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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.

##### MSC:
 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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