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

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

863
questions

5
votes

0
answers

80
views

### Solving polynomial equations in $K(h)$-local or $T(h)$-local spectra?

This is the same question as an earlier question of mine, except in a different category.
Let $Spt_{T(h)}^{fin}$ be the category of finite $T(h)$-local spectra. Let $K_0^\oplus(Spt_{T(h)}^{fin})$ be ...

2
votes

1
answer

65
views

### Induced map in k-theory by an involution

Let $T$ be a ring with involution $s:T\rightarrow T$. And let
$$h:T\otimes T^\text{op} \rightarrow T\otimes T^\text{op}$$ be the ring automorphism given by $h(a\otimes b)=s(b)\otimes s(a)$.
suppose ...

5
votes

0
answers

122
views

### Splitting of $BGL_1(KR)$

There are infinite loop space splittings $BGL_1(KO)\simeq BGL_1(KO)[0,2]\times Z$ and $BGL_1(KU)\simeq BGL_1(KU)[0,3]\times Z'$ where $Z$ and $Z'$ are 2 and 3 connected, respectively (i.e. they have ...

4
votes

1
answer

188
views

### "Ring of traces" over a ring R

$\DeclareMathOperator\RoT{RoT}$I'm interested in the following ring. Fix a (Noetherian?) base ring $R$, and consider the category of finitely generated projective $R$-modules equipped with ...

2
votes

0
answers

87
views

### Is there the specialisation map of etale K theory?

Take a smooth proper morphism of schemes $X\to S$. Fix a point $t\in S$ and a point $s\in \overline\{s\}$. For a prime $l$ which is invertible in $S$, is there the natural specialization map of etale ...

6
votes

0
answers

118
views

### The Todd class and Weyl's character formula

Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...

4
votes

1
answer

318
views

### Pullback of complex vector bundles along a retraction of compact Hausdorff spaces: a direct proof instead?

Consider a pointed compact Hausdorff space $(X,x_0)$ and a closed pointed subspace $i:A\subset X$ such that there exists a continuous map $r:X\rightarrow A$ such that $r|_A=\text{Id}_A$. Set
$$q:(X,...

2
votes

0
answers

101
views

### Does the category of stable infinity categories form a "subtractive Waldhausen" category?

In "The $K$-theory spectrum of varieties", Jonathan Campbell introduces the notion of a subtractive Waldhausen category, a slight generalization of the notion of Waldhausen category that ...

5
votes

1
answer

378
views

### The principal symbol as an element in the K-theory

This line
The symbol may naturally be thought of as an element in the K-theory
of X
appears in the nLab page on principal symbols for differential operators. What does this mean? Are they talking ...

1
vote

1
answer

165
views

### Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory

Let $X$ be an scheme. Fix a prime $l$ which is invertible in $X$. Consider the $K(1)$-localization at prime $l$ of algebraic K theory $L_1K(X)$ and $l$-adic completion of etale K theory $K^{et}(X)$.
...

8
votes

2
answers

139
views

### Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras

There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an ...

3
votes

2
answers

324
views

### Involution map, and induced morphism in K-theory

Let $T$ be a ring with involution $s:T\rightarrow T$. And let
$$h:T\otimes T^\text{op} \rightarrow T\otimes T^\text{op}$$ be the ring automorphism given by $h(a\otimes b)=s(b)\otimes s(a)$.
I was ...

1
vote

0
answers

122
views

### "Interesting" examples of exact abelian subcategories of R-Mod

A somewhat vague question: for which rings there exist "interesting" exact abelian subcategories of $R-\operatorname{Mod}$ that are closed with respect to products? Actually, I would like ...

6
votes

1
answer

284
views

### The optimal ranges for the integral homological stability of $\operatorname{GL}_n(F)$'s for a field $F$

$\DeclareMathOperator\GL{GL}$
$\DeclareMathOperator\co{H}$
$\DeclareMathOperator\ko{K}$
$\DeclareMathOperator\trd{tr-deg}$
$\DeclareMathOperator{\ch}{char}$Given a field $F$ and a homological degree $...

7
votes

1
answer

316
views

### Does a Gysin map depend on the choice of Thom class?

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom ...

6
votes

1
answer

217
views

### What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?

H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi:
...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...

1
vote

0
answers

74
views

### A question about mapping cone and resolutions

I am studying this papper
https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-44/issue-3/Betti-numbers-of-almost-complete-intersections/10.1215/ijm/1256060413.full
By Daniel ...

4
votes

0
answers

82
views

### KK-theory for commutative $C^*$-algebras

The Gelfand--Naimark theorem tells us to regard noncommutative $C^*$-algebras as "noncommutative function spaces". In that spirit $K$-theory the Grothendieck group of "noncommutative ...

1
vote

0
answers

62
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 ...

2
votes

1
answer

274
views

### Does the Grothendieck group detect the Picard group?

Let $C$ be a curve (=smooth projective curve) of genus $g$ over an algebraic closed field $\mathbb{k}$.
It is well known that the Grothendieck group $K_0(\operatorname{coh} C)$ of the category of ...

1
vote

0
answers

95
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 $K$-...

2
votes

1
answer

199
views

### Relation between free resolutions and minimal free resolutions

Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $M$ a finitely generated $R$-module. In this case, given $F_{\bullet}$ a free resolution of $M$, what is the relation between $F_{\bullet}$ and a ...

4
votes

2
answers

220
views

### Computation of cohomology of Morava $K$-Theory

Let $G$ be an elementary abelian group, so that $G = (\mathbb{Z}/p)^k$ for some $k$. We can then compute the Morava $K$-theory of $BG = (BZ/p)^k$ pretty easily: $K(n)^*(BG) = K(n)^*[[x]]/[p](x) (x)_{K(...

6
votes

0
answers

140
views

### THH and string topology

There is an equivalence $THH(S^{\infty}_+ LX) = S^{\infty}_+ FX$ where $FX$ is the free loop space (I already used the letter $L$). The circle action on $THH$ gives rise to a degree $1$ cyclic ...

1
vote

1
answer

125
views

### Grothendieck group and faithfully flat morpshim

For regular schemes $X$ and $Y$, and a faithfully flat morphism $f:Y \to X$, there is a flat pullback map of Grothendieck groups:
$$
f^*:K^0(X) \to K^0(Y).
$$
Is this map injective?

0
votes

0
answers

64
views

### How to compute $G_0(kG)$ for any field $k$

$\newcommand{\red}{\mathrm{red}}$I am trying to compute $G_0(kG)$ for any field $k$, where $G$ is the set of all monomials $x^iy^j$ such that the line through the origin and $(i,j)$ has slope between ...

2
votes

0
answers

95
views

### Does the convolution $C^*$-algebra of locally compact Hausdorff groupoids recover back the respective groupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.
Let $X$ be a locally compact Hausdorff groupoid (or Lie ...

13
votes

2
answers

485
views

### When are bundles of odd and even differential forms isomorphic?

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...

7
votes

1
answer

426
views

### Topological K-theory of Riemann surface

Let $X$ be a compact Riemann surface of genus $g$, then $K^1_{\mathrm{top}}(X)\cong\mathbb{Z}^{2g}$. Is there a explicit description of a set of basis of $K^1_{\mathrm{top}}$? (e.g., For cohomology $H^...

8
votes

0
answers

352
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 ...

4
votes

1
answer

237
views

### The third homology stability of general linear groups over finite fields

Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\...

2
votes

1
answer

147
views

### How to compute the $G$-theory groups of $k[x,y]/(xy)$ for any field $k$

I am trying to compute the $G$-theory groups of the ring $k[x,y]/(xy)$ for any field $k$. What I have tried so far are two approaches.
Approach 1: Use the $G$-theory localization sequence for $k[x,y]/(...

1
vote

0
answers

181
views

### Motivic cohomology commutes with field extension

$\DeclareMathOperator\Cor{Cor}$Let $X$ be a smooth scheme over $k$ and $k \subset F$ a field extension. Let $X_F$ be the field extension of $X$. Then there is a map
$$\varinjlim_{k\subset E \subset F} ...

5
votes

1
answer

284
views

### Is ku reflexive as a spectrum?

Let $S$ and $\rm ku$ denote the sphere spectrum and the connective K-theory spectrum, respectively.
Then is the morphism ${\rm ku} \to F(F({\rm ku},S),S)$ an equivalence?
Here $F$ denotes the mapping ...

4
votes

0
answers

176
views

### How to to understand the homology groups $H_*(\Omega_0^\infty S^\infty)$?

The original statement of the Barratt--Priddy theorem says there is an isomorphism of homology groups
$$H_*(\Sigma_\infty)\cong H_*(\Omega_0^\infty S^\infty),$$
where $\Omega_0^\infty S^\infty$ is the ...

8
votes

0
answers

144
views

### Conner-Floyd Chern classes and $E$-(co)homology of $BU$

In his book, Stable homotopy and generalised homology, Adams computes the $E$-(co)homology of $BU$ for a complex oriented cohomology theory $E$. In II.4, he first describes the $E$-homology of $BU$ as ...

5
votes

1
answer

172
views

### Computation of the torsion of K-groups related to elliptic curves

Let $E$ be an elliptic curve over $\mathbb Q$. Let $F$ be the rational function field of $E$.
The $K_2$ group of $F$ may be described by elements in $F^\times ⊗_\mathbb{Z} F^\times$ quotiented by the ...

4
votes

1
answer

236
views

### Which complexes of coherent sheaves can be presented as countable homotopy limits of perfect complexes?

Let $X$ be a noetherian scheme (actually, I need the case where $X$ is proper over an affine scheme), $C$ is an object of the derived category $D_{coh}(X)$ of coherent sheaves on $X$. Under which ...

3
votes

1
answer

242
views

### Approximation of continuous projections on a manifold by smooth idempotents

Every continuous vector bundle on a closed smooth manifold $M$ has a smooth structure. On the other hand, every vector bundle $E$ is the image of a trivial bundle $M\times\mathbb{C}^n$ under some ...

19
votes

1
answer

707
views

### What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?

Consider the following partial order. The objects are unordered tuples $\{V_1,\ldots,V_m\}$, where each $V_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V_1 \oplus \cdots \oplus V_m =...

5
votes

1
answer

310
views

### Strict graded commutativity of $\pi_*(\operatorname{THH}(A))$?

$\DeclareMathOperator\THH{THH}\DeclareMathOperator\HH{HH}$A version of the strict graded commutativity (i.e. graded commutativity & $x^2=0$ for every homogeneous element $x$ of odd degree) of $\...

4
votes

1
answer

140
views

### Injectivity of assembly in A-theory for $BC_2 = \mathbb R P^\infty$ in degree $4$

I am trying to understand the assembly map
$$\pi_i ((BC_2)_+ \wedge A( \ast )) \rightarrow A_i( BC_2 ) $$
in low degrees for the space $BC_2 = \mathbb R P^\infty$ in Waldhausen $A$-theory. I know we ...

5
votes

1
answer

248
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

156
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

132
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

174
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

169
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

357
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

134
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

162
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 ...