In the book "Cyclotomic Fields and Zeta Values", we see a claim made (page 58, first paragraph) as follows:

If $N$ is any finitely generated torsion $\Lambda(G)$-module, the structure theory shows that we have an exact sequence of $\Lambda(G)$-modules $$0 \longrightarrow \bigoplus\limits_{i=1}^{r} \frac{\Lambda(G)}{\Lambda(G)f_i} \longrightarrow N \longrightarrow Q \longrightarrow 0,$$ where $f_i \,(i=1,\ldots,r)$ is a non-zero divisor, and $Q$ is finite.

What I know about fin. gen. torsion modules over $\Lambda(G)$-modules, is that they are pseudo-isomorphic to $\bigoplus\limits_{i=1}^{r} \frac{\Lambda(G)}{\Lambda(G)f_i}$. So, why is the kernel missing in the above quoted paragraph?

Edit: $\Lambda(G)$ is the inverse limit of $\mathbb Z_p[G/H]$ as $H$ varies over open subgroups of $G$ where $G$ is $Gal(\mathbb Q(\mu_{p^\infty})^+/\mathbb Q)$, which is isomorphic to $R[[T]]$ where $R$ is is the group ring $\mathbb Z_p[\mu_{p-1}]$.

  • $\begingroup$ Does $R[[T]]$ appear somewhere in your question? $\endgroup$ – user26857 Sep 20 '17 at 20:51
  • $\begingroup$ Sorry, now edited the question. $\endgroup$ – Sameer Kulkarni Sep 21 '17 at 5:36

"Pseudo-isomorphic " is a rather "all purpose" word, with no precision about neither the map $M\to N$ nor the torsion modules $M$ and $N$ modules themselves. In your specific case, a module such as the first term in your exact sequence (denote it $E$) is usually called "elementary". The injectivity of $f: E\to N$ comes implicitly from the fact that such an elementary $E$ has no non trivial finite submodule $(*) $. Pseudo-isomorphism between two torsion modules is a symmetric binary relation, but if you switch the roles of $E$ and $N$, i.e. if you write a pseudo-isomorphism $g:N \to E$, then $\ker g$ is the maximal finite submodule of $N$, which could be non trivial.

In the appendix of your book, the structure theorem is proved by referring to Bourbaki's "Commutative Algebra". But as usual Bourbaki deals with the most general case, in which the base ring is a noetherian Krull domain $A$. "Finiteness" is then a particular case of "pseudo-nullity" : a torsion $A$-module is pseudo-null if its annihilator ideal has height $\ge 2$. To show the generalized property $(*)$, use the valuation on the localized ring $A_P$ at a height $1$ prime ideal $P$ to check that the annihilator of a non zero element of $R/P^{e}$ is contained in $P$. In the particular case of the Iwasawa algebra here, a direct proof of $(*)$ is possible, starting from the Weierstrass Preparation Theorem as in Washington's book, §$13-2$.


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