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modules – Commutation of tensor merchandise with inverse limits in a selected illustration

The key level is to generalize the issue in organize to make extra effectual the employ of limits. Consider extra typically for any $R$-module $R$ the unaffected map

$$f_{M,Y}: M otimes_R prod_{y in Y} R rightarrow prod_{y in Y} M$$

given by $m otimes (r_y) mapsto (r_y m)$. In the particular illustration $M = R^X$ this recovers the map in query, so it will suffice extra typically to show that such maps $f_{M,Y}$ are injective.

If some $xi$ lies within the kernel then by writing it as a finite sum of elementary tensors we get a finitely generated $R$-submodule $N subset M$ such that $xi$ comes from some $theta in N otimes R^Y$ after which $f_{N,X}(theta) = 0$ for the reason that goal is left-exact in $M$. Hence, it suffices to deal the illustration when $M$ is finitely generated. In illustration $M$ is finitely offered then $f_{M,Y}$ is an isomorphism as a result of right-exactness of root and goal permits one to dwindle to the illustration of finite free $M$ (which is simple). So this provides an affirmative respond when $R$ is noetherian.

Are you interested by non-noetherian $R$?

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