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Dmitry Kerner
Dmitry Kerner
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(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

(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.  )

(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

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Dmitry Kerner
Dmitry Kerner
  • 2.2k
  • 13
  • 19

what are the possible approximations for ideals

(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. )