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I am in the beginnings on learning persistence homology, and as a start I'm studying Gunnar Carlsson's survey "Topology and Data".

Theorem 2.10 states the following:

"Suppose $M_{\star}$ is a finitely generated $F[t]$-module (where $F$ is a field). Then there exist integers $i_{1}, \dots, i_{m}, j_{1}, \dots, j_{n}, l_{1}, \dots, l_{n}$ and an isomorphism

$$ M_{\star} \cong \bigoplus_{s=1}^{m} F[t](i_{s}) \oplus \bigoplus_{i=1}^{n}(F[t]/(t^{l_{t}}))(j_{t}) $$

where for any graded $F[t]$-module $N_{\star}$, the notation $N_{\star}(s)$ denotes $N_{\star}$ with an upward dimension shift of $s$, so $N_{\star}(s)_{l} = N_{l-s}$."

I've seen the classification theorems before, but I've never seen the notation he is using: specifically, I haven't seen $F[t](i_{s})$ or $(F[t]/(t^{l_{t}}))(j_{t})$ before (the multiplication by the ideal $(j_{t})$ -- if that is what it's supposed to be). In addition, what does he mean by $N_{\star}(s)_{l}$?

It seems the literature (after performing google searches) has a cleaner approach and cleaner notations. Is this really just a notation discrepancy? And if so, can you provide me with an explanation in detail?

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  • $\begingroup$ The $l$th graded component of $N_*(s)$ (which equals the $(l-s)$th graded component of $N$). (Btw, usually this denotes the $(l-s)$th graded component of $N$.) $\endgroup$ – user26857 Nov 22 '15 at 23:11
  • $\begingroup$ What about the $F[t](i_{s})$ business? $\endgroup$ – boldbrandywine Nov 22 '15 at 23:13
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    $\begingroup$ If $R$ is a graded ring, then $R(s)$ denotes the ring $R$ shifted (in degree) by $s$, that is, $R(s)_i=R_{s+i}$. For instance, $F[t](s)_i=Ft^{s+i}$.) $\endgroup$ – user26857 Nov 22 '15 at 23:15
  • $\begingroup$ So in this instance, we have a $\mathbb{Z}$-grading on $F[t]$, and thus $F[t](i_{s})$ denotes the ring $F[t]$ shifted (in degree) by $i_{s}$? $\endgroup$ – boldbrandywine Nov 22 '15 at 23:19

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