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The Chern character for Lie-Rinehart algebras. (English) Zbl 1097.14004
The author studies the Chern character in the context of Lie-Rinehart algebras. For a commutative \(k\)-algebra \(A\) a \((k,A)\)- Lie-Rinehart algebra is a \(k\)-Lie algebra and \(A\)-module \(\mathfrak g\) along with a map of \(k\)-Lie algebras and \(A\)-modules: \(\alpha : \mathfrak g \rightarrow \text{ Der}_{k}(A).\) For an \(A\)-module \(W\) a \(\mathfrak g\)-connection \(\nabla\) on \(W\) is an \(A\)-linear map \(\nabla : \mathfrak g \rightarrow\text{ End}_{k}(W)\) satisfying the Leibniz property: \({\nabla}(\delta)(aw)=a{\nabla}(\delta)(w)+{\alpha}(\delta)(a)w.\)
In such setting the author introduces (theorem 2.12) the Lie-Rinehart complex \(C^{*}({\mathfrak g},W,{\nabla})\) and a ring homomorphism \(\text{ch}^g : K_0(\mathfrak g) \rightarrow H^{*}(\mathfrak g, A).\) Here \(K_0(\mathfrak g)\) denotes the Grothendieck ring of locally free \(A\)-modules with a \(\mathfrak g\)-connection. Moreover the author shows (theorem 3.10) that this Chern character is independent of the choice of connection.

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14L15 Group schemes
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