Taking tensor algebras commutes with filtered colimits.

Lemma 10.13.4. Let $R$ be a ring. Let $M_ i$ be a directed system of $R$-modules. Then $\mathop{\mathrm{colim}}\nolimits _ i \text{T}(M_ i) = \text{T}(\mathop{\mathrm{colim}}\nolimits _ i M_ i)$ and similarly for the symmetric and exterior algebras.

Proof. Omitted. Hint: Apply Lemma 10.12.9. $\square$

Comment #828 by on

Suggested slogan: Tensor algebras commute with colimits over directed systems of modules.

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