Let $R$ be a commutative Noetherian ring. Then, for $R$-modules $\{X_i\}_i$ and $Y$, do we always have $$\mathrm{Ext}^n_R(\varinjlim X_i,Y) \cong \varprojlim \mathrm{Ext}^n_R( X_i,Y)$$ for all $n\geq 1$ ?
I would be happy even with a reference.
2026-03-26 07:55:07.1774511707
$\mathrm{Ext}$ and direct limit
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