Rigidity for equivariant K-theory

Serge Yagunov, Paul Arne Østvær

Abstract

We extend the classical rigidity results for K-theory to the equivariant setting of algebraic group actions. Following the work of Andrei Suslin on ordinary K-groups, these results can be considered as first steps toward an explicit computation of equivariant K-groups of algebraically closed fields.

Contents

1 Introduction 2

2 Proofs 3

1 Introduction

For k an infinite field and G an algebraic group over k, let Sm_{k} denote the category of smooth projective k-schemes of finite type and fix some V in the category Sch_{k}^{G} of G-equivariant k-schemes of finite type and G-maps. The latter data entails a G-action σ: V × G → V subject to the usual associative and unital identities for a group action. The abelian subcategory of G-modules on V comprising coherent O_{V} -modules and its exact subcategory of locally free O_{V} -modules, a.k.a. G-vector bundles on V, gives rise to the G-equivariant K′-theory (respectively K-theory) of V. In the event G is the trivial group, this reduces to ordinary K′- and K-theory. Throughout we consider mod-n coefficients for n relatively prime to the exponential characteristic of k and assume that G acts trivially on the base fields.

Theorem (Rigidity for rational points): The following are equivalent.

(1) For every X in Sm_{k} with trivial G-action and any two rational points x_{0} and x_{1}

on X,

\xymatrix{x_{0}^{∗} = x_{1}^{∗}: K_{∗}(G, V ×_{k} X; Z/n) \ar[r] \ar[r] & K_{∗}(G, V; Z/n).}

(2) The rational points 0 and 1 on the affine line A_{k}^{1} with trivial G-action yield equal

pullback maps

\xymatrix{K_{∗}(G, V ×_{k} A_{k}^{1}; Z/n) \ar[r] \ar[r] & K_{∗}(G, V; Z/n).}

The four step proof of the theorem consists only of checking that K_{∗}(G, V ×_{k} −; Z/n) defines a functor with weak transfers on Sm_{k} in the sense of [7].

If V is smooth, so that the naturally induced map K_{∗}(G, V ×_{k} X) → K′_{∗}(G, V ×_{k} X) is an isomorphism for all X in Sm_{k}, then the second condition in the theorem holds because K′-theory is homotopy invariant [12, Theorem 4.1], cf. Lemma 2.5. Although G acts trivially on X in the previous theorem we do not assume that G acts trivially on V. The following result generalizes [9, Main Theorem].

Theorem (Rigidity for extensions): Suppose K/k is an extension of algebraically closed fields and (V, σ) a G-equivariant smooth k-scheme of finite type. With the induced Gaction σ × id_{K} on the base change V_{K} ≡ V ×_{k} K the natural G-map V_{K} → V induces an isomorphism

\xymatrix{K_{∗}(G, V; Z/n) \ar[r]^{≅} & K_{∗}(G, V_{K}; Z/n).}

Proof. If a noetherian G-scheme X is the inverse limit of a system {X_{α}} of noetherian G-schemes with flat transition maps, then K′(G, X) ≅ colim_{α} K′(G, X_{α}), as Thomason noted in [13, §3.7] (in fact, the category of coherent G-modules on X is the direct limit of the categories of coherent G-modules on the X_{α}’s). Our claim follows now as in the proof of [7, Theorem 1.14] by combining rigidity for rational points and Lemma 2.5 (viewing Spec K as an inverse limit of smooth affine k-schemes with trivial G-actions).

The last of our main results deals with rigidity for equivariant K-theory of Hensel local rings. For non-equivariant results we refer to [1], [2] and [11].

Theorem (Rigidity for Hensel local rings): Let k be an infinite field, (V, σ) a Gequivariant smooth k-scheme, and X a smooth k-scheme of finite type. If P ∈ X(k) is a rational point, denote by O_{X,P}^{h} the corresponding Hensel local ring. Then there is a natural isomorphism

\xymatrix{K_{∗}(G, V ×_{k} O_{X,P}^{h}) \ar[r]^{≅} & K_{∗}(G, V).}

In order to prove this result we basically check that the conditions in [5] are satisfied.

In the study of equivariant K-theory over an algebraically closed field k one may ask for an explicit computation of the groups K_{∗}(G, k; Z/n). Based on Suslin’s computation of the K-groups K_{∗}(k; Z/n) [10], we expect this problem is closely related to equivariant topological complex K-theory and that an answer involves the representation theory of the algebraic group G.

2 Proofs

Proof. (Rigidity for rational points) For legibility we let P^{G} denote G-vector bundles of finite rank, BQ denote the classifying space of Quillen’s Q-construction [8], and write

F(−) ≡ π_{∗+1}(BQP^{G}(V ×_{k} −); Z/n): (Sm_{k})^{op} → Ab

for the functor K_{∗}(G, V ×_{k} −; Z/n) in the formulation of the theorem. Assuming that (2) holds, we show F(−) acquires weak transfers for the class C_{ff} of finite flat maps in the sense of [7].

Lemma 2.1: If f: X → Y is a finite flat map in Sm_{k} then the direct image functor induces a transfer map f_{∗}: F(X) → F(Y).

Proof. The assumption on f implies that F = id_{V} ×_{k} f: V ×_{k} X → V ×_{k} Y is a finite flat map. This shows [3, Théorème 3.2.1, Corollaire 1.3.2] the direct image map F_{∗} is an exact functor on the category of coherent G-sheaves on X. In this situation it suffices to verify that F_{∗} preserves locally free sheaves. The latter statement is local, so it suffices to consider affine schemes. Given a ring R, a finite flat algebra S over R and a projective S-module P, we wish to show that P is a projective R-module. In effect, we shall verify that the functor Hom_{R}(P, −) ≃ Hom_{S}(P, Hom_{R}(S, −)) is exact. This follows by combining the standard facts that S is a projective R-module according to [6, Theorem 2.9], P is a projective S-module and the composition of two exact functors is an exact functor.

Remark 2.2: Since we require transfer maps for a very restricted class of morphisms, we may avoid using higher direct images and Tor-formulas as in Quillen’s general transfer construction [8, §7].

Secondly, we need to verify the additivity, base change, and normalization conditions formulated in [7].

Remark 2.3: For our purposes it suffices to check the base change property only for closed embeddings. Therefore, our form of this property is less general than in [7].

1. Additivity: For X = X_{0} ∐ X_{1} with corresponding embeddings i_{m}: X_{m} ↪ X for m = 0, 1 and f: X → Y a map in Sm_{k}, then

f_{∗} = (fi_{0})_{∗}i_{0}^{∗} + (fi_{1})_{∗}i_{1}^{∗}.

2. Base change: For every cartesian square

\xymatrix{X′ \ar@{^{(}->}[r]^{g̃} \ar[d]_{f̃} & Y′ \ar[d]^{f} \\ X \ar@{^{(}->}[r]^{g} & Y}

where f ∈ C_{ff} and g is a closed embedding, one has g^{∗}f_{∗} = f̃_{∗}g̃^{∗}.

3. Normalization: If f is the identity map on k, then f_{∗} = id_{F(k)}.

The direct and inverse image functors are G-maps in a natural way, so one can immediately verify that all the isomorphisms appearing in the proof of the base change property below are actually G-isomorphisms. Thus the additivity condition follows immediately by using additivity of the direct image functor.

Next we show that g^{∗}f_{∗} = f̃_{∗}g̃^{∗} for the maps in the cartesian diagram. To wit, for a locally free sheaf S on Y′, the adjunction between left adjoint inverse image functors and right adjoint direct image functors furnishes canonical elements α ∈ Hom_{Y′}(S, g̃_{∗}g̃^{∗}(S)) and β ∈ Hom_{Y′}(f^{∗}f_{∗}(S), S). Moreover, there are canonical isomorphisms

Hom_{Y′}(f^{∗}f_{∗}(S), g̃_{∗}g̃^{∗}(S)) ≅ Hom_{X′}(g̃^{∗}f^{∗}f_{∗}(S), g̃^{∗}(S))
≅ Hom_{X′}(f̃^{∗}g^{∗}f_{∗}(S), g̃^{∗}(S))
≅ Hom_{X}(g^{∗}f_{∗}(S), f̃_{∗}g̃^{∗}(S)).

The image of the composite αβ in Hom_{Y′}(f^{∗}f_{∗}(S), g̃_{∗}g̃^{∗}(S)) under these isomorphisms determines a map of sheaves

\xymatrix{α̃β: g^{∗}f_{∗}(S) \ar[r] & f̃_{∗}g̃^{∗}(S).}

Using the assumptions on f and g one verifies easily that α̃β is a fiberwise isomorphism. Hence α̃β is a sheaf isomorphism. This completes the proof of the base-change property.

Finally, the normalization condition holds trivially.

In order to finish the proof it remains to note that the maps appearing in [7, §1] belong to C_{ff}, and moreover that the base change diagrams in loc. cit. are of the type above (with respect to some closed embedding). These conditions hold according to the following elementary result.

Lemma 2.4: If X is a smooth projective curve over a field and f a non-constant rational function on X, then the corresponding map f: X → P^{1} is a member of C_{ff}.

Proof. This is a special case of [4, III, Proposition 9.7].

The proof of rigidity for rational points is now complete.

The second condition in the rigidity for rational points result serves as a “replacement” for homotopy invariance. A functor F: (Sm_{k}^{G})^{op} → Ab is homotopy invariant for X if the canonical projection map p: X ×_{k} A_{k}^{1} → X induces an isomorphism p^{∗}: F (X) → F (X ×_{k} A_{k}^{1}). (Here and below G acts trivially on the affine line A_{k}^{1}.)

Lemma 2.5: (1) If F is homotopy invariant for X, then the rational points 0 and 1 on the affine line over k yield equal pullback maps

\xymatrix{F (X ×_{k} A_{k}^{1}) \ar[r] \ar[r] & F (X).}

(2) If

\xymatrix{i_{0}^{∗} = i_{1}^{∗}: F (Y ×_{k} A_{k}^{1}) \ar[r] \ar[r] & F (Y)}

holds for Y = X ×_{k} A_{k}^{1}, then F is homotopy invariant for X.

Proof. Part (1) holds since the composite map X →^{i} Y →^{p} X is the identity. In order to prove (2), contemplate the diagram

\xymatrix{F (X ×_{k} A_{k}^{1}) \ar[r]^{μ^{∗}} & F (X ×_{k} A_{k}^{1} ×_{k} A_{k}^{1}) \ar[r]^{i_{0}^{∗}} \ar[r]_{i_{1}^{∗}} & F (X ×_{k} A_{k}^{1}).}

Here μ^{∗} is induced by the product map μ: A_{k}^{1} ×_{k} A_{k}^{1} → A_{k}^{1} while the maps i_{0}^{∗} and i_{1}^{∗} are induced by the rational points 0 and 1 on A_{k}^{1}. By hypothesis, the composite maps in the diagram coincide. The map involving i_{1}^{∗} is the identity and i_{0}μ equals the composite Y →^{p} X →^{i} Y. Hence i^{∗} is inverse to p^{∗}.

As noted in the introduction, the proof of rigidity for extensions is now complete.

Proof. (Rigidity for Hensel local rings) The Additivity and Normalization properties stated in [5] coincide with the ones in this paper, while the base change diagrams in loc. cit. are also of the type considered here (with respect to some closed embedding), cf. Remark 2.3. This shows that with our formulation of the base-change property, the approach in [5] can be adopted verbatim. In [5] one also require transfer maps (trace homomorphisms) for finite separable field extensions. It remains to note that such extensions induce maps between schemes in C_{ff}.

Acknowledgements

This work was mostly done during a stay of the first author at the Max-Planck-Institut für Mathematik (Bonn). He is deeply grateful to the institute for its hospitality and perfect working conditions. The authors wish to thank the University of Oslo for financial support.

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Steklov Mathematical Institute, St. Petersburg, Russia. University of Bielefeld, SFB-701, Bielefeld, Germany. e-mail: yagunov@gmail.com

Department of Mathematics, University of Oslo, Norway. e-mail: paularne@math.uio.no