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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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