Talk:Flat module

Diagram suggested

From the german wikipedia I translated a diagram which helped me many times in my studies:

I consider adding it here (and as well on the free, projective and torsion-free module pages).
What do you think? Konrad (talk) 13:40, 5 April 2012 (UTC)

I'm a big fan of diagrams like this, so hopefully we can work something out. Firstly, I'm not sure how others feel, but the diagram does take up quite a bit of space. Can it be done more compactly with a vertical layout? Secondly, I think A should be avoided as a ring name in en.WP. I would recommend changing that in this way: "local rings and PIDs", "Dedekind rings". Lastly, while I know the reversed arrow in the middle is true, it might be more in theme with the picture to use a ring condtion instead of a module condition. That middle reversed arrow could be replaced with "Perfect rings". A second diagram could be developed with module conditions... I can't think off the top of my head if both reversed arrows can be done with module conditions :) I'll have to consult my notes... Rschwieb (talk) 14:33, 5 April 2012 (UTC)

I modified the diagram, so that no explicit name of the ring or module is mentioned. Right now I don't want to make another diagram with module-only properties (although it would be more "pure"). You're free to do it, but I think the current version would already be an improvement to no diagram at all. About the space-occupying-problem: You're right there, and I think the diagram shouldn't be on top of the page because of that. I experimented with a vertical layout but that didn't turn out to be satisfying. Konrad (talk) 14:51, 5 April 2012 (UTC)

"local" should definitely read "local ring" as there are so many uses for the word local. Did you consider changing the middle one to "perfect ring"? (You seem silent on that point so far.) Rschwieb (talk) 19:12, 5 April 2012 (UTC)

I had to think about that point a little bit, since I was unaware of the notion of perfect rings. Now I followed your suggestion. Konrad (talk) 12:16, 10 April 2012 (UTC)

I like it! Find a good spot and plant it in the article :) Rschwieb (talk) 13:58, 10 April 2012 (UTC)

Notherian local rings need not be perfect

"over a Noetherian local ring, flatness, freeness, and projectivity" are all equivalent.

I don't believe this. Q_p is a flat Z_p module, but not a free one.

I don't believe it either. I think the author must be thinking of finitely generated modules.

Dave Benson, Aberdeen. —Preceding unsigned comment added by 139.133.7.37 (talk) 22:36, 2 May 2009 (UTC)

Rings in which all flat modules are projective are called perfect rings. Commutative noetherian perfect rings are artinian (and perfect commutative integral domains are fields). I added "finitely generated" to the claim, but then it is true in much larger generality than noetherian local rings (well, even without finitely generated, there are non-noetherian perfect rings). I was not sure how to make the comment more interesting. JackSchmidt (talk) 23:41, 2 May 2009 (UTC)

There are no examples in the article.

There are no examples in the article. — Preceding unsigned comment added by 128.100.216.170 (talk) 23:07, 25 October 2012 (UTC)

You are right. I'll add one. D.Lazard (talk) 17:48, 26 October 2012 (UTC)

I guess that "the local rings of a commutative ring" refers to its localizations at prime ideals? And the R module in question is the product of all of those rings viewed as an R module? Rschwieb (talk) 20:10, 26 October 2012 (UTC)

You are right. If you find a better formulation, that is not too pedantic for the lead, it is OK. Otherwise, the examples should be moved to the body of the article and expanded. I am not willing to do that, because I am rather busy with Algebraic curve. D.Lazard (talk) 23:04, 26 October 2012 (UTC)

Removal of the newly added section "Examples"

I have removed the newly added section "Examples" for the following reasons:

• WP:TECHNICAL: This section is much too technical for its place in the article, as, in this place, examples must be accessible to every reader who is able to understand.
• WP:Verifiability: Every example must be verifiable either by providing a source or by linking to a Wikipedia article where the example is described as a flat module.
• WP:Original synthesis: There is no textbook presenting this content as examples of flat modules.
• The two first paragraphs are out of scope, as devoted to give technical examples of projective modules. They are indeed flat, but such examples are definitely not useful in an article about flat modules.
• Wrong assertion that a finitely generated module over a non-Noetherian integral domain is flat.
• The paragraphs from the third one are, at least, misplaced, as placed before the section about flat ring extensions.
• Talking of schemes here is somehow circular reasoning, as the definition of schemes is heavily based on the flatness of localization.

D.Lazard (talk) 07:53, 7 May 2021 (UTC)

---

I moved these edits further down the article, so many of your points above are addressed. Furthermore, are you asking for a citation that the module of Kahler differentials and their duals are projective modules? I can find a citation, but that should be part of the Kahler differentials article. Also, I have a citation from a paper by Cartier for the assertion about finitely generated flat modules being projective over an integral domain. Do you have a counter-example for the Non-Noetherian hypothesis? If you look at the remark after the cited lemma, he claims you only have to care about if the ring is integral or not, not if it is Noetherian.

Also, discussing schemes is not circular. You should absolutely be familiar with the algebraic and geometric interpretations of flatness, and without which, you will have a tough time fully understanding the material. Furthermore, Serre introduced flatness in GAGA which is fundamentally both algebraic and geometric. In fact, it's the initial technical movement which motivated general scheme theory as developed by Grothendieck. Kaptain-k-theory (talk) 14:55, 7 May 2021 (UTC)

You did not address the three first points, which are the major ones. I'll remove the section again, and I explain here the reasons with more details than above.

• First paragraph: It adds nothing to the assertion (appearing above in the article) that every projective module is flat. The explicit example of Kähler differentials is is too technical for being useful here, and I doubt that it is useful in Projective module. So this paragraph does not belong to this article.
• Second paragraph: Except for Cartier's lemma, this is not useful in this article, for the same reasons as for first paragraph. Moreover, this may be confusing for many readers who either do not know vector bundles or disagree with your concept of "geometric situation". The use of this phrase that is WP:Original synthesis.
• Cartier's lemma: I apologize: It true that, over an integral domain, every finitely generated flat module is projective. It would be worth to add it to section "Free or projective modules vs. flat modules". However, the result is not explicitely stated by Cartier. So, it must be said explicitly that the result follows from Cartier's lemma and the fact that a locally free module is projective. Finding a source where the result is explicitly stated would be a better solution.
• Discussion on flat morphisms: Flat morphism is linked in the preceding section. Explaining "the geometric intuition for flat morphisms of schemes" belongs to this article, not to Flat module. It could be worth to expand a little the mention of flat morphisms given in the preceding section, but this must be done without technical formulas, which are exactly the contrary of an explanation. Also for the flat example, you give a proof of flatness that requires an advanced knowledge, while it is almost immediate that the flat extension is in fact free with $\{1,x,y\}$ as a basis.

D.Lazard (talk) 10:28, 8 May 2021 (UTC)

---

• For your first point, there is not logical dependence on the definition of flat modules and the definition of Kahler modules. Without such a dependence I fail to see why one would be considered more advanced than the other. Kahler differentials are an elementary construction based on elementary questions (i.e. how can I transport differential calculus to commutative algebras). Furthermore, whenever (differential geometry) textbooks give examples of vector bundles, the typical first examples are the (co)tangent bundles and the various tensor powers thereof. I see no reason why this pattern cannot be followed here and improved upon, such as giving graded projective modules.
• I think you're taking the wording "geometric situation" too seriously. You could equivalently write "geometrically" and that's a commonly used word in the literature. I agree if you absolutely must avoid anything that resembles slightly new terminology you could rephrase the sentences, but that could be easily done without deleting all of these changes.
• You did not give a basis, only generators for the ring. I would certainly say going from a finite dimensional situation to an infinite dimensional situation adds a degree of complexity. I agree we could move some of this material over to that article and shorten up the section.

Finally, can you please just make edits for changes instead of just blindly reverting new material? This makes the wikipedia writing experience 10x more frustrating. Kaptain-k-theory (talk) 20:47, 8 May 2021 (UTC)

@Kaptain-k-theory: Welcome to Wikipedia! It is indeed frustrating to have one's work reverted. The solution to that is not to keep adding challenged material to the article -- that behavior is called edit warring, and if you continue, it is likely that you will be blocked from editing, which is even more frustrating. Instead, please discuss your proposed edits and try to come to a consensus before making additional edits to the article. Also, if you'd like other editors to weigh in, you can try WP:3O or the mathematics wikiproject. --JBL (talk) 11:16, 9 May 2021 (UTC)

Welcome to Wikipedia, Kaptain-k-theory! In my mind, Kaptain's edits don't amount to edit warring - their edits strike me as constructive attempts at expanding the article. Flatly reverting those edits is certainly not a particularly welcoming reaction to those edits.

That said, I do agree with D.Lazard's assessment of the quality of these edits:

• Kähler differentials depend on two rings (not one!), and are not in general projective. In fact this is part of a smoothness condition that is missing. I agree with D.Lazard that dwelling on these points is probably not helpful in this article: one would need to explain all this, just to get an example of a projective module, which is trivially then flat (as is mentioned above).
• The concrete example strikes me, again as D.Lazard, not particularly relevant / helpful here.
• I agree that it makes sense to elaborate on the relation between flat modules, flat ring extensions, and their geometric interpretation. But most of that geometric interpretation would belong to flat morphism, I think. Jakob.scholbach (talk) 13:06, 9 May 2021 (UTC)

Thanks for the messages Jakob.scholbach. I've had issues with this user before on other accounts and it seems like they aren't interested in making changes on pages which that consider "their's". Anything related to the pages they have worked on generally results in an instant rollback. (e.g. Zeros and poles, Hilbert polynomial, this page). It's very frustrating trying to improve Wikipedia when this user blatantly removes new material if it doesn't fit to some unwritten standard in their mind - highly frustrating. Furthermore, this user has gone to misinterpret material on my user-page and complain that it's somehow wrong. It's very unwelcoming to wikipedia when a pedant trolls through and puts up unnecessary obstructions for improving wikipedia. It would be more constructive to just go ahead and make specific edits, move material around, etc. instead of just flatly creating unnecessary barriers. Kaptain-k-theory (talk) 16:13, 10 May 2021 (UTC)

It is easy to get defensive about material you have worked hard on, especially when you believe it improves the article. But writing well is hard, and a good article has to balance many different concerns. The best writing on Wikipedia happens from collaboration between editors with a variety of perspectives. At times, that collaboration can be rough; nobody really enjoys that, but it's an occasional side effect of having multiple expert contributors.

I am of the opinion that Kähler differentials are not a good example here for reasons that have already been discussed above. I do think it would be nice to give some other examples, though. Perhaps an explicit example of a localization of a ring (e.g., $\mathbf {Z} [1/2]$ as a $\mathbf {Z}$-module, or $k[x,x^{-1}]$ as a $k[x]$-module). Maybe an étale ring extension. Oh, and how about $k(x)$ as a $k[x]$-module? That's a good one because it's flat but not free. Ozob (talk) 04:52, 11 May 2021 (UTC)

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Ozob, excellent I agree there should be more examples and non-examples. I'm still not convinced about the Kahler differentials but will compromise by including it in the projective modules page and letting people look their.

Let's move on to the most recent edit reversion. Here's the material in a block quote:

In the case of non-finitely generated flat modules over integral domains, this situation is typically studied through flat morphisms of algebras/schemes. Note that flat morphisms of schemes represent are an algebraic analogue for continuity since the fibers are deformations from each other. A typical example^{pg 3} of a flat morphism of algebras is given by the map

$\mathbb {C} [t]\to {\frac {\mathbb {C} [t,x,y]}{(xy-t)}}$ sending $t\mapsto t$

which when considered in the category of affine schemes, corresponds to a family of curves in $\mathbb {A} _{x,y}^{2}$ parameterized by $\mathbb {A} _{t}^{1}$.

I fail to see how this is original synthesis since this is material cited out of Artin's book on deformations. Furthermore, it gives context for why flat morphisms of schemes should be discussed other than the fact the definition of flat morphisms depends on the definition of flat modules.

Here's an additional citation which gives extra credence to what I've written above

• Algebraic Geometry I: Schemes - Ulrich Görtz, Torsten Wedhorn - page 428 at the beginning of the chapter the author essentially states what I have put into the quote. Combining this with Artin's book about deformations (and his remarks about flatness) there is nothing new here.

Kaptain-k-theory (talk) 18:32, 11 May 2021 (UTC)

References

I don't see how it could be that non-finitely generated flat modules over integral domains are "typically studied through flat morphisms". How would you study an infinitely generated free module this way?

I disagree with the assertion that flatness is an "algebraic analogue for continuity". It's true that the fibers of a flat morphism have various continuity properties. But this is quite a lot stronger than what we normally think of as continuity. Consider, for example: At a critical value of a smooth function between two smooth manifolds, the fiber can have the wrong dimension; but a flat morphism has constant relative dimension (under suitable finiteness hypotheses).

Finally, while the example is correct in the sense that it really is a flat morphism, I think that explaining its significance requires a lot more explanation than is in the paragraph. As it stands, it makes a true assertion but doesn't provide the reader enough context to understand its significance. Ozob (talk) 00:53, 12 May 2021 (UTC)

By the way, that flat family example already appears in the "Flat ring extensions" section, although in a somehow abstract formulation. (It's not only a good geometric example but it is a nice way to understand the difference between flat and free.) I will add a mention of the explicit case. (Incidentally, I don't know if "Noetherian" is needed; it's in the reference, but I suspect that assumption might be superficial.) To Kaptain-k-theory, the best way not to get your edit reverted is to slowly expand or modify the existing materials. Slowness and incrementalism are important in the content development in Wikipedia. -- Taku (talk) 01:22, 12 May 2021 (UTC)

On related notes, we probably need to add mentions of two things: (1) in the projective-scheme (not module!) setup, flatness is equivalent to constancy of Hilbert polynomial (different fibers share the same Hilbert polynomial) and (2) flatness by blow-up technique (it has a way cooler French name that I can't recall). -- Taku (talk) 01:50, 12 May 2021 (UTC)

You're probably thinking of "platification par éclatements", as in Gruson–Raynaud's "Critères de platitude et de projectivité." Their theorem 5.2.2 says, "Let S be a quasi-compact and quasi-separated scheme, U a quasi-compact open subscheme of S, f : X → S a morphism of finite presentation, M an O_X-module of finite type, and n an integer. Suppose that $M|_{f^{-1}(U)}$ is U-flat in dimension ≥ n. Then there exists a U-admissible blowup g : S' → S such that the strict transform of M by g is S'-flat in dimension ≥ n." (A blowup is U-admissible if it's the blowup along a finitely presented closed subscheme that's disjoint from U.) Ozob (talk) 03:16, 13 May 2021 (UTC)

---

Ozob, for the second point you're right, it should be rephrased to continuously varying family of varieties, which are like fiber bundles which allow for degenerations. This would be a good place to include a couple examples, such as a family of curves (like a family of hyperelliptic curves) and a family of elliptic curves over ${\text{Spec}}(\mathbb {Z} )$ which have degenerations at certain primes.

• By flat morphisms being a technique for studying non-finitely generated flat modules, I mean this because most examples studied, (unless they're interesting counter-examples/edge cases) come from flat morphisms with non-zero dimensional fibers.
• How could I improve it? What context is missing? Should I show what the fibers are over the degenerate point and the smooth points on the base?

Kaptain-k-theory (talk) 16:21, 12 May 2021 (UTC)

I don't think it's accurate to say that most examples arise that way.

• A standard construction of a projective resolution of a module M is to take the free module F generated by the set of elements of M; then the free module generated by the set of elements of F, etc. This is even a useful construction because it gives you functorial projective resolutions. The terms of this resolution will be flat but often non-finitely generated.
• If $R$ is a Noetherian local ring, then ${\hat {R}}$ is flat over $R$ but rarely finitely generated. This can have zero dimensional fibers, for example, $k[x]_{(x)}\to k[[x]]$.
• Torsion-free modules over a valuation ring are flat. So, for example, if R is a valuation ring, then its fraction field is flat over R. E.g., $k((t))$ is flat over $k[[t]]$. There are plenty of more exotic examples involving valuations of rank 2 or more.

Regarding context, one can write down plenty of random-looking equations, and some will be flat and some won't. Unless you explain some of the features of the example, like the fact that the fibers have constant dimension, then the reader gains no insight. So my advice would be to describe some interesting features of the family; but that level of detail might be better suited for the flat morphism article. Ozob (talk) 04:03, 13 May 2021 (UTC)

Taku Thanks for the input. I agree flatness being equivalent to constancy to the Hilbert polynomial for projective morphisms should be included. This is a very useful result and motivates Hilbert schemes! Kaptain-k-theory (talk) 16:21, 12 May 2021 (UTC)

Flat ring extensions

The section "Flat ring extension" includes now the material that was discussed in the preceding thread. So, it is time to open a new thread.

I have fixed two issues: firstly writing a quotient ring as a fraction is uncommon, and I have fixed this. Secondly, I have removed a sentence asserting that the example of a flat morphism is not free. This is wrong, as it has as a basis the monomials in x and y that are not multiple of xy.

The section has other issues, but, as some are related to the whole structure of the article, I'll discuss them later. Examples of issues related to the whole structure are the too short discussion on localizations as flat ring extensions, and the fact that the discussion on flat morphisms is done before mentioning that flatness is a local property. D.Lazard (talk) 08:51, 12 May 2021 (UTC)

That sentence does not refer to that explicit example; I have reworded it for clarification. For a finite module over a PID, flatness and freeness are equivalent. So one probably needs to use a more complicated ring as a base (to construct a non-free but flat example); but I didn’t think we need to actually do that except saying one can. Also, I don’t have an opinion on the structure. —- Taku (talk) 09:23, 12 May 2021 (UTC)

(To all, please also note a reply I made in the previous thread). —- Taku (talk) 09:29, 12 May 2021 (UTC)

For the basis, maybe there was a typo earlier? If we set $R=\mathbb {C} [x,y,t]/(xy-t)$ then as a $\mathbb {C} [t]$ module $R$ is infinite dimensional. It has the $\mathbb {C} [t]$ basis $1,x,y,x^{2},y^{2},x^{3},y^{3},\ldots$. Kaptain-k-theory (talk) 16:21, 12 May 2021 (UTC)

Article structure

Since there is nicely a lot of activity on this article, I thought I add my 2 cents via talk. I think it is great that the article contains more and more material, but a reordering of the sections seems in order. My suggestion would be this:

• Flatness and faithful flatness of module.
  • Definition, basic (immediate) examples
  • Relation to other notions (free, projective, etc.)
  • Local aspects
  • Flatness of completion
• Flat resolutions
  • Definitions
  • relation of flatness to Tor
  • flat covers
• Flatness and faithful flatness of a ring
  • Faithfully flat descent
  • Brief discussion of flat morphisms of schemes. Most of this material should go to that separate article.

It would also be nice if the article would get more references for the individual claims. It would also be ideal not to make an assumption about the rings being commutative unless this is really necessary. Jakob.scholbach (talk) 12:40, 17 May 2021 (UTC)

I mostly agree with your comments. However, reordering sections is difficult to do before rewriting section for having an encyclopedic tone and focusing on the main properties. It is this task on which I am working. I proceed section by section, and, when some material must be moved in a section that is not yet written rewritten, I enclose it in a subsection that must be moved later. This is the case, for example for "Flat morphism of schemes". I intend to move it as a subsection of section "Localization" that will contain the definition and the fact that the local property of flatness implies that a ring homomorphism is flat if and only if the associated morphism of affine schemes is flat.

About Tor and flat resolutions: I intend to create a section "Functor Tor", since the current section is not appropriate (it supposes that the reader knows already Tor, derived fuctors, ..., which certainly not the case for a reader who learn flatness from this article). The place of the section will probably be soon after the section on localization. IMO, the section on flat resolutions and flat covers must be placed much nearer the end of the article, because, as far as I know, this is not commonly used in commutative algebra. However, this must be discussed later, when the content of the sections will be clearer. D.Lazard (talk) 14:50, 17 May 2021 (UTC)

"Faithfully flat" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Faithfully flat and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 June 5#Faithfully flat until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 15:44, 5 June 2022 (UTC)

Still no examples in the article

Apparently there were some examples in the past, but they were removed. And there were many detailed discussions among experts on this talk page. It would be great if one of those experts could provide just a few very basic examples of flatness for non-experts (at the moment, there are only non-examples, and examples for faithfully flat, but not for flat; these examples should come early in the article, after the definition or alternative characterization). DG-on-WP (talk) 20:01, 16 May 2023 (UTC)

There is no section "Examples", but there are many examples in the article: in § Relations to other module properties, it is said that all free modules, all projective modules, and all torsion free modules over a principal ideal domain or a Dedekind domain are examples of flat modules. Sections § Direct sums, limits and products and § Flat ring extensions provide other examples. However, I agree that some of these examples should appear before the formal definition. I'll rewrite the introduction for fixing this. D.Lazard (talk) 10:24, 17 May 2023 (UTC)