Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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12
votes
4answers
746 views

Injective dimension of graded-injective modules

In "Existence theorems..." Van den Bergh proposes the following "pleasant excercise in homological algebra": Let $A$ be a connected graded noetherian $k$-algebra (that is, $\mathbb N$-graded with ...
7
votes
1answer
236 views

Triviality of direct multiples of flat complex vector bundles

Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more ...
10
votes
1answer
737 views

Friedlander Conjecture

I just read on the ALGTOP discussion list that Morel has announced a proof of the Friedlander conjecture. Question: Are there other applications besides the Milnor conjecture $H_*(G,F_p)=H_*(BG,F_p)$ ...
17
votes
3answers
1k views

Brauer Groups and K-Theory

Is there some a priori reason why we should expect the Brauer group of real [complex] super vector spaces to be closely related to periodicity in real [complex] K-theory? By "a priori" I mean a proof ...
11
votes
1answer
651 views

Motivic cohomology vs. K-theory for singular varieties

As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for ...
4
votes
2answers
704 views

On two spectral sequences for the cohomology of a double complex

For a (bounded) double complex (of abelian groups or vector spaces) one can consider two spectral sequences that converge to the cohomology of the totalization: one can first compute either the ...
6
votes
0answers
244 views

What are the relations in the unbounded model of K-homology?

I have posed this question to some experts at my university who would probably know the answer if there were a complete one, so my expectations are limited. It's possible that the question deserves ...
10
votes
1answer
476 views

Homology of infinite loop spaces $QX$

Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}_p$ is a Hopf algebra over the ...
0
votes
0answers
186 views

Topological K-theory of Bohr compactification of real numbers

I am interested in the K-theory of the Bohr compactification $\mathbb{R}_B$ of the real numbers. Do we have $K_0(C(\mathbb{R}_B))$ isomorphic to $K_1(C(\mathbb{R}_B))$ ? More generally, what do we ...
10
votes
2answers
1k views

Atiyah-Patodi-Singer Eta invariant and Chern-Simons form

I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant: 1) Is eta a topological invariant ...
0
votes
0answers
269 views

A modified version of K-theory for manifolds ?

If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
15
votes
1answer
1k views

Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the litterature quite hard to follow -"from experts, for experts". Voevodsky in ...
9
votes
0answers
391 views

Two constructions for BU×Z

Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$: 1) Take the groupoid of finite dimensional complex inner product spaces with isometries ...
3
votes
0answers
325 views

K-theory of differential graded modules over differential graded algebras

Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of ...
10
votes
3answers
1k views

State of the art for Gersten's conjecture for K-theory?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to ...
16
votes
2answers
992 views

Algebraic K-theory of the group ring of the fundamental group

I know of two places where $K_{*}(\mathbb{Z}\pi_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology. The first is the Wall ...
2
votes
1answer
345 views

Spectral sequence for H-space bundles

Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle. One ...
4
votes
1answer
221 views

Index vs. equivariant index (and then taking invariant part)?

Let $C$ be a smooth projective curve and let $C^{(n)}$ be its $n$th symmetric power. Let $E$ be a $S_n$-equivariant vector bundle over the Cartesian power $C^n$. Suppose that $E$ descends to a vector ...
6
votes
2answers
399 views

Somewhat general question that includes: “Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?”

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree? Example: Given a dg ...
11
votes
2answers
748 views

Twists of K-theory and tmf

I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How ...
3
votes
2answers
278 views

Extensions which define the same element of $\text{Ext}^n(M,N)$ are in fact equivalent

It is well known (and wouldn't be so-named unless it were) that: If $\xi$, $\eta$ are $n$-fold extensions of $N$ by $M$ (modules over a ring $R$) which yield the same element of $\text{Ext}^n(M,N)$, ...
0
votes
1answer
262 views

Can injective resolutions be 'enlarged' (or shrunk) to admit only injective maps from extensions?

Let $M$ and $N$ be $R$-modules for some ring $R$. There is a standard result involving the computation of $\text{Ext}^n(M,N)$, using projective resolutions, which says that you can always choose a ...
6
votes
0answers
311 views

Higher K-theory of Orlik-Solomon algebras (and possible generalizations?)

This topic of this question is a bit outside my comfort zone, and I should say that my end goal is to really understand how much "graph theory" is captured by contraction-deletion relations. It seems ...
2
votes
1answer
303 views

A question about a proof in Rosenberg's algebraic K-theory book

In corollary 4.3.13 of Rosenberg's book Algebraic K-theory and its applications, it's proved that $K_2(F)=0$ if $F$ is a finite field. The last sentence in the proof says that the central extension ...
2
votes
0answers
205 views

The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety.

There is a very important result of Bloch and Ogus: for any smooth variety $X$ and fixed $r\in \mathbb{Z}$, $r\ge 0$, $l$ is prime to the residue field characteristic, the Zariski sheafification of ...
1
vote
1answer
386 views

If V is an irreducible representation of G, what is K_{G}(T_{G}V)?

Here, G is a compact lie group. V is a finite dimensional irrepn of G. By Atiyah, every element in K_{G}(T_{G}V) is a symbol of a transversally elliptic operator on V. Of course, K_{G}(T_{G}V) is ...
5
votes
1answer
320 views

Splitting principle in equivariant cohomology

The following is a weaker version of what is called splitting principle in Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6: Let $G$ be a compact ...
1
vote
2answers
226 views

Z-torsion homology for groups

I would like to ask the following: if for a group $G$ the homology $H_n(G,\mathbb{Z})$ is $\mathbb{Z}$-torsion for every $n\geq n_0$, then what can be said concerning $\mathbb{Z}$-torsion for ...
4
votes
1answer
192 views

Do K-equivalent rings have isomorphic Nil-Terms?

Let $R,S$ be rings and $f: R \rightarrow S$ a ring homomorphism such that $f$ induces an isomorphism on the $K$-theory of the rings. The map $f$ also induces a ring homomorphism $f[t]: R[t] ...
8
votes
2answers
494 views

Grothendieck group for projective space over the dual numbers

Fix a field $k$. For a singular variety $X$, I understand that the Grothendieck group $K^0(X)$ of vector bundles on $X$ is not necessarily isomorphic to the Grothendieck group $K_0(X)$ of coherent ...
2
votes
1answer
210 views

Unitary representation acting on the K-theory of the reduced group $C^*$-algebra

Let $G$ be a group (usually infinite), $R$ a ring and $\rho: G \rightarrow Gl_n(\mathbb{Z})$ a finite-dimensional representation of $G$. Then we can define a functor from the category of projective ...
2
votes
0answers
192 views

The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.

For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram ...
2
votes
2answers
233 views

Action of centralizer on Borel-Moore homology of Springer Fibers for Affine Hecke Algebra

In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of ...
5
votes
1answer
407 views

K-Theory as a special $\lambda$-ring

I wonder if there is a nice and short proof that the $K$-theory of a topological space is a special $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly ...
5
votes
3answers
584 views

Infinite dimensional central simple algebras

When constructing the Brauer group of a field, only the finite-dimensional central simple algebras are considered (because of Artin-Wedderburn's characterization). But what happens to the ...
1
vote
0answers
175 views

Special properties of (the $\gamma$-filtration of) $K$-theory of affine varieties.

Let $A$ be a smooth affine variety of dimension $n$. Are there any facts known on $K_{\ast}(A)$ and its $\gamma$-filtration which do not hold for $K_*(V)$ for an arbitrary smooth $V$ (of the same ...
3
votes
0answers
145 views

The vanishing of homotopy invariant $K$-theory of dg-categories

In my previous question The vanishing of non-connective K-theory in negative degrees I asked when one can be sure that the negative non-connective $K$-groups of a differential graded category vanish. ...
4
votes
1answer
441 views

The vanishing of non-connective K-theory in negative degrees

In the works of Cisinski, Tabuada, and Schlichting certain non-connective K-theory groups for a differential graded category $C$ are defined. As far as I understand, $K_i(C)$ is not necessarily zero ...
1
vote
0answers
94 views

When the class of a complex is necessarily equi-dimensional

Let $P$ be a smooth projective variety. For an object $X$ of $D_{perf}(P)$ (i.e. a bounded perfect complex of $\mathfrak{O}_P$-module sheaves) we consider its class $[X]$ in $K_0(P)\otimes ...
9
votes
3answers
965 views

Is there a good definition of (topological) K-Theory over arbitrary spaces?

Hi (this is my very first question here, so please don't hurt me...) for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...
6
votes
2answers
690 views

Understanding the analytic index map of the Atiyah-Singer index theorem

Hi, I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn. I do not understand why the analytic index map ...
3
votes
1answer
420 views

Local Cohomology and Maximal-Cohen-Macaulay modules

Checking a recent article [this one, specifically section 3.1] I found the following claim (I'm paraphrasing, of course): Let $A$ be a graded connected noetherian algebra (not necessarily ...
2
votes
0answers
240 views

Six term exact sequence In E-theory

I just want to know whether the two six term exact sequences in E-theory is true for nonseparable C*-algebras. We know already if the first varible is complex number, then we get six term exact ...
3
votes
1answer
548 views

K-theory of number field

Let $R=\mathbb{Q}[e^{2\pi i /3}]$. Does $H_3(GL(R))$ have $\mathbb{Z}$-rank $1$? If so, what is the index of the map: $$ \mathbb{Z}\cong K_3(R)/{\rm Torsion} \to H_3(GL(R))/{\rm Torsion}\cong ...
2
votes
1answer
316 views

Index of elliptic operators III: H-structure invariant under a group G

In the Atiyah-Singer paper mentioned above, they introduced on p.557 a concept called $H$-structure which is used to describe the Chern character of special elements of $K(TX)$. It is roughly the ...
2
votes
0answers
58 views

Computing morphisms in localizations of $K(B)$

Let $B$ be an additive category (a small one; one can assume that it is a $\mathbb{Q}$-category, yet not much else is known about it). Given a set of objects $S$ of $K^b(B)$ (or $K(B)$), I consider ...
5
votes
0answers
203 views

Are Tate twists of t-positive motives positive with respect to the Voevodsky's homotopy t-structure?

Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). ...
2
votes
1answer
183 views

pairs of matrices up to similiarity and vector bundles over punctured torus

I would like to construct 2D vector bundles over the punctured torus, but I don't know a lot of K-theory. Over the square, there can only be the trivial bundle, but now since ...
3
votes
1answer
908 views

Where could I publish an average paper on triangulated categories?

I have a rather abstract paper on triangulated categories; I would say that it is of average size and quality. I want to find an appropriate journal to publish it; I would like it to be accepted in ...
8
votes
0answers
249 views

H-space structure on the Calkin algebra

By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...