Is the inverse limit an additive functor in the category of $R$-mod?

Asked Feb 18 '21 at 20:22

Active Feb 18 '21 at 20:22

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Is the inverse limit an additive functor in the categories of $R$-mod?

Equivalently, if we have two inverse systems $\{A_i\}$ and $\{B_i\}$ of $R$-modules sharing the same index set, do we have $$ \lim_{\leftarrow} A_i \oplus B_i = \lim_{\leftarrow} A_i \oplus \lim_{\leftarrow} B_i? $$

asked Feb 18 '21 at 20:22

k99731

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You get this for free: The (finite direct sum) is a finite product, which is a limit. Since limits commute with each other, you win. – Geoff Feb 18 '21 at 20:25
@Geoff Thank you – k99731 Feb 18 '21 at 20:35

Is the inverse limit an additive functor in the category of $R$-mod?

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