# Graham Leuschke

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## Registered User

 Name Graham Leuschke Member for 3 years Seen 15 hours ago Website Location Syracuse Age
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 15h comment Canonical ModulesThat is a hypersurface ring, so Gorenstein, so the canonical module is the ring itself. May14 comment ideals of a noetherian ring $R$ Cohen-Macaulay as $R-$modulesThe length of a maximal $R$-sequence in $P$ is usually denoted $\mathrm{depth}_P(R)$, while $\mathrm{depth}_R(P)$ would mean the length of a maximal $P$-sequence in $R$. Also, it's not true in general that $\mathrm{depth}_R(M) = \mathrm{depth}_R(\mathrm{Ann}_R(M))$; for example, if $R$ is a domain then $\mathrm{depth}_R(R) \geq 1$, while $\mathrm{depth}_R(\mathrm{Ann}_R(R))=0$. May13 comment ideals of a noetherian ring $R$ Cohen-Macaulay as $R-$modulesI think it's a cut-n-paste error, perhaps supposed to be $\mathrm{depth}_R(P) = \mathrm{dim}(R)$. In that case, $P$ must have height (at most) $1$ and $R/P$ must be a CM ring. May13 comment An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.More generally, any reduced Gorenstein local ring of dimension one will satisfy this double-annihilator property. May13 comment An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.You can localize at the ideal $(x,y)$ to get a local example. May8 accepted Height unmixed ideal May7 answered Height unmixed ideal May7 comment When does “second annihilator” of a (principal) ideal equal the ideal itself , ie $Ann_R(Ann_R(r))=Rr$?$R=\Bbb {Z}/n\Bbb {Z}$ is itself an artinian Gorenstein ring. Apr25 comment Central Element in Sklyanin Algebras?If translation by the point $\tau$ of the elliptic curve $E$ has infinite order, then the center of $A$ is $k[g]$. That's the only intrinsic description of $g$ I know. There might be more known about the two central elements in the Sklyanin algebra of GKdim 4. Paul Smith has some notes that might be useful there: math.washington.edu/~smith/Research/… Apr22 awarded ● Popular Question Apr21 awarded ● Good Question Apr9 comment $k[[x]]$ as a $(k[[x]])^p$ module for ugly fieldsOops, it's not. I should have said 'flat', which Karl already noted in the question. Apr9 comment $k[[x]]$ as a $(k[[x]])^p$ module for ugly fieldsI guess it's a direct limit of free modules, so at least projective. Mar31 comment Notation Problem, Fixed Rings and FieldsLinks for the lazy: citeseerx.ist.psu.edu/viewdoc/… and www-apr.lip6.fr/~avb/DonneesTelechargeables/…, respectively. Mar11 answered When does End(M) consist entirely of zero, zero divisors, and units? Mar8 revised On the equation defining a surfacedisplay Mar2 comment Detecting and counting free direct summandsEntirely possible I'm missing something silly. Mar2 asked Detecting and counting free direct summands Feb20 revised Question in the paper of Robert Bryant “Calibrated embeddings in the special Lagrangian and coassociative cases”backticks Feb18 comment How to check if a commutative ring is Gorenstein.Dear Sándor, of course I was not offended. You're right -- our answers complement each other. In fact, a full list of all the ways to answer this question might involve a pretty broad cross-section of commutative algebra and algebraic geometry. Feb18 comment How to check if a commutative ring is Gorenstein.I also wish Sándor [and all people 'like' and 'unlike' him] nothing but long life and happiness, but I do want to say that while my first answer is admittedly pretty black-boxy, the second one and Youngsu's comment following are both easy and instructive to do by hand, and that it's only by doing many such examples that we can get the kind of algebraic intuition that serves us [i.e. me] when we [i.e. I] meet examples that are not so susceptible to geometry. Feb18 comment How to check if a commutative ring is Gorenstein.Yes, absolutely. Feb18 accepted How to check if a commutative ring is Gorenstein. Feb18 revised How to check if a commutative ring is Gorenstein.added alt. method Feb18 revised How to check if a commutative ring is Gorenstein.added 49 characters in body Feb18 answered How to check if a commutative ring is Gorenstein. Feb3 comment Jacobian ideals referenceA student of Huneke's named Hsin-Ju Wang wrote some papers about the Jacobian, and always cited Lipman-Sathaye for the definition and basic properties. Here's one of Wang's papers: tandfonline.com/doi/abs/10.1080/00927879808826222 Feb2 revised When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?backticks again Dec4 revised Question on localization techniqueadded 8 characters in body Nov30 comment Mutually tangent ellipsoids in 3 spaceI suspect the asker wanted the interiors of the ellipsoids to be disjoint.