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A Koszul complex in quaternionic analysis and its applications

  • Yong Li
  • Yuchen Zhang
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Abstract

Let $n\geqslant 1, Ω\subset\mathbb{H}^n $ be a domain. We construct a Koszul-type complex for the ideal sheaf ℐX(k) of k-regular functions vanishing on X={(q0, q1, ⋯, qn−1)∈ Ω: q0=0} in several quaternionic variables: 0→ ℛ(k+2)xrightarrowℒ̃(k) ℛ(k+1)⊕ℛ(k+1)xrightarrowℒ(k) ℐX(k)→ 0, where kgeqslant 0, ℛ(k) is the sheaf of k-regular functions on Ω, ℒ̃(k)=(−L1(k+2),L0(k+2))T, ℒ(k)=(L0(k+1),L1(k+1)), and L0(k),L1(k) are multiplication-like operators on k-regular functions. This gives the quaternionic analogue of the classical Koszul complex. And we present the long exact sequence in cohomology for the case Ω∩{q0=0}=∅ with explicit differential connecting maps, by applying the Cauchy-Fueter complex and cohomological methods. As an application, in the special case n=1, k=1, the operator pair (L0(1), L1(1)) is shown to be surjective if and only if H3(Ω, ℝ)=0. Furthermore, a cohomological vanishing criterion is given for H1(Ω,ℐX(k)); under this criterion, every k-regular function on {q0=0}∩Ω extends to a k-regular function on Ω.

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Cite

BibTeXRIS
BibTeX
@misc{li2026koszul,
  title = {{A Koszul complex in quaternionic analysis and its applications}},
  author = {Yong Li and Yuchen Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.10338},
  eprint = {2607.10338},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.10338},
}
RIS
TY  - UNPB
TI  - A Koszul complex in quaternionic analysis and its applications
AU  - Yong Li
AU  - Yuchen Zhang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.10338
UR  - https://arxiv.org/abs/2607.10338
AB  - Let $n\geqslant 1, Ω\subset\mathbb{H}^n $ be a domain. We construct a Koszul-type complex for the ideal sheaf ℐ_X^(k) of k-regular functions vanishing on X={(q_0, q_1, ⋯, q_n−1)∈ Ω: q_0=0} in several quaternionic variables: 0→ ℛ^(k+2)xrightarrowℒ̃^(k) ℛ^(k+1)⊕ℛ^(k+1)xrightarrowℒ^(k) ℐ_X^(k)→ 0, where kgeqslant 0, ℛ^(k) is the sheaf of k-regular functions on Ω, ℒ̃^(k)=(−L_1^(k+2),L_0^(k+2))^T, ℒ^(k)=(L_0^(k+1),L_1^(k+1)), and L_0^(k),L_1^(k) are multiplication-like operators on k-regular functions. This gives the quaternionic analogue of the classical Koszul complex. And we present the long exact sequence in cohomology for the case Ω∩{q_0=0}=∅ with explicit differential connecting maps, by applying the Cauchy-Fueter complex and cohomological methods. As an application, in the special case n=1, k=1, the operator pair (L_0^(1), L_1^(1)) is shown to be surjective if and only if H^3(Ω, ℝ)=0. Furthermore, a cohomological vanishing criterion is given for H^1(Ω,ℐ_X^(k)); under this criterion, every k-regular function on {q_0=0}∩Ω extends to a k-regular function on Ω.
ER  -