Pith Number

pith:332C4CIG

pith:2026:332C4CIG6QE3MLL5YGBMGLPMBX

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A note on short and long exact sequences in the BBG construction of complexes from complexes

Snorre H. Christiansen

Cohomology of BGG sequences follows from long exact sequences of complexes even without injectivity

arxiv:2605.15933 v1 · 2026-05-15 · math.NA · cs.NA

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Record completeness

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3 Author claim open · sign in to claim

4 Citations 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

the cohomology of some Bernstein-Gelfand-Gelfand (BGG) sequences that are important for the numerical analysis of partial differential equations, can be obtained through the construction of a long exact sequence connecting cohomology groups

C2weakest assumption

That BGG constructions of complexes from complexes admit short exact sequences even when the underlying maps are not injective or surjective, allowing the long exact sequence of cohomology to be formed systematically.

C3one line summary

Demonstrates computation of cohomology in BGG sequences via long exact sequences from short exact sequences of complexes, extended to non-bijective maps with spectral sequence view.

References

19 extracted · 19 resolved · 0 Pith anchors

[1] D. N. Arnold, R. S. Falk, and R. Winther. Differential complexes and stability of finite element methods. I. The de Rham complex. InCompatible spatial discretizations, volume 142 ofIMA Vol. Math. Appl 2006

[2] D. N. Arnold, R. S. Falk, and R. Winther. Differential complexes and stability of finite element methods. II. The elasticity complex. InCompatible spatial discretizations, volume 142 ofIMA Vol. Math. 2006

[3] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications.Acta Numer., 15:1–155, 2006 2006

[4] D. N. Arnold and K. Hu. Complexes from complexes.Found. Comput. Math., 21(6):1739–1774, 2021 2021

[5] I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand. Differential operators on the base affine space and a study ofg-modules. InLie groups and their representations (Proc. Summer School, Bolyai J´ anos 1971

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:01:45.640919Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

def42e0906f409b62d7dc182c32dec0de357f02e9901f430b8beb11d4a1cadbc

Aliases

arxiv: 2605.15933 · arxiv_version: 2605.15933v1 · doi: 10.48550/arxiv.2605.15933 · pith_short_12: 332C4CIG6QE3 · pith_short_16: 332C4CIG6QE3MLL5 · pith_short_8: 332C4CIG

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/332C4CIG6QE3MLL5YGBMGLPMBX \
  | 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: def42e0906f409b62d7dc182c32dec0de357f02e9901f430b8beb11d4a1cadbc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "922ef0d9e51bd9d32352716f1425b420908f4f6df53685b60015d20984502bca",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://creativecommons.org/licenses/by-sa/4.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-05-15T13:15:31Z",
    "title_canon_sha256": "75d43929c09b1c05b932e18d8f64dd1201cbdf50f9ffd94a255946890b572ea0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15933",
    "kind": "arxiv",
    "version": 1
  }
}