Questions tagged [kt.k-theory-and-homology]

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

Filter by
Sorted by
Tagged with
5 votes
1 answer
298 views

Motivic cohomology with $\mathbb{Z}/2$ coefficients in positive characteristic

In G. M. L. Powell's note 'Steenrod operations in motivic cohomology', he stated that if $\mathrm{char}(k)=0$, $$H^{*,*}(k,\mathbb{Z}/2)=K_*^M(k)/2[\tau]$$ where $\tau\in H^{0,1}$ is the unique ...
Nanjun Yang's user avatar
2 votes
1 answer
221 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
169 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
275 views

Matrix units in von Neumann algebras, and $K_0$ groups

This question arises from trying to understand the proof of Lemma 3.1.4 in De Commer, Martos, and Nest - Projective representation theory for compact quantum groups and the quantum Baum–Connes ...
Matthew Daws's user avatar
  • 18.5k
6 votes
0 answers
183 views

Hall-Littlewood polynomials of non-dominant weights

$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let $$ R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
A. S.'s user avatar
  • 518
3 votes
1 answer
485 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
  • 1,851
1 vote
0 answers
164 views

Calculation about Chern character in a special setting

I'm confused with working out the Chern character in the following special setting. Let $E$ be a spinor bundle $$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$ over sphere $S^{2n}$, where $\rho$ ...
Radeha Longa's user avatar
4 votes
1 answer
183 views

The upper bounds on rank $ 2 $ real matrices

Let $ A_{n}(F) $ be the collection of all skew-symmetric matrices over the field $ F $ ($\operatorname{char} F \neq 2 $). Let M be a subspace of $ A_{n}(F) $ such that all non zero elements have rank ...
Sky's user avatar
  • 913
1 vote
2 answers
362 views

Pushforward of structure sheaf along a torsor for a finite group

Let $\pi : P \to X$ be a torsor for a discrete, finite group $G$ of size $\#G = N$ on a scheme $X$. I want to compare $\pi_* \mathcal O_P$ with $\mathcal{O}_X$. Locally but not globally, $\pi_* \...
Leo Herr's user avatar
  • 1,084
2 votes
0 answers
120 views

When semi-simple subcategories "extend" to hearts of t-structures?

Let $A$ be a semi-simple abelian subcategory of a triangulated category $C$ that "generates" $A$ (that is, $C$ equals its own smallest triangulated subcategory that is closed under direct ...
Mikhail Bondarko's user avatar
1 vote
0 answers
116 views

On infinite global dimensions of "slightly non-commutative" rings

Assume $R$ is a commutative Noetherian ring of finite Krull dimension; $R'$ is a not commutative ring that contains $R$ in its center and also finitely generated as an $R$-module. If the (left) global ...
Mikhail Bondarko's user avatar
3 votes
0 answers
148 views

Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
Mikhail Bondarko's user avatar
12 votes
0 answers
364 views

Can Quillen-Lichtenbaum recover Borel's computation?

Borel famously used analysis on symmetric spaces to compute the rationalised algebraic $K$-theory groups of rings of integers $\mathcal{O}_F$ in number fields, e.g. $K_i(\mathbb{Z}) \otimes \mathbb{Q}...
skupers's user avatar
  • 7,923
1 vote
0 answers
198 views

Pushforward of sheaves along finite etale map

Suppose $\pi : Y \to X$ is a finite 'etale map of degree d. I want a formula for $\pi_* \mathcal O_Y$. I'm happy with a formula in $K$ theory. There is a $S_d$-torsor $P \to X$ of local isomorphisms $...
Leo Herr's user avatar
  • 1,084
1 vote
1 answer
89 views

Problem concerning about an $n$-subspace of $ A_{n}(F) $

Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...
Sky's user avatar
  • 913
20 votes
2 answers
1k views

Does Waldhausen K-theory detect homotopy type?

Recall that $A(X)$, the K-theory of a connected, pointed space X, is defined as the K-theory spectrum of the ring spectrum $\Sigma^\infty_+ \Omega X$ (or via a plethora of alternative definitions). Is ...
Connor Malin's user avatar
  • 5,191
5 votes
0 answers
159 views

Uniqueness of complex topological $K$-theory as an $S$-algebra

This might be well-known or trivial, but I could not figure out how to fill in the details: For an $S$-algebra $K$ denote its associated multiplicative cohomology theory by $h^*_K$. Suppose that I ...
Ulrich Pennig's user avatar
4 votes
0 answers
215 views

K theoretic pushforward along gerbes

I have a nontrivial gerbe $\pi : \mathscr{G} \to X$ banded by a cyclic group $G = \mathbb{Z}/r$. I'm working over $\mathbb{C}$. I want to describe $\pi_\ast$ and relate the fundamental class $[\...
Leo Herr's user avatar
  • 1,084
0 votes
0 answers
119 views

Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant

Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$. Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$ So $A$ is a Banach algebra. Can we equip $A$ ...
Ali Taghavi's user avatar
2 votes
0 answers
151 views

construction of $K_0$-group and Karoubian completion

Let $A$ be a ring. The $K_0$ group of $A$ can be defined in most old fashioned way as the Grothendieck group of the set of isomorphism classes of its finitely generated projective $R$ modules, ...
user267839's user avatar
  • 5,948
3 votes
1 answer
281 views

Sheaf of chain complexs glued by chain homotopy equivalences

Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose: For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-...
Hang's user avatar
  • 2,719
7 votes
3 answers
1k 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
  • 441
1 vote
0 answers
108 views

Question on the classification of Cuntz algebras via their extension groups and via their K-theory

I've recently been reading Kenneth Davidson's book on C*-algebras by example. One thing that particularly interested me was the classification of the Cuntz algebras by looking at the extensions of the ...
Owen Tanner's user avatar
2 votes
0 answers
142 views

About the algebraic structure of the $G$-equivariant $KK$-theory

Let $ G $ be a second countable locally compact group. Let $ A $ and $ B $ be two $G$-$C^*$-algebras. Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $. Could you tell me ...
Bradley04's user avatar
  • 487
5 votes
1 answer
406 views

L-theory of additive category

Reading some articles in the field, I found the following statement: Proposition: Let $\mathcal{B}$ be an additive category and $\mathcal{A}$ a full additive subcategory of $\mathcal{B}$. If $\mathcal{...
cellular's user avatar
  • 983
6 votes
1 answer
435 views

Algebraic K-theory and intersection theory (Bloch's formula)

It seems to be a well known fact that algebraic K-theory can be used to understand intersection theory, at least for varieties (or stacks!) over a field. A first glimpse of this result seems to be ...
Thiago's user avatar
  • 221
6 votes
0 answers
160 views

Elliptic operators with with same index but non homotopic symbols

Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$. Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold. In Atiyah-Singer "the index of ...
Overflowian's user avatar
  • 2,523
12 votes
1 answer
306 views

Group ring with infinite stable rank

In searching for a counterexample in homological stability, I came across the following question: Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ ...
user124543's user avatar
3 votes
1 answer
274 views

Algebraic K-theory of a category containing all perfect complexes

Let $R$ be a ring and let $\mathcal{C}$ be the category of perfect $R$-complexes. Suppose that $$S=\bigoplus_{i=1}^{\infty}R$$ Let us define $\mathcal{D}$ the smallest thick category generated by $S$. ...
LGO's user avatar
  • 169
6 votes
1 answer
435 views

Stable Adams operations

I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not ...
Avishkar Rajeshirke's user avatar
8 votes
1 answer
645 views

Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible

Let $A_{n}(\mathbb{Q}) $ denote the $n$ times $n$ skew symmetric matrices over the rational number field. Let $N$ be a subspace of $A_{n}(\mathbb{Q}) $. If all the non-zero matrices in $N$ are ...
Sky's user avatar
  • 913
3 votes
0 answers
206 views

Central division algebras over $ \mathbb{Q} $

Quaternions over $ \mathbb{Q} $ are an example of a Central Division algebra over $\mathbb{Q} $ for which the basis elements $\{ i,j,ij \} $ other than $1$ are represented by skew-symmetric matrices ...
Sky's user avatar
  • 913
11 votes
1 answer
504 views

Problems concerning subspaces of $M_{n}(\mathbb{Q}) $

Let $M_{n}(\mathbb{Q}) $ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M_{n}(\mathbb{Q}) $.Then if all the non-zero matrices in $N$ are invertible, what is ...
Sky's user avatar
  • 913
6 votes
1 answer
283 views

Irreducible representations of the symmetric group on homology of simplicial complex

I am following Wall's paper A note on symmetry of singularities and I have some questions regarding representation theory and the homology of some objects: Consider an action of $\Sigma_k$ on a finite ...
MathBug's user avatar
  • 258
7 votes
2 answers
556 views

What is a most natural categorification of a vector space?

Few days ago I became excited when I learned from an answer to Examples of simple vertex operator algebras (VOAs) that The irreducible modules of the rank $d$ free boson are naturally parametrized by ...
მამუკა ჯიბლაძე's user avatar
4 votes
0 answers
286 views

Is there algebraic $K$-theory of a group independent of the base ring?

Given a ring R and a group $G$, I can consider the group ring $R[G]$ and then take the algebraic $K$-theory $K(R[G])$. This the $K$-theory of the category $\operatorname{Rep}_R(G)$. As a variant, one ...
David Corwin's user avatar
  • 15.1k
6 votes
1 answer
2k views

Algebraic K-theory "with proper support"

I would like to know what is the "correct" algebraic $K$-theory "with proper support". I suppose that the answer should be found in the condensed world, which is mostly inspired ...
Z. M's user avatar
  • 1,948
10 votes
0 answers
309 views

Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$

First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
Francis Baer's user avatar
3 votes
0 answers
184 views

About the representation ring of a compact group

A question stuck in my mind when I was reading the paper "The representation ring of a compact Lie group" by Segal. He says on page one that I confine myself to the case of a compact Lie ...
Mehmet Onat's user avatar
  • 1,161
5 votes
1 answer
379 views

What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?

The cohomology ring of $\mathrm{SL}_2(\mathbb C)$ as a topological group is straightforward (it's generated by a Chern class), but what is known in the discrete case? I'm particularly interested in $H^...
Calvin McPhail-Snyder's user avatar
5 votes
2 answers
401 views

K-theory of a coconnective dga

I have seen somewhere that if a differential graded algebra $A$ is connective (homologically graded), then the Grothendieck group $K_{0}(A)=K_{0}(H_{0}(A))$. Suppose that $A$ is a differential graded ...
Let's user avatar
  • 511
3 votes
0 answers
196 views

What is the multiplicative structure of K-theory $\Omega$-spectrum $KU$?

The K-theory $\Omega$-prespectrum $KU$ has spaces $KU_{2i}=BU\times \mathbb{Z}$ and $KU_{2i+1}=U$, according to Bott periodicity we have a $\mathbb{Z}_2$-graded cohomology theory $\tilde{K}^*(X)$, ...
cyber's user avatar
  • 71
5 votes
0 answers
167 views

Which t-structure extend from subcategories of compact objects uniquely?

Let $T$ be a compactly generated triangulated category, that is, $T$ is closed with respect to small coproducts and equals its own smallest triangulated subcategory closed with respect to coproducts ...
Mikhail Bondarko's user avatar
2 votes
0 answers
106 views

Special case of Elliott's Theorem

Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...
Peg Leg Jonathan's user avatar
5 votes
1 answer
249 views

Integral homology of braid groups as a ring

Let $Br_k$ denote the braid group on $k$ strands. In Corollary A.4 of "Homology of Iterated Loop Spaces" (Page 348), Cohen-Lada-May compute $H_i(Br_k;\mathbb Z)$ as an abelian group for each ...
qqqqqqw's user avatar
  • 915
19 votes
1 answer
798 views

Diffeomorphism groups of h-cobordant manifolds

Do we have specific examples of h-cobordant smooth manifolds $M$ and $M'$ such that $\operatorname{BDiff}(M) \not \simeq \operatorname{BDiff}(M')$? Perhaps something can be said in terms of K-theory ...
Connor Malin's user avatar
  • 5,191
11 votes
2 answers
817 views

Solving polynomial equations in spectra?

Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general ...
Tim Campion's user avatar
  • 60.6k
3 votes
0 answers
113 views

$K$-theory with respect to two different choices of quasi-isomorphisms

This question is related to another question asked here. Let's assume we have an exact category $C$ that consists of specific vector bundles on a variety. Furthermore assume $C$ is idempotent complete ...
user127776's user avatar
  • 5,831
10 votes
2 answers
580 views

Stable rank one and corners of $C^\ast$-algebras

Thanks to a result of Herman and Vaserstein in [3], Rieffel's notion of stable rank [4] coincides with the Bass stable rank [1] for every $C^\ast$-algebra $A$: we denote it by $\mathrm{sr}(A)$ and we ...
Julien's user avatar
  • 650
5 votes
1 answer
567 views

Locally trivializing a G vector bundle?

In §1.6 of Atiyah's K-theory, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ ...
Nikhil Sahoo's user avatar
  • 1,175

1 2
3
4 5
19