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

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

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5 votes
0 answers
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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 ...
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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 ...
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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. ...
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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,...
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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 $K^{et}(X)$. ...
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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 ...
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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 ...
  • 97
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 $...
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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 ...
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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 ...
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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 ...
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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 $K$-...
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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 ...
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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(...
  • 432
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 ...
  • 432
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?
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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 ...
  • 71
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 ...
  • 2,119
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$ ...
  • 499
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^...
  • 5,643
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 ...
  • 5,643
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]/(...
  • 71
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} ...
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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 ...
  • 346
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 ...
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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 ...
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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 ...
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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 ...
  • 1,779
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 =...
  • 903
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 $\...
  • 1,001
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 ...
  • 1,583
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 ...
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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 ...
  • 17.6k
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^{\...
  • 458
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}...
  • 1,779
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 ...
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