Oriented Chow groups, Hermitian K-theory and the Gersten conjecture

We show that the oriented Chow groups of Barge–Morel appear in the E ₂-term of the coniveau spectral sequence for Hermitian K-theory. This includes a localization theorem and the Gersten conjecture (over infinite base fields) for Hermitian K-theory. We also discuss the conjectural relationship between oriented and higher oriented Chow groups and Levine’s homotopy coniveau spectral sequence when applied to Hermitian K-theory.     

Topological K-Theory of Algebraic K-Theory Spectra

Let KX denote the algebraic K-theory spectrum of a regular Noetherian scheme X. Under mild additional hypotheses on X, we construct a spectral sequence converging to the topological K-theory of KX. The spectral sequence starts from the étale homology of X with coefficients in a certain copresheaf constructed from roots of unity. As examples we consider number rings, number fields, local fields, smooth curves over a finite field, and smooth varieties over the complex numbers.     

The Milnor–Chow homomorphism revisited

The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor–Chow homomorphism between Milnor K-theory and higher Chow groups for essentially smooth (semi-)local k-algebras A with infinite residue fields. It implies the exactness of the Gersten resolution for Milnor K-theory at the generic point. Our method uses the Bloch–Levine moving technique and some properties of the Milnor K-theory norm for fields. Furthermore we give new applications. Supported by Studienstiftung des deutschen Volkes and Deutsche Forschungsgemeinschaft.     

Some filtrations on higherK-theory and related invariants

We lift Bloch's higher Chow construction from the level of simplicial sets to the level of simplicial spaces. We construct a simplicial space that becomes isomorphic to the Bloch/Chow complex when the functor ₀ is applied in each degree. The homotopy groups of this space are theE ₂-terms in an Atiyah-Hirzebruch spectral sequence converging to algebraicK-theory. TheseE ₂-terms map nontrivially to the expected higher Chow groups. We define and compute several intermediate invariants associated to our simplicial space.     

Homotopy spectral sequences and obstructions

For a pointed cosimplicial spaceX ^•, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX ^•. In this paper,X ^• is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here. Supported in part by NSF Grant DMS-8602432.     

K-theory and intersection theory revisited

Another proof that the product structure on K-theory may be used to define the product structure on the Chow ring of a smooth variety over a field is presented. The virtue of this proof is that it is essentially a formal argument using natural properties of Quillen's spectral sequence, the K-theory product, cycle classes, and the classical intersection product.     

The spectral sequence relating algebraic K-theory to motivic cohomology

Beginning with the Bloch-Lichtenbaum exact couple relating the motivic cohomology of a field F to the algebraic K-theory of F, the authors construct a spectral sequence for any smooth scheme X over F whose E₂ term is the motivic cohomology of X and whose abutment is the Quillen K-theory of X. A multiplicative structure is exhibited on this spectral sequence. The spectral sequence is that associated to a tower of spectra determined by consideration of the filtration of coherent sheaves on X by codimension of support.     

A Quillen-type spectral sequence for the K-theory of varieties with isolated singularities

We construct a spectral sequence to compute the algebraic K-theory of any quasiprojective scheme X, when X has isolated singularities, using an explicit flasque resolution of the K-theory sheaves. This is a generalization of Quillen's construction for nonsingular varieties. The explicit resolution makes it possible to relate K-theory to intersection theory on singular schemes.Partially supported by NSF grants.Dedicated to A. Grothendieck on his sixtieth birthday     

The Farrell–Hsiang method revisited

We present a sufficient condition for groups to satisfy the Farrell–Jones Conjecture in algebraic K-theory and L-theory. The condition is formulated in terms of finite quotients of the group in question and is motivated by work of Farrell–Hsiang.     

Semi-Topological K-Homology and Thomason''s Theorem

In this paper, we introduce the 'semi-topological K-homology' of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasi-projective complex variety Y coincides with the connective topological K-homology of the associated analytic space Y ^an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the 'Bott inverted' semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X ^an. In combination with a result of Friedlander and the author, this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason's celebrated theorem that 'Bott inverted' algebraic K-theory with /n coefficients coincides with topological K-theory with /n coefficients.     

Sur la catégorie Db,G(X) pour G réductif fini

In this Note, we are interested in the G-equivariant derived category of a smooth projective scheme over an algebraically closed field k, on which a reductive finite group G is acting. We compare the G-equivariant derived category of X with the derived category of the quotient by giving a descent criterion. The result generalizes a theorem of Lønsted in G-equivariant K-theory on curves (K. Lønsted, J. Math. Kyoto Univ. 23 (4) (1983) 775–793). We also give an equivariant version of Be??linson's equivalence of categories (Funct. Anal. Appl. 12 (1979) 214–216) and treat the exemple of the projective line. To cite this article: S. Térouanne, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Higher <Emphasis Type="Italic">K</Emphasis>-theory of group-rings of virtually infinite cyclic groups

 F.T. Farrell and L.E. Jones conjectured in [7] that Algebraic K-theory of virtually cyclic subgroups V should constitute `building blocks' for the Algebraic K-theory of an arbitrary group G. In [6], they obtained some results on lower K-theory of V. In this paper, we obtain results on higher K-theory of virtually infinite cyclic groups V in the two cases: (i) when V admits an epimorphism (with finite kernel) to the infinite cyclic group (see 2.1 and 2.2(a),(b)) and (ii) when V admits an epimorphism (with finite kernel) to the infinite dihedral group (see 3.1, 3.2, 3.3). Received: 18 April 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 19D35, 16S35, 16H05.     

A Descent Theorem in Topological K-Theory

We prove the Lichtenbaum–Quillen conjecture in the topological context: in other words, real K-theory can be deduced from complex K-theory via the usual descent spectral sequence. More precise results are proved, however, and new applications are stated. The main ingredients in the proof are Atiyah's KR-theory and the definition of higher K-groups via Clifford algebras.     

Gersten conjecture for equivariant K-theory and applications

For a reductive group scheme G over a regular semi-local ring A, we prove the Gersten conjecture for the equivariant K-theory. As a consequence, we show that if F is the field of fractions of A, then K^G₀(A) @ K^G₀(F){K^G_0(A) \cong K^G_0(F)}, generalizing the analogous result for a dvr by Serre (Inst Hautes études Sci Publ Math 34:37–52, 1968). We also show the rigidity for the K-theory with finite coefficients of a Henselian local ring in the equivariant setting. We use this rigidity theorem to compute the equivariant K-theory of algebraically closed fields.     

The Bass Conjecture and Group von Neumann Algebras

The precise relationship between the Bass conjecture for the Hattori–Stallings trace for projective ZG-modules and the map from reduced K-theory of ZG to reduced K-theory of the von Neumann algebra, NG, of G, of G is determined. As a consequence it is shown this map is zero for all groups G. It is also shown that the map induced on K-theory from the inclusion of NG to the ring of closed, densely defined operators affiliated to NG is an isomorphism. Together with the above result, this gives some positive evidence for the validity of the Division Ring Conjecture for torsion free groups.     

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ _K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q. Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999     

MilnorK-theory is the simplest part of algebraicK-theory

We identify the MilnorK-theory of a field with a certain higher Chow group.     

On A(X) → TC(X) for some X

Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic trace map K(ℤ[C_p])→ TC(ℤ[C_p]).     

Techniques,computations, and conjectures for semi-topological <Emphasis Type="Italic">K</Emphasis>-theory

We establish the existence of an Atiyah-Hirzebruch-like spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.Mathematics Subject Classification (2000): 19E20, 19E15, 14F43The first author was partially supported by the NSF and the NSAThe second author was supported by the Helen M. Galvin Fellowship of Northwestern UniversityThe third author was partially supported by the NSF and the NSA     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].