2
$\begingroup$

For a ring $R$, a left $R$-module $F$ is flat if the functor $- \otimes_R F$ brings an exact sequence of right $R$-modules to an exact sequence of right $R$-modules.

It is faithfully flat if the functor $- \otimes_R F$ brings a sequence of right $R$-modules to an exact sequence of right $R$-modules if and only if the original sequence is exact.

It is a "standard fact" that the sum of a faithfully flat module and a flat module is faithfully flat. Can someone please explain why this is true.

$\endgroup$
2
$\begingroup$

The tensored sequence splits as two exact sequences, and it suffices to use the exactness of one to recover the exactness of the untensored sequence.

$\endgroup$
  • $\begingroup$ Yes, of course! Thanks for your help. $\endgroup$ – Dyke Acland Aug 18 '16 at 15:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.