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In Eisenbud and Atiyah-Macdonald, a composition series of an $R$-module $M$ is defined to be a chain of submodules so that no extra submodules can be inserted in the chain. Eisenbud goes on to say that this is equivalent to saying that successive quotients in the chain are isomorphic to $R/\mathfrak{m}$ for some maximal ideal $\mathfrak{m}\subseteq R$. This is corroborated with what Wikipedia says: every simple $R$-module is of the form $R/\mathfrak{m}$ for a maximal ideal $\mathfrak{m}\subseteq R$.

However, in Gathmann's notes (page 166), it is said that a composition series (when $R$ is Noetherian and $M$ is finitely-generated) is a chain of submodules of $M$ such that successive quotients are of the form $R/\mathfrak{p}$ for prime ideals $\mathfrak{p}\subseteq R$. Gathmann goes on to say that the composition series is not unique, but for any prime $\mathfrak{p}\subseteq R$, the number of times $\mathfrak{p}$ appears in the series is invariant.

Why is there a discrepancy between Gathmann's definition and the standard definition? And is there a reference for Gathmann's claim that given his definition of composition series, the number of times a prime ideal appears in the series is invariant?

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I have typically seen other authors avoid the words "composition series" when referring to the concept that Gathmann is discussing (the only exception here is Bourbaki's commutative algebra). In my experienc, this is most commonly presented is as follows:

Suppose $M$ is a finitely generated module over a noetherian ring $R$. Then there is a finite filtration $$0= M_0 \subset M_1\subset\cdots\subset M_n=M$$ such that $M_{i+1}/M_i\cong R/\mathfrak{p}_i$ for some prime ideal $\mathfrak{p}_i\subset R$.

The invariance of the number of times that some specific $\mathfrak{p}$ appears is usually only claimed for minimal $\mathfrak{p}$, and that's not so hard to show - localize at $\mathfrak{p}$ and take the length, which works because any step in the filtration where $\mathfrak{p}\neq\mathfrak{p}_i$ will give $(M_{i+1}/M_i)_\mathfrak{p}=0$, and $R_\mathfrak{p}/\mathfrak{p}_\mathfrak{p}$ is a field.

For instance, this is how Vakil treats it in exercise 6.5.J in the August 29, 2022 version of his notes, and Hartshorne does a graded version of this as proposition I.7.4. For a reference for the commutative algebra underlying this result, see Bourbaki's Commutative Algebra, chapter IV, section 1, number 4, or Matsumura's Commutative Algebra, page 51.

As to why there's a discrepancy in the definitions, they're just used for different things. Finite length modules are nice, but most modules we see in algebraic geometry aren't of finite length, and we still want to have something good to say about them like the finite length case. One reason finite length modules aren't quite sufficient to do geometry with is that a module being finite length would mean that its associated sheaf is supported at only finitely many points on $\operatorname{Spec} R$, which is not quite permissive enough for us - we want to be able to think about vector bundles, for instance, which have to be supported everywhere unless they're trivial.

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    $\begingroup$ Just a note: Prime filtration is a more general definition in this context, as a prime filtration of a module of finite length is exactly a composition series. $\endgroup$ May 3, 2023 at 4:53
  • $\begingroup$ Finite length modules are also exactly the "irrelevant" ones in projective geometry $\endgroup$
    – math54321
    May 3, 2023 at 15:46
  • $\begingroup$ A small followup question: by localizing the filtration at $\mathfrak{p}$, one can show actually the number of times any prime $\mathfrak{p}$ (not necessarily minimal) appears in the filtration is invariant, correct? $\endgroup$ May 5, 2023 at 3:41
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    $\begingroup$ @AndrewPaul It requires a bit more work and I haven't checked myself in a while, but I'm pretty sure that's true, yes. $\endgroup$
    – KReiser
    May 5, 2023 at 5:50

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