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I'm having a doubt in the proof of Corollary 2.5 from Atiyah-Macdonald's Introduction to Commutative Algebra (page 21).

The proof is simple: apply Proposition 2.4 (basically a version of Cayley-Hamilton's theorem) with the identity module homomorphism: from $$ \phi^n+a_1\phi^{n-1}+...+a_n \text{ and } \phi(x)=x, $$ we get $$ x + a_1 x + a_2 x + ... + a_{n-1}x+a_n = x(1+a_1+...+a_{n-1})+a_n. $$

Proof follows by picking $x=1+a_1+...+a_n$ as if you could put in evidence the factor $x$ from the previous expression. But you can't. There is no $x$ multiplying the last $a_n$.

Can someone go through some trouble and explain what I'm missing? Thanks in advance.

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    $\begingroup$ Sorry, what is the statement of corollary 2.5? $\endgroup$
    – jgon
    Mar 5, 2020 at 3:02
  • $\begingroup$ "If M is a finitely generated A-module, I an ideal of A such that $I.M=M$, then there exists $x\in A$ such that $x-1 \in I$ and $xM=0$" $\endgroup$
    – Marra
    Mar 5, 2020 at 3:11

1 Answer 1

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I'm having a bit of trouble parsing what you're saying. I did look up the version in A-M. in A-M, $x$ is an element of $A$, defined to be the sum of the coefficients of the characteristic polynomial of $\phi=\textrm{id}_M$.

In detail the proof goes as follows:

Let $A$ be a commutative ring, $M$ a finitely generated $A$ module. If $I\subseteq A$ is an ideal such that $IM=M$, then there exists $x\in A$, $x\equiv 1 \pmod{I}$ such that $xM=0$.

Proof

Let $\phi=\textrm{id}_M$. $\phi(M)=M=IM$, so $\phi(M)\subseteq IM$. Then by 2.4 (generalized Cayley-Hamilton kinda) $\phi$ satisfies a polynomial equation $$\phi^n + a_1\phi^{n-1}+\cdots + a_n=0,$$ with $a_j\in I^j$. Let $x=1+a_1+\cdots + a_n$. Certainly $x\equiv 1\pmod{I}$. Then if $m\in M$, $$xm = (1+a_1+\cdots a_n)m = m + a_1m + \cdots a_nm = \phi^n(m) + a_1\phi^{n-1}(m)+\cdots a_nm=0.$$ Hence $x$ annihilates $M$, as desired.

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    $\begingroup$ Exactly. You first equation with $\phi$ should be equal to zero. The last element $a_n$ has no $x$ multiplying it. Now you need to compare this equation to your last one, but in the last one there is $m$ times $a_n$ . Why can you use the first to reduce that the second is zero if they have this difference? $\endgroup$
    – Marra
    Mar 5, 2020 at 9:40
  • $\begingroup$ @Marra, please try to use $x$ to mean what it means in A-M and my answer. I suspect what you're asking about now has to do with $a_n$ being the constant term of the polynomial equation. The point then is the mapping into the endomorphism ring sends an element $a$ of $A$ to the endomorphism $\phi(m)=am$. Thus $a_n$ is acting as it should. I'll try to come back and edit my answer later, to add this explanation in and expand on it, but hopefully this is helpful. $\endgroup$
    – jgon
    Mar 5, 2020 at 15:50
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    $\begingroup$ I think I got it. Each $a_n$ is actually the A-module homomorfism $x\mapsto a_n\cdot x$ is that right? $\endgroup$
    – Marra
    Mar 6, 2020 at 12:02
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    $\begingroup$ @Marra Sorry for the slow reply, but yes, exactly. $\endgroup$
    – jgon
    Mar 9, 2020 at 3:51

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