On operations and characteristic classes

In this paper exterior products are used to define operations and characteristic classes with values in the K-theory of an abelian category with tensor and exterior products. We apply the general construction to define Chern and Segre classes with values in algebraic K-theory and the K-theory of connections.

💡 Research Summary

The paper develops a unified framework for defining operations and characteristic classes in the K‑theory of any abelian category that possesses both a tensor product and exterior powers. After recalling that classical Chern and Segre classes are usually constructed in cohomology or Chow groups, the author observes that the λ‑operations arising from exterior powers provide a natural analogue in K‑theory, but that a systematic treatment beyond the category of vector bundles has been lacking.

Section 1 introduces the notion of an “abelian tensor‑exterior category” 𝔄. The category is required to have a bifunctor ⊗ that is exact in each variable and a family of exterior power functors ∧ⁿ (n ≥ 0) satisfying the usual identities (∧⁰ ≅ 𝟙, ∧¹ = Id, and the commutation ∧ⁿ∘∧ᵐ ≅ ∧ᵐ∘∧ⁿ). Typical examples include the category of finite‑dimensional modules over a ring, the category of coherent sheaves on a scheme, and the category of vector bundles equipped with a connection.

In Section 2 the author defines λ‑operations on the Grothendieck group K₀(𝔄) by setting λⁿ(