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Lie algebroid cohomology and Lie algebroid extensions. (English) Zbl 1453.32025

Summary: We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid \(\mathcal{Q}\) over a scheme \(X\), and a coherent sheaf of Lie \(\mathcal{O}_X\)-algebras \(\mathcal{L}\), we determine the obstruction to the existence of extensions \(0 \rightarrow \mathcal{L} \rightarrow \mathcal{E} \rightarrow \mathcal{Q} \rightarrow 0\), and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.

MSC:

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14F40 de Rham cohomology and algebraic geometry
14H70 Relationships between algebraic curves and integrable systems
18G40 Spectral sequences, hypercohomology
55N35 Other homology theories in algebraic topology
55T05 General theory of spectral sequences in algebraic topology
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