Tagged Questions

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

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16
votes
2answers
971 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
343 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
219 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
396 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
731 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
277 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
261 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
308 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
295 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
199 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
380 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
318 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
223 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
2answers
191 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
485 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
205 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
190 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
231 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
398 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
572 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
167 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
429 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
946 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
676 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
413 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
238 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
544 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
315 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
199 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 ...
2
votes
1answer
896 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
243 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 ...
1
vote
1answer
255 views

Products on the K-theory of graded C*-algebras

One can define products on the K-theory of graded C*-algebras as in http://web.me.com/ndh2/math/Papers_files/Higson,%20Guentner%20-%202004%20-%20Group%20C*-algebras%20and%20K-theory.pdf on page 152, ...
0
votes
1answer
232 views

transversally elliptic operator, fundamental class, K-homology

How transversally elliptic pseudo-differential operator naturally induces a K-homology class in KK(A, C), where the algebra A is the crossed product algebra C(M) ⋊ G, where M is compact manifold and G ...
6
votes
1answer
407 views

Projective modules over free groups

Consider the ring of Laurent polynomials $R := \mathbb{Z}[s,s^{-1}]$ with integer coefficients. Are all projective $R$-modules free? (Let's say left modules by convention.) More generally, let $G$ ...
4
votes
2answers
791 views

Survey of Algebraic K-Theory Since 1980?

I just came across Charles Weibel's Development of Algebraic K-Theory until 1980, and found it really helpful. Is there been anything analogous which surveys the developments in the last 30 years? ...
1
vote
1answer
320 views

In K-homology K(X), if the Dirac operator D is invertible, does [D] represent zero element?

When X = pts, we know that the index of [D] equal to 0. What about X is not a point. Thanks
15
votes
2answers
913 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 ...
8
votes
1answer
342 views

Heller operator without left adjoint?

Suppose given a noetherian ring $R$. On the stable category $R\text{-}\underline{\text{mod}} := R\text{-mod}/R\text{-proj}$, we have the Heller operator $$ \Omega : R\text{-}\underline{\text{mod}} ...
22
votes
2answers
1k views

Where do all these projection formulas come from?

I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples. Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, ...
4
votes
1answer
385 views

Is Margolis's axiomatisation conjecture still alive?

The construction of the category of finite spectra is easy, but there are different constructions of the whole homotopy category of spectra, all of which leading to the same result up to an ...
4
votes
0answers
644 views

$Ext$ functor, filtered complexes and spectral sequences

Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
0
votes
1answer
260 views

The stabilized homotopy category of graded C* algebra

Hi everyone On page 147 of the note "Group C*-Algebras and K-theory" by N.Higson and E.Guentner there are something about the stabilized homotopy category of graded C* algebra, which is a category ...
9
votes
1answer
552 views

Equivalent definitions of topological K-theory over locally compact spaces

Hello everyone My question is the following: Given a locally compact space $X$, assume it is also connected for simplicity, its K-theory ring is defined as $\tilde{K}(X^+)$, where $X^+$ is the one ...
8
votes
1answer
419 views

Is there an intrinsic definition of the topological index map in $K$-theory?

In the language of $K$-theory, the Atiyah-Singer index theorem says that for a compact manifold $X$ the topological index map $\text{t-index}: K(TX) \to K(T\mathbb R^n) \simeq \mathbb Z$ induced by ...
5
votes
2answers
979 views

Is every homology theory given by a spectrum?

Let $E$ be a spectrum. For any CW complex $X$, define $h_*=\pi_i(E\wedge X)$. Then we know that $h_*$ form a homology theory. In other words, there functors satisfy the homotopy invariance, maps a ...
3
votes
3answers
1k views

Homology or cohomology?

How do people call an additive functor from a triangulated category $C$ to an abelian one that converts distinguished triangles into long exact sequences. Does one usually call a covariant functor ...