Pith Number

pith:OCN54RRC

pith:2026:OCN54RRCTTL647375XET6FROWY

not attested not anchored not stored refs pending

A projective resolution of the symplectic Steinberg module

Urshita Pal

The symplectic Steinberg module admits an explicit projective resolution over Sp_{2n}(R).

arxiv:2605.06499 v2 · 2026-05-07 · math.AT · math.GR · math.GT

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\pithnumber{OCN54RRCTTL647375XET6FROWY}

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Record completeness

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4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We construct a projective resolution of this symplectic Steinberg module as an 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.

C2weakest assumption

The construction of the projective resolution is valid and produces a genuine resolution of the Steinberg module as an Sp_{2n}(R)-module for general number rings R (with the cohomology application restricted to Euclidean R).

C3one line summary

Constructs a projective resolution of the symplectic Steinberg module St^ω_{2n}(K) for Sp_{2n}(R), analogous but more involved than Lee-Szczarba's for SL_n, and applies it to compute top cohomology of principal level-p congruence subgroups over Euclidean rings.

Receipt and verification

First computed	2026-05-20T00:03:14.467818Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

709bde46229cd7ee7f7fedc93f162eb61bc2d440d008a6f899a7080068762df8

Aliases

arxiv: 2605.06499 · arxiv_version: 2605.06499v2 · doi: 10.48550/arxiv.2605.06499 · pith_short_12: OCN54RRCTTL6 · pith_short_16: OCN54RRCTTL64737 · pith_short_8: OCN54RRC

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/OCN54RRCTTL647375XET6FROWY \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 709bde46229cd7ee7f7fedc93f162eb61bc2d440d008a6f899a7080068762df8

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "427bb4932f9156606f6dfaff890ee10387d2a1a2cad9ec6ac1e610d7fcb2cb20",
    "cross_cats_sorted": [
      "math.GR",
      "math.GT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AT",
    "submitted_at": "2026-05-07T16:17:56Z",
    "title_canon_sha256": "e9d80c42724bceca9dd65c2c48f368588d66fda4d04131ddee0a931bc3216634"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.06499",
    "kind": "arxiv",
    "version": 2
  }
}