Comments on: Localisation and ring depth http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/ Because my LiveJournal is too silly Mon, 15 Mar 2010 18:50:59 +0100 http://wordpress.org/?v=2.8.6 hourly 1 By: Craig http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1350 Craig Thu, 14 Sep 2006 15:17:44 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1350 Sorry, I got sidetracked and forgot to keep up with the comments... I must confess that I didn't actually read the maths in too much detail as I was having a bad-math-week and I couldn't bring myself to do any extra! Sorry, I got sidetracked and forgot to keep up with the comments… I must confess that I didn’t actually read the maths in too much detail as I was having a bad-math-week and I couldn’t bring myself to do any extra!

]]> By: Michi http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1276 Michi Wed, 06 Sep 2006 13:53:30 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1276 Well, they are kinda. A graded ring is - in the cases I'm interested in - always a graded algebra over some field. Thus, the regular sequences exhibiting lower bounds on depth are all parts of homogenous systems of parameters, but it may happen that some of the parameters occuring end up being zero-divisors, so it may happen that not all of them occur. The "usual" definitions tend to be for the depth of a module over a specific ring, with the more or less tacit understanding that the ring as such has depth given as the depth of the ring viewed as the obvious module over itself. Well, they are kinda. A graded ring is – in the cases I’m interested in – always a graded algebra over some field. Thus, the regular sequences exhibiting lower bounds on depth are all parts of homogenous systems of parameters, but it may happen that some of the parameters occuring end up being zero-divisors, so it may happen that not all of them occur.

The “usual” definitions tend to be for the depth of a module over a specific ring, with the more or less tacit understanding that the ring as such has depth given as the depth of the ring viewed as the obvious module over itself.

]]> By: folk angel http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1275 folk angel Wed, 06 Sep 2006 13:40:37 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1275 have failed to find a definition of depth of graded ring on internut, only for algebras: http://planetmath.org/encyclopedia/HomogeneousSystemOfParameters.html or modules: http://eom.springer.de/d/d031180.htm but my brain is too atrophied these days to work out if they're relevant. have failed to find a definition of depth of graded ring on internut, only for algebras:

http://planetmath.org/encyclopedia/HomogeneousSystemOfParameters.html

or modules:

http://eom.springer.de/d/d031180.htm

but my brain is too atrophied these days to work out if they’re relevant.

]]> By: Michi http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1274 Michi Wed, 06 Sep 2006 13:19:27 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1274 (administrative note: Since it doesn't accurately reflect reading times anyway, I have now removed that particular bit) (administrative note: Since it doesn’t accurately reflect reading times anyway, I have now removed that particular bit)

]]> By: Michi http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1273 Michi Wed, 06 Sep 2006 13:18:57 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1273 Atiyah-MacDonald doesn't treat depth at all (I've been staring at it and swearing while working with this), my Eisenbud is in Jena - some 8 hours by train from here, and the local maths library doesn't carry anything that helped me. The definition is basically what I boiled down from how I think about it, and how I remember it being defined. I am certain of the formulation in terms of length of longest sequence of sequential non-zero divisors, but not certain as to what extent my formulation captured all possible subtleties. There is another formulation of it in terms of vanishing of Ext-modules - basically, the depth (when finite) is the degree of the first non-vanishing Ext. Had I had a decent reference while writing this, my uncertainty would have been nowhere near the currently exhibited levels. Atiyah-MacDonald doesn’t treat depth at all (I’ve been staring at it and swearing while working with this), my Eisenbud is in Jena – some 8 hours by train from here, and the local maths library doesn’t carry anything that helped me.

The definition is basically what I boiled down from how I think about it, and how I remember it being defined. I am certain of the formulation in terms of length of longest sequence of sequential non-zero divisors, but not certain as to what extent my formulation captured all possible subtleties.

There is another formulation of it in terms of vanishing of Ext-modules – basically, the depth (when finite) is the degree of the first non-vanishing Ext. Had I had a decent reference while writing this, my uncertainty would have been nowhere near the currently exhibited levels.

]]> By: folk angel http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1272 folk angel Wed, 06 Sep 2006 13:04:41 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1272 Well, it's not much use to you, but it took me 10 mins to get my head around your definition of the depth of a (graded) commutative ring* - it sounds plausible enough but as your uncertain as to its accuracy where did you get it from? (presumably not Atiyah-MacDonald - and I can't find it to verify in my limited remaining supply of scary yellow maths books (Eisenbud being the most relevant looking) or by my usual cheat of looking it up on mathworld). Thanks for waking my brain up though and good luck with the Phud. *then it is 3 years since I did any maths and whilst I did used to know some things about commutative rings, I don't recall depth being one of them. Well, it’s not much use to you, but it took me 10 mins to get my head around your definition of the depth of a (graded) commutative ring* – it sounds plausible enough but as your uncertain as to its accuracy where did you get it from? (presumably not Atiyah-MacDonald – and I can’t find it to verify in my limited remaining supply of scary yellow maths books (Eisenbud being the most relevant looking) or by my usual cheat of looking it up on mathworld).

Thanks for waking my brain up though and good luck with the Phud.

*then it is 3 years since I did any maths and whilst I did used to know some things about commutative rings, I don’t recall depth being one of them.

]]> By: Michi http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1271 Michi Wed, 06 Sep 2006 08:34:34 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1271 Nah, I'd rather just find out how to turn the thing off instead then. It's an artifact of the preview plugin I'm using. How much time DID you use to grok what I was writing? Nah, I’d rather just find out how to turn the thing off instead then. It’s an artifact of the preview plugin I’m using.

How much time DID you use to grok what I was writing?

]]> By: Craig http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/comment-page-1/#comment-1270 Craig Wed, 06 Sep 2006 07:31:19 +0000 http://blog.mikael.johanssons.org/archive/2006/09/localisation-and-ring-depth/#comment-1270 Hi Michi, does the "estimated 2.12 minutes" reading time when you click to continue reading the rest of your post take into account the time needed to understand the maths therein? :) Perhaps readers should register and complete a short test to assess their mathematical knowledge and then you could calculate more accurately... Hi Michi, does the “estimated 2.12 minutes” reading time when you click to continue reading the rest of your post take into account the time needed to understand the maths therein? :)

Perhaps readers should register and complete a short test to assess their mathematical knowledge and then you could calculate more accurately…

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