Regularity and cohomology of Pfaffian thickenings

Let $S$ be the coordinate ring of the space of $n\times n$ complex skew-symmetric matrices. This ring has an action of the group ${\text{GL}}_n(\mathbb{C})$ induced by the action on the space of matrices. For every invariant ideal $I\subseteq S$, we provide an explicit description of the modules ${\text{Ext}}_S^{\bullet }(S\mathrm{/}I,S)$ in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusion of invariant ideals $I\supseteq J$, we compute the (co)kernel of the induced map $$\text{Ext}$^j_S(S/I,S)\to $\text{Ext}$^j_S(S/J,S)$forall$j\geq 0$. As a consequence, we show that if an invariant ideal $I$ is unmixed, then the induced maps ${\text{Ext}}_S^j(S\mathrm{/}I,S)\to H_I^j(S)$ are injective, answering a question of Eisenbud-Musta\c{t}\u{a}-Stillman in the case of Pfaffian thickenings. Finally, using our Ext computations and local duality, we verify an instance of Kodaira vanishing in the sense described in the recent work of Bhatt-Blickle-Lyubeznik-Singh-Zhang.