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

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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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