A note on the cohomology of pullback Lie algebroids

@misc{meinrenken2026note,
  title = {{A note on the cohomology of pullback Lie algebroids}},
  author = {Eckhard Meinrenken},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.23859},
  eprint = {2607.23859},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.23859},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A note on the cohomology of pullback Lie algebroids
AU  - Eckhard Meinrenken
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2607.23859
UR  - https://arxiv.org/abs/2607.23859
AB  - We give a direct differential-geometric proof of a result of Crainic on the cohomology of pullbacks of Lie algebroids under surjective submersions: If the fibers are connected and have vanishing cohomology through degree n, pullback with coefficients in any representation is an isomorphism through degree n and is injective in degree n+1. The proof reduces to a vanishing result for leafwise cohomology and uses an exhaustion-and-patching argument due to Buchdahl. No local triviality or finite-type assumption on the fibers is required. We also prove the analogous result for holomorphic Lie algebroids.
ER  -