All Questions
Tagged with kt.k-theory-and-homology reference-request
105 questions
0
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0
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80
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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}...
- 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'...
- 2,907
13
votes
1
answer
2k
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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 ...
- 155
1
vote
0
answers
124
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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_\...
- 38.6k
2
votes
0
answers
129
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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 ...
- 653
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votes
0
answers
1k
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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 ...
- 2,356
3
votes
0
answers
166
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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 ...
- 16.9k
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 ...
- 451
55
votes
8
answers
7k
views
Applications of Grothendieck-Riemann-Roch?
I am currently trying to learn a bit about Grothendieck-Riemann-Roch...
To try to get a better feeling for it, I am looking for examples of nice applications of GRR applied to a proper morphism $X \...
- 21k
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 ...
- 31
3
votes
0
answers
137
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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 ...
- 2,806
2
votes
0
answers
134
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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 ...
- 4,600
80
votes
2
answers
7k
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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 ...
1
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0
answers
83
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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$ ...
- 449
36
votes
5
answers
6k
views
What is the equivariant cohomology of a group acting on itself by conjugation?
This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group.
$G$ acts on itself by conjugation. One has the equivariant ...
- 13.2k
4
votes
1
answer
617
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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}...
- 1,903
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 ...
user39380
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?...
- 16.9k
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 $K$-...
- 448
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 ...
- 789
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 ...
- 21
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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–...
user39380
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 $\textbf{Perf}(R)$. ...
- 43
12
votes
1
answer
458
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 ...
- 433
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 ...
- 709
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, ...
- 685
18
votes
5
answers
4k
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"Must read" papers in algebraic K-theory?
I'm mainly interested (graduate student) in surgery theory and geometric topology.
If I have a chance to suggest "must read" papers in geometric topology for beginner,
I'm very glad to suggest "...
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 ...
- 247
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 ...
- 2,305
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-...
- 855
7
votes
1
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448
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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 ...
- 333
14
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3
answers
3k
views
References for equivariant K-theory
I want a good introduction to localization in equivariant $K$-theory. This introduction can be simple in several ways:
I only care about torus actions.
I only care about $K^0$.
I only care about very ...
- 156k
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 $\...
- 16.9k
4
votes
2
answers
522
views
Equivariant K-theory of $S^1$-action on $S^2$
Is there any references for the structure of the equivariant K-theory $K_{S^1}(S^2)$ where the action of $S^1$ on $S^2$ is defined to be rotation about the $z$-axis? What is the ring structore of $K_{...
- 9,019
4
votes
0
answers
147
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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 $\mathbb{A}^1$-...
- 613
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_*...
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 ...
- 1,173
11
votes
1
answer
2k
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A survey for various $K$-homology theories and their relationship
The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory....
- 593
4
votes
1
answer
564
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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 ...
user119470
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 ...
- 16.9k
11
votes
1
answer
490
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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 ...
- 2,971
5
votes
1
answer
141
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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 ...
- 1,903
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 ...
- 16.9k
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 ...
- 1,805
6
votes
1
answer
874
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 (...
- 785
5
votes
1
answer
792
views
$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$
Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$?
These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...
- 593
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-...
- 528
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 $...
- 16.9k
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 ...
- 18.4k
16
votes
2
answers
1k
views
Proof of Bott Periodicity in twisted K-theory
I have a question about the Proof of Bott Periodicity in twisted K-theory
by Atiyah and Segal in their paper Twisted K-theory.
Following their notation, to prove Bott periodicity in this context it ...
