Dual Shellability of Admissible Set and Cohen-Macaulayness of Local Models
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908539317911552 |
|---|---|
| author | He, Xuhua Yu, Qingchao |
| author_facet | He, Xuhua Yu, Qingchao |
| contents | We prove Görtz's combinatorial conjecture \cite{Go01} on dual shellability of admissible sets in Iwahori-Weyl groups, proving that the augmented admissible set $\widehat{\mathrm{Adm}}(μ)$ is dual shellable for any dominant coweight $μ$. This provides a uniform, elementary approach to establishing Cohen-Macaulayness of the special fibers of the local models with Iwahori level structure for all reductive groups-including residue characteristic $2$ and non-reduced root systems-circumventing geometric methods. Local models, which encode singularities of Shimura varieties and moduli of shtukas, have seen extensive study since their introduction by Rapoport-Zink, with Cohen-Macaulayness remaining a central open problem. While previous work relied on case-specific geometric analyses (e.g., Frobenius splittings \cite{HR23} or compactifications \cite{He13}), our combinatorial proof yields an explicit labeling that constructs the special fiber by sequentially adding irreducible components while preserving Cohen-Macaulayness at each step, a new result even for split groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11581 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dual Shellability of Admissible Set and Cohen-Macaulayness of Local Models He, Xuhua Yu, Qingchao Algebraic Geometry Number Theory 14G35, 11G18 We prove Görtz's combinatorial conjecture \cite{Go01} on dual shellability of admissible sets in Iwahori-Weyl groups, proving that the augmented admissible set $\widehat{\mathrm{Adm}}(μ)$ is dual shellable for any dominant coweight $μ$. This provides a uniform, elementary approach to establishing Cohen-Macaulayness of the special fibers of the local models with Iwahori level structure for all reductive groups-including residue characteristic $2$ and non-reduced root systems-circumventing geometric methods. Local models, which encode singularities of Shimura varieties and moduli of shtukas, have seen extensive study since their introduction by Rapoport-Zink, with Cohen-Macaulayness remaining a central open problem. While previous work relied on case-specific geometric analyses (e.g., Frobenius splittings \cite{HR23} or compactifications \cite{He13}), our combinatorial proof yields an explicit labeling that constructs the special fiber by sequentially adding irreducible components while preserving Cohen-Macaulayness at each step, a new result even for split groups. |
| title | Dual Shellability of Admissible Set and Cohen-Macaulayness of Local Models |
| topic | Algebraic Geometry Number Theory 14G35, 11G18 |
| url | https://arxiv.org/abs/2509.11581 |