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.
Let G be a connected reductive algebraic group acting on a scheme X. The G-equivariant coherent sheaves on X are central to the study of X. These sheaves often have computable invariants, since the group action allows the use of tools such as localization theorems. Also, equivariant sheaves are an important source of sheaves on quotients by group actions, since if a quotient X → Y exists, then the sheaf of invariant sections of an equivariant sheaf on X is a coherent sheaf on Y . It is natural to ask which coherent sheaves on X admit G-actions. One positive result is due to Mumford, who proved that if G is connected and X is normal, and L is any invertible sheaf on X, then some power of L is G-linearizable [MFK,Corollary 1.6]. On the other hand, it is easy to find examples of coherent sheaves which do not admit Gactions. For example, PGL(2) acts on P 1 but the sheaf O P 1 (1) does not admit an action of PGL(2) (see [MFK,p. 33]).
Merkurjev proved that from the point of view of K-theory, there is no obstruction to equivariance, as long as the fundamental group of G is torsion-free (see [Mer]). Let G(X) (resp. G(G, X)) denote the Grothendieck group of coherent sheaves (resp. G-equivariant coherent sheaves) on X. There is a forgetful map G(G, X) → G(X). Let R = R(G) denote the representation ring of G, and I ⊂ R the augmentation ideal, that is, the ideal of virtual representations of rank 0. The The author was supported by the National Science Foundation.
Grothendieck group G(G, X) is an R-module. Merkurjev showed that if π 1 (G) is torsion-free, then the forgetful map induces an isomorphism
If π 1 (G) is not torsion-free, this map can fail to be an isomorphism. For example, the fundamental group of PGL( 2) is Z/2Z, and the class v = [O P 1 (1)] ∈ G(P 1 ) is not in the image of G(PGL(2), P 1 ). However, if we tensor with Q, this class is in the image. Indeed,
). This element is in the image of the forgetful map since v 2 is the class of O P 1 (2), which has a G-action.
This phenomenon holds more generally:
Theorem 1.1. Let G be a connected reductive algebraic group acting on a scheme X.
Merkurjev proves his theorem by using a spectral sequence relating equivariant and ordinary K-theory. The approach taken in this paper is different, and makes use of Brion’s analogue of Theorem 1.1 for Chow groups, along with the equivariant Riemann-Roch theorem proved by Edidin and the author. This use of Riemann-Roch explains the rational coefficients in the statement of our theorem.
We remark that Theorem 1.1 remains true even if G is not reductive, provided that G has a Levi factor L (which is automatic in characteristic 0), since then the forgetful maps from G-equivariant K-theory and Chow groups to the corresponding L-equivariant groups are isomorphisms. Also, we expect that a topological version of Theorem 1.1 holds for equivariantly formal spaces (since for these spaces the map from equivariant cohomology to ordinary cohomology is surjective). Finally, the completion theorem of [EG3] should have implications in this setting.
We work over an algebraically closed field k, and assume that the G-actions are locally linear-that is, the schemes on which G acts can be covered by G-invariant quasi-projective open subsets. This assumption is automatically satisfied for normal schemes. (We work in this setting in order to apply Brion’s results, which are proved under these hypotheses. We remark that that Merkurjev’s results, suitably stated, remain valid when the ground field is not algebraically closed.) Also, to make use of functorial properties of Riemann-Roch (see [Ful,Theorem 18.3(4)]) we will assume that our schemes can be equivariantly embedded in smooth schemes.
In this section we recall some basic facts about K-theory, Chow groups, and Riemann-Roch, in the equivariant and non-equivariant settings. We prove a result comparing topologies on equivariant Chow groups, and also prove a compatibility result between Riemann-Roch and forgetful maps. Both of these results are used in the proof of the main theorem. Our main references for equivariant Chow groups and equivariant Riemann-Roch will be [EG1] and [EG2], where more details can be found. If M is an abelian group, we write
Because we want to index Chow groups by codimension, we will assume all schemes and algebraic spaces considered are equidimensional; our results are valid without this assumption, but we would have to index Chow groups by dimension.
We begin with some definitions. Let G be a linear algebraic group acting on an algebraic space X. Let G(G, X) (resp. G(X)) denote the Grothendieck group of G-equivariant coherent sheaves (resp. coherent sheaves). There is a forgetful map
which takes the class of a G-equivariant coherent sheaf to the class of the same sheaf, viewed nonequivariantly. If we need to keep track of the space involved, we will denote this by For X . Note that G(G, X) is a module for the representation ring
This definition is independent
