Pith Number

pith:33TEYHPL

pith:2025:33TEYHPLQEX3FE2ZJZKEOHUXA6

not attested not anchored not stored refs resolved

Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras

David Pask, Promit Mukherjee, Roozbeh Hazrat, Sujit Kumar Sardar

For row-finite k-graphs without sources, the graded zeroth homology of the infinite path groupoid is isomorphic as a Z[Z^k]-module to the graded Grothendieck group of the Kumjian-Pask algebra, preserving positive cones.

arxiv:2507.19879 v2 · 2025-07-26 · math.KT · math.OA · math.RA

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{33TEYHPLQEX3FE2ZJZKEOHUXA6}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

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

For a row-finite k-graph Λ without sources, there exists a Z[Z^k]-module isomorphism between the graded zeroth (integral) homology H_0^{gr}(G_Λ) of the infinite path groupoid G_Λ and the graded Grothendieck group K_0^{gr}(KP_k(Λ)) of the Kumjian-Pask algebra KP_k(Λ), which respects the positive cones (i.e., the talented monoids).

C2weakest assumption

The k-graph Λ is row-finite and has no sources; this assumption is used to define the infinite path groupoid G_Λ and to ensure the Kumjian-Pask algebra is well-behaved for the homology and K-theory constructions (abstract, first paragraph).

C3one line summary

Establishes a Z[Z^k]-module isomorphism between graded H_0 of the groupoid and graded K_0 of the Kumjian-Pask algebra for k-graphs, shows preservation under graph moves, and provides a lifting criterion for homomorphisms.

References

47 extracted · 47 resolved · 0 Pith anchors

[1] G. Abrams, P. Ara, M. Siles Molina: Leavitt Path Algebras, Lecture Notes in Mathematics, vol. 2191, Springer Verlag,

[2] G. Abrams, E. Ruiz, M. Tomforde: Recasting the Hazrat conjecture: Relating shift equivalence to graded Morita equivalence, Algebr. Represent. Theory 27 (2024), 1477–1511. 2, 4, 34, 36, 42 2024

[3] P. Ara, J. Bosa, E. Pardo, A. Sims: The groupoids of adaptable separated graphs and their semigroups , Int. Math. Res. Not. IMRN (2021), 15444–15496. 3, 28 2021

[4] P. Ara, C. B¨ onicke, J. Bosa, K. Li: The type semigroup, comparison, and almost finiteness for ample groupoids , Ergod. Th. & Dynam. Sys. 43(2) (2023), 361–400. 3, 26 2023

[5] P. Ara, R. Hazrat, H. Li, A. Sims: Graded Steinberg algebras and their representations , Algebra Number Theory 12(1) (2018), 131–172. 3, 8, 9 2018

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T14:03:19.795296Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

dee64c1deb812fb293594e54471e9707b6fc07160e49297b163eb333e22f8d25

Aliases

arxiv: 2507.19879 · arxiv_version: 2507.19879v2 · doi: 10.48550/arxiv.2507.19879 · pith_short_12: 33TEYHPLQEX3 · pith_short_16: 33TEYHPLQEX3FE2Z · pith_short_8: 33TEYHPL

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/33TEYHPLQEX3FE2ZJZKEOHUXA6 \
  | 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: dee64c1deb812fb293594e54471e9707b6fc07160e49297b163eb333e22f8d25

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2e5200302cbbd3b56a6c54e4f374470a735f939a8bd86b82fdbc621f1b49a3bd",
    "cross_cats_sorted": [
      "math.OA",
      "math.RA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.KT",
    "submitted_at": "2025-07-26T09:20:11Z",
    "title_canon_sha256": "f882df143a87c233b8a958132d9badc2224064e690c369c9712aed00efe98469"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2507.19879",
    "kind": "arxiv",
    "version": 2
  }
}