Comparing K-theories for Complex Varieties

The semi-topological K-theory of a complex variety was defined in [FW2] with the expectation that it would prove to be a theory lying “part way” between the algebraic K-theory of the variety and the topological K-theory of the associated analytic space, and thus would share properties with each of these other theories. In this paper, we realize these expectations by proving among other results that (1) the algebraic K-theory with finite coefficients and the semi-topological K-theory with finite coefficients coincide on all projective complex varieties, (2) semi-topological Ktheory and topological K-theory agree on certain types of generalized flag varieties, and (3) (by building on a result of Cohen and Lima-Filho) the semi-topological K-theory of any smooth projective variety becomes isomorphic to the topological K-theory of the underlying analytic space once the Bott element is inverted. To illustrate the utility of our results, we observe that a new proof of the QuillenLichtenbaum conjecture for smooth, complete curves is obtained as a corollary. In the recent paper [FW-2], the authors introduced “semi-topological K-theory” K ∗ (X) for a complex quasi-projective algebraic variety X, showed that the natural map from algebraic to topological K-theory K∗(X) → K−∗ top(X) factors through this new theory, and showed that K ∗ (X) is related to FriedlanderLawson morphic cohomology L∗H∗(X) as algebraic K-theory is related to motivic cohomology and topological K-theory is related to integral singular cohomology. A few computations (projective smooth curves, projective spaces) were provided, but the general behavior of this theory remains inaccessible. In this paper, we introduce a new theory K∗(∆top ×X) which is less geometric in character but more accessible to computation. The relevance of K∗(∆top ×X) to our goal of understanding K ∗ (X) is that these theories agree whenever X is projective (and possibly in general). As we shall see, K∗(∆top ×X) admits further computations and satisfies Mayer-Vietoris for smooth varieties. More remarkable, we prove for any n > 0 the existence of natural isomorphisms for all quasi-projective varieties X K∗(X;Z/n) ' K∗(∆top ×X;Z/n). (0.1) Starting from a somewhat different point of view, R. Cohen and P. Lima-Filho have considered “holomorphic K-theory” K−∗ hol(X) of a projective algebraic variety X [CL]. This theory appears to be equivalent to K ∗ (X) when the latter is restricted to weakly normal, projective varieties. Using a result of [CL], we prove that K ∗ (X)[1/β] ' K−∗ top(X) (0.2) ∗Both authors were partially supported by the N.S.F. and the N.S.A. Typeset by AMS-TEX 1 2 ERIC M. FRIEDLANDER AND MARK E. WALKER ∗ for X a smooth, projective variety, where β ∈ K 2 (SpecC) denotes the Bott element. We conclude that our K ∗ (X) fits tightly between algebraic and topological K-theory. In the first section of this paper, we define the spectrum-valued theory K(∆top× X) whose homotopy groups are the groups K∗(∆top × X) mentioned above. We establish maps involving this new theory, algebraic K-theory, and semi-topological K-theory, and we establish the existence of the natural weak equivalence K(∆top ×X) ∼ −→ K(X) (0.3) when X is projective and weakly normal. In section two we employ (0.3) and a result of Panin [P] to conclude that the natural map K(X)→ Ktop(X) is a weak equivalence for a class of varieties which includes quotients of certain classical algebraic groups by parabolic subgroups and bundles of such over smooth, complete curves. This result can be viewed as a generalization of a stable version of results of Kirwan [K] comparing the algebraic and topological mapping spaces from Riemann surfaces to Grassmann varieties. In degree 0, this has been proven by Cohen and Lima-Filho [CL]. In section three we establish the weak equivalence (0.1) which in conjunction with (0.2) implies the weak equivalence K(X;Z/n) ∼ −→ K(X;Z/n), (0.4) for n > 0 and X a weakly normal projective variety. An interesting consequence of (0.4) in conjunction with [FW-2; 7.5] is a new proof of the Quillen-Lichtenbaum conjecture for smooth, complete curves. In section four, we use the weak equivalence (0.4), the previously mentioned result of [CL], and a result of R. Thomason [T; 4.11] to establish (0.2): “Bottinverted” semi-topological K-theory of any smooth, projective variety X coincides with the topological K-theory of X. This settles affirmatively a conjecture made in [FW-2]. Section five focuses primarily on the theory K(∆top × −). In it, we establish that the Mayer-Vietoris property for open covers is satisfied by this theory, giving long exact sequences which may prove useful for studying K(∆top ×−). Using the techniques developed in this section, we also show that the natural map K q (C)→ K −q top(C ) is an isomorphism for q ≥ 0 whenever C is a (possibly singular) complete curve. All varieties considered in this paper are quasi-projective over the complex field C. We thank Vladimir Voevodsky for his suggestion which led to our formulation of K(∆top ×X). §1. K(∆top × X) In this first section, we introduce a naturally defined Ω-spectrum K(∆top×X) for any quasi-projective variety X and verify that this Ω-spectrum is weakly equivalent COMPARING K-THEORIES FOR COMPLEX VARIETIES 3 to the Ω-spectrumK(X) of [FW-2] wheneverX is projective and weakly normal. Although we view K(X) as the primary object of interest, K(∆top ×X) proves to be a more convenient object of study for it appears to be better behaved when applied to varieties which are not projective. We begin by briefly recalling the construction of K(X). For any quasiprojective varietyX, we consider the set of continuous algebraic maps Mor(X,Grassm(P )) (i.e., morphisms from the weak normalization of X to the Grassmann variety Grassm(P ) of m+ 1 planes in C). As shown in [FW-1], this set has a natural topology; we let Mor(X,Grassm(P )) denote the resulting topological space. If X is projective, this topology has the simple description as the subspace topology of the set of all continuous maps from X to Grassm(P ) equipped with the compact-open topology. We consider the space Mor(X,Grass(P∞))an ≡ ∐ m≥0 lim −→ N Mor(X,Grassm(P )). External Whitney sum Grassm(P )×Grassm′(P ′ )→ Grassm+m′(P ′ ) determines a product on this space which is enhanced to admit the action of a certain E∞-operad I, consisting of spaces I(n), n ≥ 0. The space I(n) is the collection of n-tuples of linear maps C∞ → C∞ which induce an injection (C∞)n ↪→ C∞ (cf. [FW-1] for further details). As with any space admitting an action by an E∞-operad, we have an associated Ω-spectrum Ω∞Σ∞Mor(X,Grass(P∞))an whose 0 term is the homotopy-theoretic group completion ofMor(X,Grass(P∞))an, Mor(X,Grass(P∞))an −→ [Mor(X,Grass(P∞))an]+. This homotopy-theoretic group completion is the semi-topological K-theory space of X, K(X) ≡ [Mor(X,Grass(P∞))an]+. The space K(X) becomes somewhat more comprehensible once one observes that it is homotopy equivalent to infinite mapping telescope of a self-map α :Mor(X,Grass(P∞))an →Mor(X,Grass(P∞))an, which is defined by taken external sum with a chosen very ample line bundle O(1). We proceed to introduce K(∆top ×X) following a suggestion of V. Voevodsky. Given a compact Hausdorff space T , we write V ar for the category whose objects are continuous maps T → U, for U ∈ Sch/C (i.e., for U a quasi-projective variety). A morphism in V ar from T → U to T → V an is a morphism of complex varieties V → U causing the evident triangle to commute. Given a contravariant functor F from Sch/C to the category of sets, abelian groups, spaces, spectra, etc and given a compact topological space T , define F (T ) ≡ lim −→ (T→Uan)∈V arT F (U). 4 ERIC M. FRIEDLANDER AND MARK E. WALKER ∗ In particular, let K : (Sch/C) → (spectra) be a functor giving algebraic K-theory (for example, chosen as in [FS]). Then given a quasi-projective variety X, we define K(∆top ×X) as K(∆top ×X) ≡ lim −→ (∆top→U)∈V ar ∆d top K(U ×X), where ∆top stands for the standard d-dimensional topological simplex. Since V ar T is a directed category for any T , we have πqK(∆top ×X) ∼= Kq(∆top ×X). Definition 1.1. Let K(−) be an Ω-spectrum valued contravariant functor on (Sch/C) giving algebraic K-theory. Define K(∆top ×X) ≡ |d 7→ K(∆top ×X)|, the geometric realization of the indicated simplicial Ω-spectrum. By Kq(∆top ×X) we mean πqK(∆top ×X). We begin our analysis of K(∆top×X) by verifying the following useful property. Lemma 1.2. Let F : (Sch/C) → (spaces) be a functor. Then the natural map induced by the projection F (∆top)→ F (∆top ×∆•) induces an isomorphism in homology, where ∆• denotes the standard cosimplicial variety which in degree d is ∆ ≡ SpecC[x0, . . . , xd]/( ∑ i xi − 1). Proof. We first verify that Hr ◦ F (∆top ×−) is homotopy invariant for any r ≥ 0, arguing as in [FV; 4.1]. Let ψi : ∆ → ∆ × ∆ be the linear map sending the j vertex vj of ∆ to vj × 0 if j ≤ i and to vj−1 × 1 otherwise. Then ψi induces a continuous map ψ∗ i : F (∆ n top × ∆ × X) → F (∆ top × X) for any quasi-projective variety X, since a continuous map g : ∆top → U determines g ◦ ψ j : ∆ n+1 top → ∆top ×∆top → (U ×∆). Thus, the maps sn for n ≥ 0,