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Algebraic K-Theory and Etale Cohomology

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Algebraic K-Theory and Etale Cohomology ANNALES SCIENTIFIQUES DE L’É.N.S. R. W. THOMASON Algebraic K-theory and etale cohomology Annales scientifiques de l’É.N.S. 4e série, tome 18, no 3 (1985), p. 437-552 <http://www.numdam.org/item?id=ASENS_1985_4_18_3_437_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1985, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scient. EC. Norm. Sup., 46 serie, t. 13, 1980, p. 437 a 552. ALGEBRAIC K-THEORY AND ETALE COHOMOLOGY BY R. W. THOMASON Let F be a prime power, and let X be a separated regular noetherian scheme in which / is invertible. If (9^ contains a primitive Fth root of unity, there is canonical element P in the second algebraic K-group of X with coefficients Z/F, K//; (X), which Bocksteins to the primitive root as an F torsion class in Ki(X). The graded ring K//»(X) may be localized by inverting this P. Under a few mild extra hypotheses on X, the main Theorem 4.1 yields a strongly converging Atiyah-Hirzebruch type spectral sequence that computes K/ft(X) [p~1] in terms of the etale cohomology of X: (o.i, ^•-{"u\z'^q•2'} - K/^X)^-]. The result may be reformulated in terms of the etale or topological K-theory of Dwyer and Friedlander as giving an isomorphism between the localized algebraic and topological K-groups as in Theorem 4.11: (0.2) K/^ (X) [p -1] ^ K/l^ (X). The result holds even without the assumption that ^x contains Fth roots of unity, when P is defined as in Appendix A. There is a variant of the result for singular schemes in terms of algebraic G-theory and topological K-homology as in 2.48, 4.15-4.16. The result is quite remarkable in that it expresses a deep and subtle link between algebraic geometry and the topology of varieties. The groups K/^(X) are defined in terms of the category of algebraic vector bundles on X, and reflect the delicate algebraic- geometric structure of X. They carry much subtle information about intersection theory on X, and on Euler characteristics in coherent cohomology of algebraic vector bundles on X. On the other hand, K/fiic^X) is a much cruder invariant depending only on the underlying topology of X. As X runs through the moduli of K3 surfaces over the complex numbers, the rank of the image of K/^(X) in K/^^X) under the forgetful map is known to take on all values from 3 to 22, despite the fact that all such surfaces are diffeomorphic ([103], IX). Nothing in the definition of K/ft(X) involves or evokes the etale topology, or leads one to expect that etale cohomology can be constructed out of K//*(X). For varieties over the complex numbers, K/ft^X) can be defined in a ANNALES SCIENTIFIQUES DE L'feCOLE NORMALE SUPERIEURE. - 0012-9593/85/03437 116/S 13.60 © Gauthier-Villars 438 R. w. THOMASON manner parallel to K/l^ (X), with the category of topological vector bundles playing the role of algebraic vector bundles. However, the two categories of vector bundles are quite different: not every topological vector bundle is algebraizable, algebraic vector bundles may be isomorphic as topological bundles without being isomorphic algebraically, and not every short exact sequence of algebraic vector bundles splits algebraically, though it must split topologically. Thus K/;*(X) and K/fii^X) look quite different. However, the theorem says that they are also quite alike, in that they differ only by P-torsion, hence only in "codimension at least one". As inverting p takes a direct limit over groups in higher degrees as in (0. 3), (0.3) K/^Xnp-^hm^K/^X) ^K/^p(X) ^ . ..), the Theorem says that algebraic K-theory asymptotically approaches topological K- theory in high degrees. It also says that the category of algebraic vector bundles on X knows about the non-algebraizable topological vector bundles. There are many applications of the main result. As etale cohomology is usually easy to calculate, it is usually possible to calculate the groups K/ft(X)[p~1], which are at least close to K//* (X) if not identical to it. Examples for curves, semi-simple algebraic groups, and smooth hypersurfaces in projective space are given in paragraph 4, along with a few arithmetic examples. As another application, it is possible to use Theorem 4.1 to ^ show directly that K/^(X) [p~1] has all the formal properties used in applications of the Dwyer-Friedlander topological K-theory. Thus it may be used to replace this construction for regular schemes, and so avoid the gruesome technicalities of etale homotopy theory. Theorem 4.11 shows that this gives exactly the Dwyer-Friedlander groups. If/: X -> Y is a proper map between regular varieties, there is an obvious commutative diagram K//*(X)^K/r*(X)[P-1] I/ I/ (0.4) + • t ' K/^(Y)^K/R,(Y)[P-1] The groups on the right of (0.4) can be computed via Theorem 4.1. This makes it possible to compute the map/* on the right of (0.4). This/* may be identified to the Gysin map in topological K-theory. Thus (0.4) solves the Riemann-Roch problem as a variant of Grothendieck's Riemann-Roch Theorem. For a generalization to singular varieties and a fuller discussion, see 4.16-4.17. This generalizes those higher Riemann- Roch theorems of Gillet and of Shekhtman that deal with the Chern character from algebraic K-theory to Q^ etale cohomology. An application related to the above is my proof of Grothendieck's absolute cohomologi- cal purity conjecture for Q^ etale cohomology, as discussed in 4.18. 4® SERIE — TOME 18 — 1985 — N° 3 ALGEBRAIC K-THEORY 439 Known connections between zeta functions and etale cohomology may be reformulated in terms of K/^(X) [p~1] thanks to 4.1. This is important for calculations in arithmetic cases as in 4.7 and 4.8. It also sheds a bit of light on the results and conjectures of Beilinson, Coates, Lichtenbaum, Mazur, Soule, and Wiles. Most of my results concern the mod / algebraic K-groups introduced by Browder. As in A. 5, there is a universal coefficient sequence (0.5) 0 ^ K, (X) (x) Z/F ^ K//; (X) ^ r-torsion in K, _, (X) ^ 0. See A. 12 for the ?-adic version. From this one sees that most of the information in K*(X) is encoded in the /-adic K*(X)^ or in the system of K/^(X) as v increases: only the uniquely /-divisible subgroups of K*(X) are irretrievably lost. There is good reason to lose something, for it is well-known that the groups K* (X) violate Lefschetz's principle and that Mayer-Vietoris for closed covers and the homotopy axiom fail for singular X. Suslin has recently shown that the groups K/^ (X) satisfy Lefschetz's princi- ple [117], and Weibel has proved Mayer-Vietoris and the homotopy axiom for K//*(X) and singular affine X [139]. Thus these pathologies disappear mod l\ Similarly, etale cohomology exhibits pathology unless restricted to torsion coefficient sheaves. Suppose X is a projective variety over the complex numbers. Then the classical topology allows one to define an integral Kl^X), and there is a Dwyer-Friedlander or forgetful map to it from K*(X). Using Hodge theory, Gillet ([44], 5.5) has shown that this map has torsion image in degrees above 0. As K^X) is often torsion-free, this map is often zero in positive degrees. In contrast, the map from K//*(X) to K/^^X) is a localiza- tion by 4.1, and so is highly non-trivial. In fact, a more delicate version of 4.1 shows that this map is surjective in sufficiently high degrees [129]. In integral terms, this means that the copies of Z in K^X) for large n correspond not to Z's in K^(X), but rather via the universal coefficient theorem to torsion groups Q/Z in K^_i(X). This is like the classical relation of H1 ( ; Z) of a curve to the Tate module of torsion points on the Jacobian of the curve. This relation is much easier to see working mod lv. However, a rather messy integral form of the key descent theorem is given in 2. 50, and a simple rational descent Theorem is proved in 2.15-2.18. The general outline of the proof of the main Theorem is this: First, the machinery of homological algebra must be generalized to a "homotopical algebra" that applies not only to chain complexes, but to the spaces and spectra that occur in Quillen's definition of algebraic K-theory. The foundation for this was laid by Puppe and Quillen, and a superstructure is built on it in paragraphs 5 and 1. There is a close analogy between ordinary homological algebra and this generalization. The analog of a chain complex is a spectrum in the sense of algebraic topology. The analogs of homology groups of a chain complex are homotopy groups of a spectrum. Short exact sequences of complexes correspond to fibration sequences of spectra; they yield long exact sequences of homology or homotopy groups.
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