saturation $J:\mathfrak{m}^\infty$

What are the other ideals naturally related to $J$ that are often "computable"? (I'm interested primarily in various determinantal ideals, Fitting ideals, annihilator-of-cokernels etc.)

ps. Of course, the same question holds for modules, but then it's more difficult

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

You can try to compute the reflexification / S2ification in normal ambient rings (and for some slight generalizations of normal rings). This just recovers the codimension 1 part of the ideal (the intersection of the height 1 primary components of the ideal).

For some ideals this coincides with the saturation at the maximal ideal.

Improve this answer
$\endgroup$
6
  • $\begingroup$ sorry for being ignorant. What are the simple ways to compute these objects? e.g. given some complicated enough matrix, considered as a morphism of free modules. I'd like to bound (from above) its annihilator-of-cokernel. Or suppose the ideal is defined as the annihilator of the quotient of two (complicated) modules of high rank. Are there some simple ways to compute reflexifications/S2-fications in these cases? $\endgroup$
    – Dmitry Kerner
    Commented Dec 14, 2014 at 20:18
  • $\begingroup$ Reflexification isn't so bad. It's just applying the functor $Hom_R(\bullet, R)$ to the ideal, twice. This can be quite quick in a computer (or at least, usually it isn't the thing that usually kills you in a computer). $\endgroup$
    – Karl Schwede
    Commented Dec 15, 2014 at 1:46
  • $\begingroup$ In my case this is not just one ideal, but an infinite series. Thus the computer is of no use. Are there some shortcuts to compute/estimate the reflexification? Say, maybe it's enough to consider only projections to some DVR's/PID's? Also, what is ("usually") bigger the reflexification or the integral closure? $\endgroup$
    – Dmitry Kerner
    Commented Dec 15, 2014 at 20:03
  • $\begingroup$ Reflexification will generally be bigger. I don't think you can see it via projecting to DVRs (unless you want an infinite collection of DVRs, those corresponding to height one primes of your ring). Can you say more about your particular collection of ideals. $\endgroup$
    – Karl Schwede
    Commented Dec 15, 2014 at 20:11
  • $\begingroup$ If I understand correctly, to understand the integral closure one also needs all the possible projections to DVR's. That's ok for me. My typical scenarios are: the annihilator-of-cokernel for a matrix (a map of free modules) $\endgroup$
    – Dmitry Kerner
    Commented Dec 18, 2014 at 15:20

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .