Timeline for Rank of a module
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 4, 2011 at 12:20 | vote | accept | ashpool | ||
| Jul 1, 2010 at 1:31 | comment | added | Graham Leuschke | Yes, kwan, that's true. Projectivity is a very strong restriction. | |
| Jun 30, 2010 at 23:07 | comment | added | ashpool | Graham, if the ring is local or is an integral domain, extending scalars to the field of fraction and extending scalars to the residue field actually gives the same dimension if the module is finitely generated and projective. This is because in both cases Spec of the ring is connected. | |
| Jun 30, 2010 at 17:25 | comment | added | ashpool | I guess I used the term quotient field not as a standard terminology but as an adjective followed by a noun: $A/\mathfrak{m}$ is a field which is a quotient. I admit it is very nonstandard but I tend to associate the word quotient with "modding out" rather than "fraction." | |
| Jun 30, 2010 at 16:19 | comment | added | Keenan Kidwell | In my experience, "quotient field" and "field of fractions" are used interchangeably to mean the localization of a domain R at the set R0 . I have never seen either term used to mean a ring modulo a maximal ideal. I would say your use of "quotient field" is non-standard. The relationship is: if $P\in Spec(R)$, then the quotient field of the domain $R/P$ is the residue field of the local ring $R_P$. At any rate, as I said above, if you want to talk about the rank of a module $M$ over a domain $R$, you want the dimension $M\otimes_RK$, where $K$ is the quotient field of $R$. | |
| Jun 30, 2010 at 13:45 | comment | added | ashpool | @Keenan Kidwell Sorry I should have clarified what I meant by quotient field. I just meant the ring modded out by the maximal ideal. Now I recall some authors call field of fractions quotient field. | |
| Jun 30, 2010 at 13:41 | comment | added | ashpool | @Graham I think we are having some disagreement in definitions. Your definition of quotient field is what I call field of fractions. My definition of quotient field is your definition of residue field. My definition of residue field (of a prime ideal) is the field of fraction of the ring modded out by the prime ideal. So in my definition, residue field coincides with quotient field for maximal ideals. | |
| Jun 30, 2010 at 13:09 | comment | added | Graham Leuschke | No, kwan, that's not true. The quotient field is obtained by inverting all non-zero elements: for the localized polynomial ring k[x]_(x), it is the rational function field k(x). For example, it contains 1/x and 1/(1+x^2). The residue field is obtained by killing the maximal ideal: for k[x]_(x), it is just k. It contains only scalars, and is annihilated by x. | |
| Jun 30, 2010 at 13:08 | comment | added | Keenan Kidwell | @kwan This is very much false. Consider $\mathbb{Z}_p$, the ring of $p$-adic integers. This is a discrete valuation ring. It's residue field is $\mathbb{F}_p$, the finite field of $p$ elements, but its quotient field is $\mathbb{Q}_p$. Note that, in this case, the residue field and quotient fields do not even have the same characteristic. The definition of rank you're thinking of is for modules over a domain, and it is the dimension of the extension of scalars to the quotient field of the domain. | |
| Jun 30, 2010 at 10:53 | comment | added | Graham Leuschke | Don't extend scalars to the residue field; extend them to the quotient field. | |
| Jun 30, 2010 at 9:05 | answer | added | Georges Elencwajg | timeline score: 43 | |
| Jun 30, 2010 at 8:51 | comment | added | Sasha | The minimal number of generators is not a good invariant. For example, it is not additive in exact sequences. E.g. let the ring be Z, and consider the short exact sequence $0 \to Z \to Z \to Z/nZ \to 0$ with $n \ne 0$. | |
| Jun 30, 2010 at 2:23 | comment | added | ashpool | If the ring is local, the minimal number of generators of a finitely generated module coincides with the rank (dimension of the vector space obtained by extending scalars to the residue field) by Nakayama's lemma. | |
| Jun 30, 2010 at 1:41 | comment | added | Graham Leuschke | That's not the definition of rank in any book I know, it's the definition of minimal number of generators. Minimal number of generators is only defined locally because it's only well-defined locally -- you'd like every list of generators such that no element can be deleted to have the same length, and that fails (badly) for non-local rings. Rank is defined over domains (or, more complicatedly, as an n-tuple over for generically free modules over arbitrary rings) by relating it to vector-space dimension of a localization. | |
| Jun 30, 2010 at 1:06 | history | asked | ashpool | CC BY-SA 2.5 |