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Bakker, Brunebarbe, Tsimerman showed in \cite{bakker2022minimal} that thedefinable structure sheaf $\mathcal{O}_{\mathbb{C}^n}$ of $\mathbb{C}^n$ is acoherent $\mathcal{O}_{\mathbb{C}^n}$-module as a sheaf on the site$\underline{\mathbb{C}^n}$, where the coverings are finite coverings bydefinable open sets. In general, let $\mathbb{K}$ be an algebraically closedfield of characteristic zero. We give a more model-theoretic proof of thecoherence of $\mathcal{O}_{\mathbb{K}^n}$ as a sheaf of$\mathcal{O}_{\mathbb{K}^n}$-module on the site $\underline{\mathbb{K}^n}$using spectral topology on the type space $S_n(\mathbb{K})$.

A model theoretic proof for o-minimal coherence theorem