Tensor product and inverse limit

Asked Feb 17 '15 at 17:42

Active Feb 17 '15 at 17:42

Viewed 585 times

Let $(R,m)$ be a Noetherian local ring. Let $M$ be an $R$-module, and let $\{N_i\}$ be an inverse system. I am curious to know if there is a condition whereby the natural map

$M \otimes_R \varprojlim{N_i} \to \varprojlim{(N_i\otimes_R M)}$

is injective. If it would help, one could further assume that $N_i\otimes_R M$ has finite dimension as an $R/m$-vector space.

asked Feb 17 '15 at 17:42

thebowler_300

Condition on what? $R$? $M$? $N_i$? The maps in the inverse system? – RghtHndSd Feb 17 '15 at 17:56
That's the question. Are there conditions on these where this natural map is injective? – thebowler_300 Feb 17 '15 at 19:14

Tensor product and inverse limit

0 Answers0