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?