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Homotopy theory, homological algebra, algebraic treatments of manifolds.
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Definitions of the module $R/(x_0^\infty,x_1^\infty,\ldots,x_{n-1}^\infty)$
I strongly guess that, you mean, the maps of your direct system, $$R/I_t\rightarrow R/I_{t+1},$$ whose colimit is, $$R/I^\infty,$$ are the multiplication by $x_0\ldots x_{n-1}$. So I think that your …