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Let $M_i$ be modules over a ring $R$. It's an easy exercise to show that p.dim$( \oplus_{i =1}^n M_i) = \sup \{ $p.dim$(M_i) \} $. Does projective dimension also distribute over filtered colimits?

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No: Filtered colimits of projectives comprise the flat modules, and these are not necessarily projective.

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