Talk:Length of a module

Section "Zero and poles of an analytic function"

I have tagged this section with {{cleanup}}, because it requires a major edit, and this template allows a cleared display of the reasons of the tag than other related templates. Nevertheless the displayed reason of the tag must be rather short, and the result is that it has been misinterpreted, see User talk:D.Lazard#Length of a module. So, I give here the detail of the issues of the section.

The section has multiple issues.

• WP:TECHNICAL: The natural audience of this section is people that try to understand the subject of this article and have a minimal knowledge of the title of the section. Thus, one cannot suppose that the reader knows Weierstrass factorization theorem, which is generally taught a long time after the definition of the order of a zero or a pole. So, considering this theorem as soon as the second sentence makes the section too technical for most readers. The use, later in the section, of advanced tools of algebraic geometry is also much too technical.
• WP:Original synthesis: The use of Weierstrass factorization theorem for defining the order of a zero or a pole is certainly possible but seems to be an original synthesis, as, generally, this is the opposite that is done, that is using orders of zero and poles for proving Weierstrass theorem.
• WP:Scope: The scope of the section should clearly be to give an example of use of the subject of the article outside pure abstract algebra. This is not the case here, as, once the confusing technicalities are removed, all examples reduce to say that the order of a pole or a zero is the valuation of a polynomial or a series in some discrete valuation ring; and to say that this valuation is the length of the quotient of the discrete valuation ring by the corresponding principal ideal. This is clearly not useful for this article, as no example is given to show that it is advantageous to use the quotient by a principal element rather than the valuation of this element.
• Confusing: It is possible that something relevant for this article is hidden behind this confusing presentation, but I am unable to imagine what it is.

So, my opinion is to delete this section. But, as the discussion has been opened, I'll given time for the discussion before deleting it. Nevertheless I'll retag the section with more appropriate tags. D.Lazard (talk) 13:39, 27 May 2020 (UTC)

For (1), where else should this content go? On a page for intersection multiplicity? That's fine, but no article has been written yet, so I think including this material on this article is should be the default until such a page is made. Moreover, it's not like this is on the first part of the page, so a beginner could gloss over this or come back later. Also, lot's of references are given with the basic idea for the surrounding ideas, so it's conceivable a beginner could get something out of this section. In (2) I really don't think this is an original synthesis because it is implied and discussed briefly in these two sets of notes: https://faculty.etsu.edu/gardnerr/5510/notes/VII-5.pdf and https://faculty.math.illinois.edu/~r-ash/CV/CV6.pdf . For (3) this is more of an opinion, and the definition I gave is the one written by Fulton in his book on intersection theory. Also, this is used in the Serre Intersection Formula. For (4), I'm not sure how to help you with this, do you have any specific questions, or maybe gaps you need to fill? I'm more than happy to help fill that, and include links in the article, but I need specific complaints. Also, I will add an explicit example soon showing this extension. The idea is basically the same, given a hypersurface defined by the vanishing locus of a polynomial h, the order just gives the difference of the degree of their power in the numerator and denominator of some fraction of polynomials from the ring of the variety. Wundzer (talk) 19:10, 27 May 2020 (UTC)

It seems that you make a confusion (at least in your phrasing) between a new definition, and the proof that a older definition is a special case of a more general definition. It is this confusion that motivated my tags {{original synthesis}} and {{confusing}}. Reading the article again, it appeared that these two issues could be, and have been resolved simply by changing the first sentence of your contribution, and restructuring (changing headings, and section levels).

Nevertheless, several problem remains. Firstly, your contribution suggests that a multiplicity is always equal to the multiplicity of a root of a polynomial or to the order of a power series. This is wrong and may mislead a nonexpert reader. IMO, this problem cannot be solved independently of the second one.

The second issue is the problem of scope. You wrote "I think including this material on this article is should be the default until such a page is made". The relevant page exists. It is Multiplicity (mathematics), where the example of the order of a zero or pole is already given. A link to Length of a module could be added there. I suggest to move your contribution there after having made it less technical by using advanced terminology only when it cannot be avoided. D.Lazard (talk) 10:57, 28 May 2020 (UTC)

I'm still not sure there exist meromorphic functions whose multiplicity of zeros or poles does not come from a factorization of polynomials. Can you elaborate on this further? Also, I think this presentation of the theorem makes it clearer why I am picking out the multiplicity of polynomials.

I can move the material on intersection multiplicity over to the Intersection multiplicity page. In order to make this material clearer, I can explain the reasoning behind the codimension 1 hypothesis and the reason why this formula is for varieties and not schemes. Also, I can give the intuition for why local rings should be considered in the first place in this context. I'm not sure what other technical points make it difficult for a beginner other than the ones I listed above. Do you have any suggestions? Wundzer (talk) —Preceding undated comment added 17:20, 28 May 2020 (UTC)

I wonder which kind of intuition you can give for explaining why local ring should be considered. The intuition is that a multiplicity is a local property, and local ring have been introduced for eliminating everything that is not local to the studied point. This is the reason for which I disagree with the intrroduction of Weierstrass factorization theorem in these articles: Weierstrass theorem is a global theorem about all zeros and poles together. It is thus confusing to consider it in articles about local properties. By the way, you seem to use the term "intersection multiplicity" in a context where there is no intersection, or, at least, you use it without making explicit the involved intersection. This is another way to confuse the reader. So, please, make an effort for making your contribution clear and comprehensible for people who have not the same background as you. D.Lazard (talk) 20:17, 28 May 2020 (UTC)