Lemma 17.21.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ Y$-modules. Then $f^*\text{T}(\mathcal{F}) = \text{T}(f^*\mathcal{F})$, and similarly for the exterior and symmetric algebras associated to $\mathcal{F}$.

**Proof.**
Omitted.
$\square$

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