We study duality and finiteness for locally analytic representations arising from compactly supported coherent cohomology of Drinfeld spaces. We show that Grothendieck--Serre duality for equivariant vector bundles realizes Bernstein--Zelevinsky duality of locally analytic representations. Following Fargues' strategy for $\ell$-adic sheaves, we obtain explicit resolutions of these representations by compactly induced representations. The resolutions imply that the cohomology is of compact presentation, a notion that generalizes finite presentation. As an application, we deduce finiteness and duality for extension groups of supercuspidal locally analytic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$.
This is joint work with Zhenghui Li and Benchao Su.