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Relation between Chow groups and K theory

I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence $$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
KAK's user avatar
  • 613
3 votes
2 answers
342 views

Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism

I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map. More precisely, I'...
kindasorta's user avatar
  • 2,907
1 vote
0 answers
124 views

Computing the induced homomorphisms of derived functors using acyclic resolutions

Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
Benjamin Steinberg's user avatar
2 votes
0 answers
129 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
13 votes
1 answer
2k views

Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
Song Ye's user avatar
  • 155
9 votes
0 answers
1k views

Some questions about Clausen's third IHES lecture on Efimov K-theory

I have some questions about the last theorem stated by Clausen at https://youtu.be/2xNG4rHUC6U?si=yw9eYiygLegH0nQK&t=4319. I'm not very familar with the definitions, so please correct me about any ...
Andy Jiang's user avatar
  • 2,356
3 votes
0 answers
84 views

Cohomological counterpart of the K-theory of the Roe C*-algebra in non-periodic systems

In crystalline insulating quantum systems the dynamics of electrons is governed by a Schrödinger operator which is periodic with respect to a Bravais lattice $\Gamma\cong \mathbb{Z}^d$ and whose ...
Rosencrantz's user avatar
3 votes
0 answers
137 views

Cyclic K-theory as cyclic nerve in a letter of Goodwillie

Kaledin mentioned in https://arxiv.org/abs/2004.04279 Remark 11.5 that, in a letter to Waldhausen by Goodwillie in 1988, Goodwillie showed that the cyclic K-theory can be computed by the geometric ...
Z. M's user avatar
  • 2,806
3 votes
0 answers
166 views

Which "tensor" endofunctors on triangulated categories are essentially exact?

Assume that $T$ is a symmetric monoidal triangulated category, and $X$ is an object in it. Then the functors $X\otimes -$ and $-\otimes X: T\to T$ are not necessarily exact since they send ...
Mikhail Bondarko's user avatar
2 votes
0 answers
134 views

Algebra of finite width matrices

$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
Denis T's user avatar
  • 4,600
1 vote
0 answers
83 views

Hochschild homology computation of certain type

I know that in general Hochschild homology is not very computable. However, this certain type of Hoshschild homology shows up and I feel like there could be existing result. Let $k$ be a field and $A$ ...
Li Guanyu's user avatar
  • 449
1 vote
0 answers
68 views

Metric and connection on virtual bundles

Let $E=[E^+]-[E^-]$ be an element of the Grothendieck group $K(X)$ of a compact Kahler manifold $X$. Does it exist a way to define more "geometric" structures on $E\in K(X)$ such that a ...
BinAcker's user avatar
  • 789
1 vote
0 answers
104 views

Higher equivariance

The Atiyah-Segal completion theorem states that $K(BG) = \mathrm{Rep}(G) = K_G(*)$, when the left hand side is completed with respect to the augmentation ideal. In some sense, $G$-equivariant $K$-...
taf's user avatar
  • 448
8 votes
0 answers
440 views

Poincaré duality for topological $K$-theory

Let $n$ be an even number. Let $X$ be a $n$-dimensional complex projective manifold with $H^{2m+1}(X,\mathbb{Z})=0$, for all $0\leq m\leq n-1$. $H^{2m}(X,\mathbb{Z})$ is a free $\mathbb{Z}$-module ...
user avatar
2 votes
1 answer
267 views

Semi-orthogonal decompositions over singular schemes

Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?...
Mikhail Bondarko's user avatar
2 votes
0 answers
181 views

Is there a degeneration formula for Gromov-Witten K-theoretic invariants?

By Gromov-Witten K-theoretic invariants (call them KGW) I mean the invariants defined by Givental and Lee. I expect the formula expresses the KGW of the generic fiber of a given degeneration in terms ...
jimmy's user avatar
  • 21
4 votes
1 answer
617 views

Computation of KO theory of a point

I have some basic questions about real K-theory (I mean $KO$-theory). Question 1: I have seen the table $$ KO^{-i}(\mathrm{pt})= \begin{cases} \mathbb{Z},& i=0\\ \mathbb{Z}_2,& i=1\\ \mathbb{Z}...
geometricK's user avatar
  • 1,903
7 votes
3 answers
2k views

Complex structure on $S^4$

I have heard that there is a proof of non-existence of complex structure on the 4-sphere $S^{4}$ using only the topological K-theory (complex $KU$ and real $KO$). Moreover this argument can not be ...
lab's user avatar
  • 451
3 votes
0 answers
180 views

$k$-invariants of $KO$ and $ko$ and differentials in the AHSS spectral sequence

Let $KO$ and $ko$ denote real $K$-theory and connective real $K$-theory. It appears to be a well done result that the $k$-invariants can be used to determine the early differentials in the Atiyah-...
Sam's user avatar
  • 855
0 votes
1 answer
215 views

Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$

Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$ I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact ...
user90041's user avatar
  • 709
4 votes
1 answer
259 views

Induced map in K-theory by a "trivial" bimodule

Let $R$ be a ring (not necessary commutative) and let $P_{\bullet}$ be a perfect $R$-bimodule (chain complex). I will denote the category of perfect right $R$-chain complexes by $\textbf{Perf}(R)$. ...
M. Cousto's user avatar
4 votes
0 answers
147 views

Alternative definitions of Weibel's homotopy K-theory

Consider a sort of $\mathbb{A}^1$-homotopy-stable algebraic $K$-theory for rings constructed as follows. For $K_0$ we take a symmetrization subject to natural direct sum operation of $\mathbb{A}^1$-...
Kolya Ivankov's user avatar
4 votes
0 answers
213 views

When does the canonical $t$-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...
Mikhail Bondarko's user avatar
2 votes
1 answer
213 views

A question on the ring structure of topological K-theory and Chern character

Let $X$ and $Y$ be compact spaces (or closed manifolds if you want). I have two questions relating the ring structure of topological $K$-theory. To motivate my questions let me give some background ...
Ho Man-Ho's user avatar
  • 1,173
7 votes
1 answer
448 views

Reference request for K-Theory linearization

I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that. In Waldhausen's paper Algebraic K ...
Noah Riggenbach's user avatar
13 votes
1 answer
700 views

Reference for the algebro-geometric proof of Matsumoto theorem

Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$ The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks ...
cll's user avatar
  • 2,305
4 votes
1 answer
1k views

Who was the first to capitalize Real?

For example in Atiyah's $KR$-theory there is the notion of a Real vector bundle in contrast to complex or real vector bundles. I am also familiar with the notion of a Real $C^*$-algebra and there are ...
JBantje's user avatar
  • 247
12 votes
2 answers
341 views

Which $K$-groups $K(C^*_r(G))$ are computed?

We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map for possible computation of the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ for a topological, locally compact, ...
hänsel's user avatar
  • 685
10 votes
3 answers
725 views

Reduction mod $n$ of symplectic group

Let $g,n$ be positive integers, is there a reference that $\mathrm{Sp}(2g,\mathbb{Z})\to\mathrm{Sp}(2g,\mathbb{Z}/n\mathbb{Z})$ is surjection? The only reference I could find is lemma 5.16 in Deligne–...
user avatar
5 votes
1 answer
141 views

Finding a proof within a paper: reduced $K$-theory of Higson compactification of $[0,\infty)$ is uncountable

Emerson and Meyer's Paper "Dualizing the Coarse Assembly Map" (2006) states the following Proposition (5.1): Let $X = [0,\infty)$ be the ray with its Euclidean metric coarse structure. Then the ...
geometricK's user avatar
  • 1,903
4 votes
1 answer
564 views

Borel regulator and Bloch-Beilinson regulators

Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$-theory of algebraic number rings, and the Bloch-Beilinson regulator from motivic cohomology to real ...
user avatar
6 votes
0 answers
236 views

Fundamental class in equivariant K-theory

I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory. The setup I'm interested in is the following: suppose $V$ is a vector space equipped with ...
clementine's user avatar
80 votes
2 answers
7k views

Vladimir Voevodsky's works

Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...
6 votes
1 answer
241 views

For which exact couples do associated spectral sequences degenerate at $E_1$?

It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My ...
Mikhail Bondarko's user avatar
6 votes
0 answers
101 views

Squaring operation in KO theory

There's an operation $\Lambda^2: KO^0(X) \to KO^0(X)$ such that $\Lambda^2(x+y)-\Lambda^2(x)-\Lambda^2(y)=xy$, which comes from the antisymmetric power. Similarly, there's a $\Lambda^2: KO^4(X) \to KO^...
Yuji Tachikawa's user avatar
3 votes
1 answer
203 views

"Direct" calculation of $K_0$ for surfaces, 3-folds

I apologize in advance for the vagueness of my question, but I am looking for sources (if they exist) where $K_0$ (the Grothendieck group of coherent sheaves) is computed "by hand" for some low-...
A. S.'s user avatar
  • 528
3 votes
1 answer
159 views

K-group properties of quasi-diagonal $C^*$-algebras

Let $A$ be a separable unital quasidiagonal $C^*$-algebra. What can be said about the $K$-theory of $A$, for example some properties? Especially, are there some criterions to decide whether or not $K_*...
Kashvi Ramaprasad's user avatar
9 votes
0 answers
371 views

Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?

First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion). I am telling this since as it is, the title contains a meaningful question, but it ...
მამუკა ჯიბლაძე's user avatar
11 votes
1 answer
490 views

Reference to Chern classes in algebraic k theory

I am reading P. Schneider's paper, Introduction to the Beilinson conjectures. Section 4 in this paper is something very formal about Chern classes. Personally I find some terminologies in the paper a ...
Wenzhe's user avatar
  • 2,971
18 votes
1 answer
1k views

Which motivic cohomology groups of complex numbers are non-torsion?

I would like to know which motivic cohomology groups of complex numbers are non-zero and ("better") non-torsion, i.e., for which $(i,j)$ the $i$th cohomology of the complex ${\mathbb{Q}}(j)$ over $\...
Mikhail Bondarko's user avatar
5 votes
0 answers
520 views

Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence

Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
Mikhail Bondarko's user avatar
8 votes
0 answers
411 views

Equivariant K-theory of projective representation on complex projective space

Let $G$ be a group and let $V$ be a complex projective representation of $G$, so that $G$ acts on the projectivization $\mathbb{P}(V)$. Is there any way to calculate the $G$-equivariant complex $K$-...
Akhil Mathew's user avatar
  • 25.6k
23 votes
0 answers
647 views

Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
მამუკა ჯიბლაძე's user avatar
3 votes
1 answer
352 views

Generators K-theory of Cuntz algebras

The Cuntz algebra $O_n$ is the C*-algebra generated by n isometries $S_1$, ..., $S_n$ such that $S_i^* S_j=\delta_{i,j}$ and $\sum_{i=1}^nS_iS_i^*=1$. Cuntz proved that this algebra has the following ...
John N.'s user avatar
  • 743
6 votes
1 answer
674 views

Good references for K-theory of modular curves?

The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$. I have some background in $K$-theory and also some background in ...
The Thin Whistler's user avatar
13 votes
1 answer
459 views

Algebraic K-theory of a ring

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is: What is the list of rings such that all their algebraic $K$-theory groups are known? I ...
sphere's user avatar
  • 443
6 votes
1 answer
875 views

reference request for mod p and p-adic K-theory

Is there a good reference that explains mod p K-theory and p-adic or p-complete K- theory? All I know about K-theory is the topological K-theory of "vector bundles and k-theory" in Switzer's book (...
usr0192's user avatar
  • 785
18 votes
0 answers
734 views

How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
Pablo's user avatar
  • 11.3k
3 votes
0 answers
188 views

Can one complete a morphism of commutative triangles to a "commutative cube" in a triangulated category?

This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?. I am deeply grateful for the contributions there; they roughly say that ...
Mikhail Bondarko's user avatar
1 vote
0 answers
70 views

On (universal) additive functors making a given complex contractible: examples?

Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those ...
Mikhail Bondarko's user avatar