Cohen-Macaulayness of Local Models via Shellability of the Admissible Set

TL;DR

This paper proves the Cohen-Macaulay property of local models of Shimura varieties by establishing shellability of admissible sets, providing a characteristic-free, explicit construction that confirms conjectures and covers previously unresolved cases.

Contribution

It introduces a dual EL-shellability proof for admissible sets, offering a new, intrinsic, and characteristic-free approach to Cohen-Macaulayness of local models.

Findings

Proves dual EL-shellability of admissible sets in the Iwahori-Weyl group.

Establishes Cohen-Macaulayness for special fibres of local models with parahoric level.

Provides an explicit shelling procedure preserving Cohen-Macaulayness.

Abstract

We prove that for any dominant cocharacter $\mu$ and any parahoric level $K$, the augmented admissible set $\widehat{\Adm(\mu)^K}$ in the Iwahori-Weyl group is dual EL-shellable. This resolves a conjecture of G\"ortz and provides a new proof of the Cohen-Macaulay property for the special fibres of local models with parahoric level structure. In particular, the result settles the previously open cases of residue characteristic $2$ and non-reduced root systems. This approach is characteristic-free and intrinsic to the structure of admissible sets. Moreover, our construction yields an explicit shelling, which translates into an inductive, component-by-component building procedure for the special fibre that preserves Cohen-Macaulayness at each step. As a consequence, we obtain the Cohen-Macaulayness of many local models of Shimura varieties considered in the literature, most notably those satisfying the He-Pappas-Rapoport description, as well as the local models characterized by Scholze-Weinstein and constructed by Ansch\"utz-Gleason-Louren\c{c}o-Richarz. Via the usual local model diagram, these results imply the Cohen-Macaulay property for the corresponding integral models of Shimura varieties whenever available. This gives a new proof that the integral models constructed by Kisin-Pappas-Zhou are Cohen-Macaulay.