K-Theory of Surfaces at the Prime 2

We prove that for smooth surfaces over real closed fields, and a class of smooth projective surfaces over a real number field, the map between mod 2 algebraic and étale K-theory is an isomorphism in sufficiently large degrees. For a class of smooth projective surfaces over a real closed field, including rational surfaces, complete intersections and K3-surfaces over the real numbers, we explicate the abutment of the mod 2 motivic cohomology to algebraic K-theory spectral sequence.     

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.     

On a spectral sequence for equivariant K-theory

We apply the “homotopy coniveau” machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin–Graham) and an Atiyah–Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch–Lichtenbaum and Friedlander–Suslin. The first-named author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and support of the NSF via grants DMS-0140445 and DMS-0457195.     

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.     

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.     

Adams Operations for Bivariant K-Theory and a Filtration Using Projective Lines

We establish the existence of Adams operations on the members of a filtration of K-theory which is defined using products of projective lines. We also show that this filtration induces the gamma filtration on the rational K-groups of a smooth variety over a field of characteristic zero.     

The K-Theory of C *-Algebras with Finite Dimensional Irreducible Representations

We study the K-theory of unital C^*-algebras A satisfying the condition that all irreducible representations are finite and of some bounded dimension. We construct computational tools, but show that K-theory is far from being able to distinguish between various interesting examples. For example, when the algebra A is n-homogeneous, i.e., all irreducible representations are exactly of dimension n, then K_*(A) is the topological K-theory of a related compact Hausdorff space, this generalises the classical Gelfand-Naimark theorem, but there are many inequivalent homogeneous algebras with the same related topological space. For general A we give a spectral sequence computing K_*(A) from a sequence of topological K-theories of related spaces. For A generated by two idempotents, this becomes a 6-term long exact sequence.     

The Higher K-Theory of Complex Varieties

Let X be a smooth complex variety, and let F be its function field. We prove that (after localizing at the prime 2) the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite. We also describe the torsion in the K-groups of F and X.     

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.     

A Note on K-Theory of Azumaya Algebras

For an Azumaya algebra A which is free over its centre R, we prove that K-theory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that K _i (A, ?/m) = K _i (R, ?/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semilocal rings, and K-theory of graded central simple algebras indexed by a totally ordered abelian group.     

K-Theory of Algebraic Tori and Toric Varieties

It is proved that under certain conditions the group K _n (X) of a smooth projective variety X over a field F is a natural direct summand of K _n (A) for some separable F-algebra A. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the groupK ₀(T) for an algebraic torus T is given.     

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 Higher K-Theory of a Complex Surface

Let X be a smooth complex variety of dimension at most two, and let F be its function field. We prove that the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite. We also describe the torsion in the K-groups of F and X.     

Symmetric K-Theory Spectra of C*-Categories

We define K-theory groups and symmetric K-theory spectra associated to ₂-graded C ^*-categories and show that the exterior product of K-theory groups can be expressed in terms of the smash product of symmetric spectra.     

Bases for Cooperations in K-Theory

Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K _*(K) of cooperations in K-theory and for the invariants under conjugation.     

On Cyclotomic Numbers and the Reduction Map for the K-Theory of the Integers

We apply the recently proven compatibility of Beilinson and Soulé elements in K-theory to investigate density of rational primes p, for which the reduction map K _2n+1() K_{2n+1}(F_p)is nontrivial. Here n is an even, positive integer and F_p denotes the field of p elements. In the proof we use arithmetic of cyclotomic numbers which come from Soulé elements. Divisibility properties of the numbers are related to the Vandiver conjecture on the class group of cyclotomic fields. Using the K-theory of the integers, we compute an upper bound on the divisibility of these cyclotomic numbers.     

Nondurable K-Theory Equivalence and Bousfield Localization*

In this paper we consider what happens when Adams self maps are modified by adding certain unstable maps. The unstable maps which are added are trivial after a single suspension. We can choose the modification so that the maps are still K-theory equivalences but the loops on the map are no longer K-theory equivalences. As a corollary we note that the maps are K-theory equivalences but not v ₁-periodic equivalences. Another consequence is the behavior of the cobar spectral sequences for generalized homology theories. Tamaki shows that in certain cases a cobar-type spectral sequence for generalized homology theories is well behaved. The maps we construct give an example where despite the connectivity of the spaces the cobar spectral sequence is still poorly behaved. Finally we use our maps to construct spaces whose Bousfield class is distinct from the cofiber of the Adams map but which becomes the same after one suspension.     

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.     

K-Theory and analytic isomorphisms

Ifs is a central nonzerodivisor of a ringA, letB denote thes-adic completion ofA. By a theorem of Karoubi, there is a long exact sequence relating theK-theory ofA, B, A[s ^–1], andB[s ^–1]. This sequence was first exploited by Vorst in his thesis. We give two applications of the Karoubi sequence: (1) an example of a 2-dimensional normal domain withNK ₀0, answering a question of Murthy, and (2) a complete computation ofK ₂ of an (affine) seminormal curve over an algebraically closed field.Supported by NSF Grant MCS-79-03537     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite coefficients.