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The algebraic Chern-Weyl theory exists, . You can find expositions of it in, say, Jean-Louis Loday's Cyclic Homology or in Max Karoubi's Homologie Cyclique Astérisque. It constructs characteristic classes for finitely generated projective modules, and more generally on elements of $K_0$, with values in the cyclic homology of the coordinate algebra, and/or the usual variations of this homology theory. This precisely generalizes the usual case., in view of the Swan-Serre theorem that states that finitely generated projective modules over $C^\infty(M)$ are exactly the same thing as vector bundles over a manifold $M$, and the computation of the (periodic) cyclic homology of $C^\infty(M)$, which turns out to be de Rham cohomology of $M$.

One can also get characteristic cyclic classes on the higher algebraic $K$-groups, and this works more or less automatically not only for commutative algebras but for algebras in general.

If you want non-affine schemes, you can generalize everything with sufficient care to that context.

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The algebraic Chern-Weyl theory exists, You can find expositions of it in, say, Jean-Louis Loday's Cyclic Homology or in Max Karoubi's Homologie Cyclique Astérisque. It constructs characteristic classes for projective modules, and more generally on elements of $K_0$, with values in the cyclic homology of the coordinate algebra, and/or the usual variations of this homology theory. This precisely generalizes the usual case.

One can also get characteristic cyclic classes on the higher algebraic $K$-groups, and this works more or less automatically not only for commutative algebras but for algebras in general.

If you want non-affine schemes, you can generalize everything with sufficient care to that context.