A projective resolution of the symplectic Steinberg module
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910224399466496 |
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| author | Pal, Urshita |
| author_facet | Pal, Urshita |
| contents | Borel--Serre proved that for a number ring $R$ with fraction field $K$, the symplectic group $\text{Sp}_{2n}(R)$ is a virtual duality group of degree quadratic in $n$, and that the symplectic Steinberg module $\text{St}^ω_{2n}(K)$ is its dualizing module. We construct a projective resolution of this symplectic Steinberg module as an $\text{Sp}_{2n}(R)$-representation, that is similar in form to a resolution of Lee--Szczarba for the special linear group, but whose construction is more involved. When $R$ is a Euclidean number ring, we use this resolution to compute the top degree cohomology of principal level-$p$ congruence subgroups of $\text{Sp}_{2n}(R)$, for primes $p \in R$ such that the natural map $R^\times \to (R/(p))^\times$ is surjective. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_06499 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A projective resolution of the symplectic Steinberg module Pal, Urshita Algebraic Topology Group Theory Geometric Topology Borel--Serre proved that for a number ring $R$ with fraction field $K$, the symplectic group $\text{Sp}_{2n}(R)$ is a virtual duality group of degree quadratic in $n$, and that the symplectic Steinberg module $\text{St}^ω_{2n}(K)$ is its dualizing module. We construct a projective resolution of this symplectic Steinberg module as an $\text{Sp}_{2n}(R)$-representation, that is similar in form to a resolution of Lee--Szczarba for the special linear group, but whose construction is more involved. When $R$ is a Euclidean number ring, we use this resolution to compute the top degree cohomology of principal level-$p$ congruence subgroups of $\text{Sp}_{2n}(R)$, for primes $p \in R$ such that the natural map $R^\times \to (R/(p))^\times$ is surjective. |
| title | A projective resolution of the symplectic Steinberg module |
| topic | Algebraic Topology Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2605.06499 |