The forgetful map in rational K-theory

Let G be a connected reductive algebraic group acting on a scheme X. Let R(G) denote the representation ring of G, and let I be the ideal in R(G) of virtual representations of rank 0. Let G(X) (resp. G(G,X)) denote the Grothendieck group of coherent sheaves (resp. G-equivariant coherent sheaves) on X. Merkurjev proved that if the fundamental group of G is torsion-free, then the map of G(G,X)/IG(G,X) to G(X) is an isomorphism. Although this map need not be an isomorphism if the fundamental group of G has torsion, we prove that without the assumption on the fundamental group of G, this map is an isomorphism after tensoring with the rational numbers.

💡 Research Summary

The paper studies the relationship between equivariant G‑theory and ordinary G‑theory for a connected reductive algebraic group G acting on a scheme X. Let R(G) be the representation ring of G and I⊂R(G) the augmentation ideal consisting of virtual representations of rank 0. There is a natural forgetful homomorphism
  F : G(G,X) → G(X)
that simply forgets the G‑action on a coherent sheaf. Merkurjev proved that when the fundamental group π₁(G) is torsion‑free, the induced map
  G(G,X)/I G(G,X) → G(X)
is an isomorphism. However, if π₁(G) contains torsion the map can fail to be an isomorphism; a classical example is the action of PGL(2) on ℙ¹, where the class of 𝒪(1) does not lie in the image of the forgetful map, although after tensoring with ℚ it does.

The main result of this work removes the torsion‑free hypothesis at the cost of rational coefficients. Theorem 1.1 states that for any connected reductive G acting on any scheme X, the map
  G(G,X)/I ⊗ ℚ → G(X) ⊗ ℚ
is an isomorphism; equivalently, the forgetful map becomes surjective after rationalization.

The proof combines two sophisticated tools. First, Brion’s theorem on equivariant Chow groups shows that the J‑adic completion of the equivariant Chow ring CH⁎G(X) (where J is the ideal generated by positive‑degree elements of CH⁎G(pt)) coincides with the direct product ∏{i≥0} CH_i^G(X) ⊗ ℚ. Second, Edidin–Graham’s equivariant Riemann–Roch map
  τ_G : G(G,X) → ⊕{i} CH_i^G(X) ⊗ ℚ
is compatible with the ordinary Riemann–Roch map τ : G(X) → CH⁎(X) ⊗ ℚ and with the forgetful homomorphisms. Proposition 2.3 establishes a commutative diagram linking τ_G, τ, and the forgetful maps.

A technical lemma (Lemma 3.1) about completions of modules over a Noetherian ring R with respect to an ideal I is proved: for any R‑module M, the natural map M/IM → ĤM/IĤM (where ĤM denotes the I‑adic completion) is an isomorphism, even when M is not finitely generated. This result allows the authors to identify the quotient G(G,X)/I with the I‑adic completion of G(G,X) modulo I, and similarly for Chow groups.

Putting these ingredients together, the authors first show that the completed equivariant Riemann–Roch map (\widehat{τ}_G) is an isomorphism of (\widehat{R})-modules (where (\widehat{R}) is the I‑adic completion of R(G)). Using the Chern character isomorphism (\widehat{R} ≅ \widehat{S}) (S = CH⁎_G(pt)), they deduce an isomorphism
  (\widehat{G(G,X)}/I ≅ (⊕_i CH_i^G(X))/J).
Lemma 3.1 then yields an isomorphism (G(G,X)/I ≅ \widehat{G(G,X)}/I). Combining with Brion’s identification of the J‑adic completion of CH⁎_G(X) with the product of the equivariant Chow groups, they obtain an isomorphism
  (G(G,X)/I ≅ CH⁎_G(X)/J).

Finally, the ordinary Riemann–Roch map τ is known to be an isomorphism (G(X) ≅ CH⁎(X)) after tensoring with ℚ. Since the diagram involving τ, τ_G, and the forgetful maps commutes, the induced map (G(G,X)/I → G(X)) becomes an isomorphism after rationalization. Consequently, the forgetful homomorphism is surjective on rational K‑theory, and the kernel is precisely the augmentation ideal I.

The paper thus shows that any obstruction to lifting a coherent sheaf to a G‑equivariant sheaf is torsion in nature; after passing to rational coefficients, every class in ordinary G‑theory comes from an equivariant class modulo the augmentation ideal. This result extends Merkurjev’s theorem to the general case and highlights the deep interplay between equivariant K‑theory, equivariant Chow groups, and Riemann–Roch theory.