Algebraic Ktheory

General Introduction
See also Abstract Ktheory and much more.
See also Motivic cohomology
See also Motivic homotopy theory
<]]> 
Search results
<]]> 
Online References
Weibel's book project
Lectures notes of Schlichting
Weibel's Ktheory handbook chapter: K0691
Short course by Levine
Gillet: Ktheory and Intersection theory. Treats relation to Chow groups, coniveau filtration, Gersten's conjecture and more. Also the stable homotopytheoretic viewpoint on algebraic Ktheory, and Ktheory as sheaf hypercohomology, and other basic things. Extremely nice article!
A survey by Arlettaz: Algebraic Ktheory of rings from a topological viewpoint
Garkusha on a general construction recovering KaroubiVillamayor and Quillen Ktheory for rings.
The Development of algebraic Ktheory before 1980, by Charles A. Weibel: K0343
Opérations sur la Kthéorie algébrique et régulateurs via la théorie homotopique des schémas, by Joël Riou
Some notes by Rosenberg on relations with topology and analysis.
Something on the plus constuction by Berrick
Weibel: The development of algebraic Ktheory before 1980
Brown and Gersten: Algebraic Ktheory as generalized sheaf cohomology (1973)
For and of a variety, see de Jong's notes on Algebraic de Rham cohomology
Thomason's ICM talk 1990: The local to global principle in algebraic Ktheory.
<]]> 
Paper References
See Kuku in Handbook of algebra Vol 4, for a survey of a number of different types of algebraic Kth including Volodin Kth.
Bass: Algebraic Ktheory (1968) in Kth folder. Treats K0 and K1 or rings. Background on rings and projective modules. Reciprocity laws (ch VI and XIII). Finiteness thms for rings of arithmetic type (Ch X).
Read Quillen\'s original paper!! Here is a review/summary. See also the sequel by Grayson. There is also a survey article by Swan which seems to summarize much of Quillen\'s paper, possibly taking into account later developments.
Geisser in the Ktheory handbook is excellent.
Grayson has a survey in the Motives volumes.
ThomasonTrobaugh: Higher algebraic Ktheory of schemes and of derived categories. In the Grothendieck Festschrift, Vol III.
Milnor: Intorduction to algebraic Ktheory (1971)
Rosenberg: Algebraic Ktheory and its applications.
Soulé et al: Lectures on Arakelov Geometry: First chapter treats Chow groups, with product induced from isomorphism with Kgroups. Good quick review of algebraic Ktheory.
Many papers by Gillet, including stuff on Chern classes in a very general setting.
Grayson Hangzhou lectures in Kth folder: Brief intro to algebraic Ktheory, very readable, with end remarks on motivic cohomology.
Handbook of Kth?
Friedlander's ICTP lectures in Kth folder: Very nice introduction to algebraic Ktheory.
Levine Morelia lecture notes: Basic intro to algebraic Ktheory of rings and schemes.
Manin: Lectures on the Kfunctor in algebraic geometry. In Kth folder. Among other things, he covers monoidal transformations and RiemannRoch.
Milnor: Introduction to algebraic Kth (in Kth folder). Covers mainly K2, note that these notes came before Quillen's work.
Rosenberg in Kth folder: Basic intro to algebraic Ktheory of rings: concrete approaches to K0, K1, K2, negative Ktheory, plus and Q constructions, cyclic homology.
Srinivas: Algebraic Kth book, Kth folder. Comprehensive introduction to algebraic Ktheory of rings and schemes. Advanced topics include the MerkurjevSuslin thm and localization for singular varieties.
Weibel: The Kbook. Looks like a very good introduction to algebraic Ktheory. See his webpage for the latest version, I have a version from Sep 2011.
<]]> 
Definition
First define Kgroups of an exact cat. Then the cat of locally free sheaves of finte rank give Kgroups, and if is noetherian, the cat of coherent sheaves gives Ggroups. If is regular, the two kinds of groups coincide.
From the definition, Kgroups are contravariant functors from schemes to abelian groups. Ggroups are contravariant for flat maps only.
For Ktheory of rings, one brief introduction is BurgosGil, chapter 9. Using the CartanSerre theorem, can obtain information about the Kgroups of by studying the homology of ("the homology of an Hspace is a Hopf algebra").
The 5 papers by Neeman on Ktheory of triangulated categories: K0507, K0508, K0509, K0510, K0511. Perhaps the Ktheory handbook is a better source for this material.
On Voevodsky's algebraic Ktheory spectrum BGL, by Ivan Panin, Konstantin Pimenov, and Oliver Roendigs: Under a certain normalization assumption we prove that the Voevodsky's spectrum BGL which represents algebraic Ktheory is unique over the integers. Following an idea of Voevodsky, we equip the spectrum BGL with the structure of a commutative ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over the integers. We pull this structure back to get a distinguished monoidal structure on BGL for an arbitrary Noetherian base scheme. K0838
http://mathoverflow.net/questions/1006/motivationinterpretationforquillensqconstruction
<]]> 
Properties
Charles A. Weibel, Homotopy algebraic $K$theory (pp. 461488) (1987, some proceedings)
http://mathoverflow.net/questions/984/algebraicktheoryandtensorproducts
arXiv:1301.3815 Universality of KTheory fra arXiv Front: math.AG av José Luis González, Kalle Karu We prove that graded Ktheory is universal among oriented BorelMoore homology theories with a multiplicative periodic formal group law.
<]]> 
Standard theorems
See this review for different views on the localisation sequence
<]]> 
Open Problems
<]]> 
Connections to Number Theory
Kolster: Ktheory and arithmetic (Kth folder). Notes on zeta values of rings of integers, algebraic Kgroups, etale cohomology, Iwasawa theory, motivic cohomology, Vandiver's conjecture.
Interesting paper by Dwyer and Mitchell.
See all papers by Morrow for higher local fields and arithmetic surfaces. For example: arXiv:1211.1533 Ktheory of onedimensional rings via proexcision fra arXiv Front: math.KT av Matthew Morrow This paper studies "proexcision" for the Ktheory of onedimensional (usually semilocal) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite characteristic, give the first results towards Geller's conjecture in mixed characteristic, and we establish various finiteness results for the Kgroups of singularities (covering both orders in number fields and singular curves over finite fields).
http://mathoverflow.net/questions/10204/anyreasonwhyk23zhasorder65520
For Ktheory of (rings of integers of) global fields, see Weibel's survey in the Handbook of Ktheory. Here is something about nontorsion elements. Here is something about even Kgroups of Q and relations to cyclotomic conjectures, by Banaszak and Gajda
arXiv:1002.2936 Splitting in the Ktheory localization sequence of number fields from arXiv Front: math.KT by Luca Caputo Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence to split: these conditions involve coinvariants of twisted pparts of the pclass groups of certain subfields of the fields F(\mu{p^n}) for n\in N. We also compare our conditions with the weaker condition WK^{et}_{2i}(F)=0 and give some example.
MR1760901 (2001i:11082) Bloch, Spencer J.(1CHI) Higher regulators, algebraic $K$theory, and zeta functions of elliptic curves.
arXiv:0910.4005 The extended Bloch group and algebraic Ktheory from arXiv Front: math.KT by Christian K. Zickert We define an extended Bloch group for an arbitrary field F, and show that this group is canonically isomorphic to K3^ind(F) if F is a number field. This gives an explicit description of K3^ind(F) in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3manifold with finite volume and invariant trace field k has a fundamental class in K_3^ind(k) tensor Z[1/2].
arXiv:1101.5477 On special elements in higher algebraic Ktheory and the LichtenbaumGross Conjecture from arXiv Front: math.KT by David Burns, Herbert Gangl, Rob de Jeu We conjecture the existence of special elements in odd degree higher algebraic Kgroups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of Artin Lfunctions of finite dimensional complex representations. We prove this conjecture in certain important cases and also provide other evidence (both theoretical and numerical) in its support.
arXiv:1208.2137 Wild Kernels and divisibility in Kgroups of global fields from arXiv Front: math.NT by Grzegorz Banaszak In this paper we study the divisibility and the wild kernels in algebraic Ktheory of global fields $F.$ We extend the notion of the wild kernel to all Kgroups of global fields and prove that QuillenLichtenbaum conjecture for $F$ is equivalent to the equality of wild kernels with corresponding groups of divisible elements in Kgroups of $F.$ We show that there exist generalized Moore exact sequences for even Kgroups of global fields. Without appealing to the QuillenLichtenbaum conjecture we show that the group of divisible elements is isomorphic to the corresponding group of \' etale divisible elements and we apply this result for the proof of the $lim^1$ analogue of QuillenLichtenbaum conjecture. We also apply this isomorphism to investigate: the imbedding obstructions in homology of $GL,$ the splitting obstructions for the Quillen localization sequence, the order of the group of divisible elements via special values of $\zeta_{F}(s).$ Using the motivic cohomology results due to Bloch, Friedlander, Levine, Lichtenbaum, Morel, Rost, Suslin, Voevodsky and Weibel, which established the QuillenLichtenbaum conjecture, we conclude that wild kernels are equal to corresponding groups of divisible elements.
<]]> 
Computations and Examples
[arXiv:1207.2225] Ktheory of toric varieties revisited fra arXiv Front: math.AT av Joseph Gubeladze After surveying higher Ktheory of toric varieties, we present Totaro's old (c. 1997) unpublished results on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric varieties. Apart from its independent interest, in retrospect, Totaro's observations motivated some (old) and complement other (very recent) results. We also offer a conjecture on the nilgroups of affine monoid, extending the nilpotence property. The conjecture holds true for K_0.
See Rognes for a computation in which he mentions Ktheory as an infinite loop space.
Geisser and Levine: The Ktheory of fields in characteristic p (Invent. Math., 2000)
Oriented Cohomology and Motivic Decompositions of Relative Cellular Spaces , by Alexander Nenashev and Kirill Zainoulline
For the Ktheory of finite fields, see Jardine.
For tori and toric varieties, see Panin and Merkurjev
Something on henselization of local ring: K113
Higher Ktheory of complex surfaces: K0328
KTheory of nonlinear projective toric varieties, by Thomas Huettemann: K0752
Panin: On a theorem of Hurewicz and Ktheory of complete DVRs.
Daniel R. Grayson, On the $K$theory of fields (pp. 3155) (1987, some proceedings)
Panin, I. A. Algebraic $K$theory of Grassmannian manifolds and their twisted forms. (Russian) Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 7172, translation in Funct. Anal. Appl. 23 (1989), no. 2, 143144.
See many articles by Hesselholt, Angeltveit, Gerhardt, also more recent than the Kth handbook.
Articles by Weibel et al using cdh techniques?
arXiv:1006.3413 Algebraic Ktheory of the first Morava Ktheory from arXiv Front: math.KT by Christian Ausoni, John Rognes We compute the algebraic Ktheory modulo p and v_1 of the Salgebra ell/p = k(1), using topological cyclic homology.
arXiv:1101.1866 On the algebraic Ktheory of Z/p^n from arXiv Front: math.KT by Vigleik Angeltveit We study the algebraic Ktheory groups of the ring Z/p^n using the cyclotomic trace map to the topological cyclic homology spectrum TC(Z/p^n). We prove that Kq(Z/p^n) is finite for all n \geq 2 and q \geq 1 and that the order satisfies K{2i1}(Z/p^n)/K{2i2}(Z/p^n)=p^{(n1)i}(p^i1)$ for all i \geq 2. We also determine the group Kq(Z/p^n) for all n \geq 2 and q \leq 2p2.
We approach TC(Z/p^n) by filtering Z/p^n by powers of p and studying several spectral sequences related to this filtration.
<]]> 
History and Applications
<]]> 
Some Research Articles
arXiv:0907.2710 Algebraic Ktheory, A^1homotopy and RiemannRoch theorems from arXiv Front: math.AG by Joël Riou. In this article, we show that the combination of the constructions done in SGA 6 and the A^1homotopy theory naturally leads to results on higher algebraic Ktheory. This applies to the operations on algebraic Ktheory, Chern characters and RiemannRoch theorems.
Gabber: Ktheory of Henselian local rings and Henselian pairs (1992)
Thomason: Bott stability in algebraic Ktheory (1986)
Panin: The Hurewicz theorem and Ktheory of complete discrete valuation rings (1986)
Landsburg: Some filtrations on higher Ktheory and related invariants (1992)
Knudson on the Ktheory of elliptic curves
Kahn on reciprocity laws
On the Ktheory of local fields, by Lars Hesselholt and Ib Madsen
Detecting Ktheory by cyclic homology, by Wolfgang Lueck and Holger Reich K0753
kinvariants for Ktheory of curves over global fields, by Dominique Arlettaz and Grzegorz Banaszak: K0770
Berrick and Casacuberta on a universal space for the plus construction. See also K0314
The FarrellJones isomorphism conjecture for finite covolume hyperbolic actions and the algebraic Ktheory of Bianchi groups, by Ethan Berkove, F. Thomas Farrell, Daniel JuanPineda, and Kimberly Pearson: K0288
Hughes and Prassidis "We formulate and proove a geometric version of the Fundamental Theorem of Algebraic KTheory which relates the Ktheory of the Laurent polynomial extension of a ring to the Ktheory of the ring."
Weibel and Pedrini: Higher Ktheory of complex varieties, and The higher Ktheory of Real Curves: K0429
Twoprimary algebraic Ktheory of pointed spaces, by John Rognes: K0236
Algebraic Ktheory of topological Ktheory, by Christian Ausoni and John Rognes: K0405
Huettemann: Algebraic KTheory of NonLinear Projective Spaces
On the Ktheory of complete regular local F_palgebras, by Thomas Geisser and Lars Hesselholt: K0435
Some remarks concerning modn Ktheory, by Eric M. Friedlander and Mark E. Walker: K0451
On the Ktheory of stable generalized operator algebras, by Hvedri Inassaridze and Tamaz Kandelaki: K0497
Ktheory of semilocal rings with finite coefficients and étale cohomology, by Bruno Kahn
Anderson, Karoubi, Wagoner: Relations between higher algebraic Ktheories. (1973). In LNM 341.
Weibel: A Quillentype spectral sequence for the Ktheory of varieties with isolated singularities (1988). Abstract: We construct a spectral sequence to compute the algebraic Ktheory of any quasiprojective scheme X, when X has isolated singularities, using an explicit flasque resolution of the Ktheory sheaves. This is a generalization of Quillen's construction for nonsingular varieties. The explicit resolution makes it possible to relate Ktheory to intersection theory on singular schemes. Key words: Quasiprojective scheme  spectral sequence  sheaves  homotopy Cartesian cubes
Suslin: On the Ktheory of local fields.
Bass: Algebraic Ktheory
Suslin and Wodzicki: Excision in algebraic Ktheory (1992)
J. F. Jardine, The homotopical foundations of algebraic $K$theory (pp.\ 5782) (1987)
Bloch: The Postnikov tower in algebraic Ktheory
arXiv:1009.3235 On the Algebraic Ktheory of Monoids from arXiv Front: math.KT by Chenghao Chu, Jack Morava Let $A$ be a not necessarily commutative monoid with zero such that projective $A$acts are free. This paper shows that the algebraic Kgroups of $A$ can be defined using the +construction and the Qconstruction. It is shown that these two constructions give the same Kgroups. As an immediate application, the homotopy invariance of algebraic Ktheory of certain affine $\mathbb{F}1$schemes is obtained. From the computation of $K2(A),$ where $A$ is the monoid associated to a finitely generated abelian group, the universal central extension of certain groups are constructed.
<]]> 
Other Information
Henri Gillet  University of Illinois, Chicago Title: KCorrespondences and complexes of Motives Abstract: I shall discuss how to use algebraic Ktheory to construct an enrichment of the category of varieties over the category of chain complexes of rational vector spaces, and how to use this to define maps between the homological weight complexes of singular varieties (such as pull back with respect to morphisms of finite tordimension). (Joint work with C. Soulé)
"Kth is not an invariant of the triangulated cat, but it is an invariant of the inftyone cat. See example by Schlichting I think, and observation of ToenVezzosi."
http://mathoverflow.net/questions/11404/whatisaneulersystemandthemotivationforit
http://mathoverflow.net/questions/39499/whenisthektheorypresheafasheaf
From http://www.maths.soton.ac.uk/pure/researchabstract.phtml?keyword=Ktheory
The higher algebraic Ktheory of geometric objects such as algebraic varieties or, more generally, schemes over Noetherian rings, was contructed by Quillen (circa 1970). Almost immediately exciting applications of algebraic Ktheory we found in algebraic geometry by Bloch, who showed how to recover the Chow groups of a variety from its algebraic Ktheory sheaf. Generalised to a form called the BlochOgusGabber Theorem, the discovery of Quillen and Bloch has become
motivic complexes'' which would, for algebraic varieties in characteristic zero, unify algebraic Kgroups, Chow groups, singular cohomology, de Rham cohomology as well as the cohomology theories constructed for schemes by Grothendieck and his school in the 1960's. The existence of various categories of higher Chow groups''  sometimes known as ``motivic cohomology''. Then Suslin and Voevodsky used the BlochLichtenbaum spectral sequence to calculate the Ktheory of curves and surfaces over an algebraically closed field. Finally Voevodsky (c.1997) devised an entirely new construction of motivic cohomology which enabled him to determine the $2$adic part of the Ktheory of any field admitting the resolution of singularities. Voevodsky's work establishes (at the prime $p=2$) the 1973 conjecture of Quillen and Lichtenbaum, which relates Ktheory with mod $p^{n}$ coefficients to Grothendieck's \'{e}tale cohomology with mod $p^{n}$ coefficients. Incidentally, Snaith and his collaborators proved the surjectivity half of the QuillenLichtenbaum conjecture for general smooth, projective schemes (Inventiones 1982). In arithmeticalgebraic geometry one tries to relate all these things to algebraic geometry coming from curves over number fields  for example, Wiles' work is in this area. Snaith has worked on the connections between Galois actions on motivic objects  the central one being given by Kgroups  and special values of Lfunctions. This is an extremely active area and the maojor open questions are: the BirchSinnertonDyer conjecture, the BrumerStark conjecture, the Lichtenbaum conjecture, the QuillenLichtenbaum conjecture, the Beilinson conjectures, the Kato conjecture, the Tate conjecture, the Hodge conjecture, the CoatesSinnott conjecture and one of mine  the ChinburgSnaith conjecture concerning the "Wiles unit". Snaith uses connective topological Ktheory to study Chow groups in arithmeticalgebraic geometry as above and to study the famous problem of the existence/nonexistence of framed manifolds of ArfKervaire invariant one (a problem that is the natural successor to JF Adams' "Hopf invariant one" work.Unrecognised command "\'"one of the fundamental results of modern algebraic geometry''. For example, the connection between algebraic Ktheory and Chow groups led Soul\'{e} and Beilinson to make a number of very refined, precise conjectures about the existence of<]]> 
Comments Posted
<]]> 
Comments
There are no comments.