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I am struggling to understand a key step in a proof of the Artin-Rees lemma, which I have put in a red box below. I don't really see how we can pass from a finite direct sum to an infinite one. I've tried writing out both sides of the equality to get from one to the other, but I'm just not seeing it...

I'm sure I'm just missing something, but any help would be much appreciated, thanks!

Here is the proof given:

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Here is the preceding section on how to make an $R$-module $M$ into an $R[t]$-module $M[t]$:

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Tim
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    A couple of observations first: prove that $N_n I^j \subseteq N_{n+j}$, "expand out the product" and observe that the first piece is contained in the second so we may forget about it. Now, can you try to coax the second piece into what you need? – knsam May 21 '15 at 03:33
  • @knsam Glad to see you've come to my rescue again! I can show that $N_nI^j\subseteq N_{n+j}$, but beyond that I'm not really grasping it. The first piece is an infinite direct sum, and the second is a finite one, so I don't really understand how they can be compatible... – Tim May 21 '15 at 14:38
  • Sorry, I had not replied quicker. My hint is misleading, so please accept my apologies. But I will now post a descriptive answer. – knsam May 22 '15 at 11:44

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Here is a small computation that will help:

We are given:

\begin{align*} &\qquad \left(\bigoplus_{n=0}^k N_nt^n \right)\left(R + \sum_{j=1}^\infty I^jt^j\right) \\ &= \left(\bigoplus_{n=0}^k N_nt^n \right)\left(I^0t^0 + \sum_{j=1}^\infty I^jt^j\right) \\ &= \left(\bigoplus_{n=0}^k N_nt^n \right)\left(\sum_{j=0}^\infty I^jt^j\right) \end{align*} We expand out the product to get: \begin{align*} \sum_{n=0}^k \sum_{j=0}^\infty N_nI^jt^{n+j} \end{align*}

In the sum that remains, we make the change of variable $n+ j = i$ to get: $$\sum_{i=0}^{\infty} \left(\sum_{n=0}^{\min(i,k)}N_nI^{i-n}\right)t^i$$

This resulting sum over $i$ is also direct by the definition of $M[t]$.

knsam
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  • Great, thank you again! I must admit, I completely forgot that direct sums are just normal sums with trivial intersection. I always see direct sum and think Cartesian product, or direct product... Will have to do some revision on all that stuff now! – Tim May 22 '15 at 14:02
  • @Tim: Glad to know that helped. I rearranged my answer a bit in order not to belabour the point. – knsam May 22 '15 at 14:36