Questions tagged [kt.k-theory-and-homology]
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
925
questions
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 ...
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?...
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 ...
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 ...
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
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
3
votes
1
answer
485
views
Computation of KO theory of a point
I have some basic questions about real K-theory (I mean $KO$-theory).
Question 1: I have seen the table
$$
KO^{-i}(\mathrm{pt})=
\begin{cases}
\mathbb{Z},& i=0\\
\mathbb{Z}_2,& i=1\\
\mathbb{Z}...
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 $\rho$ ...
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 ...
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_* \...
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 ...
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 ...
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 ...
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}...
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 $...
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 ...
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 ...
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 ...
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 $[\...
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 $A$ ...
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, ...
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 $\mathcal O_X(U)$-...
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 ...
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 ...
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 ...
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{...
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 ...
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 ...
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 $\mathbb{Z}[G]$ ...
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 $S$.
...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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^...
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 ...
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 $\tilde{K}^*(X)$, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $G$ ...