Questions tagged [kt.k-theory-and-homology]
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
43 questions from the last 365 days
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109
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Calculi of pseudodifferential operators and K-theory
I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...
- 319
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80
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Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence
$$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 613
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94
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Action of Adams operations on $K_*(K)$
Any k-theory cooperation $K_*(K)$ can be uniquely expressed as some finite Laurent series $f(u,v)$ with certain generators $u,v$ arising from the left and right action of $\pi_*(K)$. That I understand....
4
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2
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256
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Waldhausen S-construction for exact categories
Let $\mathcal{C}$ be an exact category. Then, we can consider $\mathcal{C}$ as a Waldhausen category, where the cofibrations are admissible monomorphisms. By Waldhausen $S$-construction we know that $...
- 167
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129
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Is $K_0(\mathrm{Vect}(X))\to K_0'(X)$ injective for a proper variety $X$?
Let $X$ be an integral scheme, proper over an algebraically closed field $k$. Let $\mathrm{Vect}(X)$ be the exact category of finite locally free $O_X$-modules. Let $K_0(\mathrm{Vect}(X))$ be its ...
- 615
7
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265
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Connecting Harris's proof of periodicity to the Bott element
$\newcommand\ku{\mathrm{ku}}$In Denis Nardin's lectures on stable homotopy theory, he reproduces Bruno Harris's proof of Bott periodicity from group completion and goes on to identify the Bott element ...
- 1,337
2
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0
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205
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What role does homotopy play in Karoubi's K-Theory?
In Karoubi's book K-Theory An Introduction, he defines the groups $K^{p,q}(\mathcal{C})$ for a pseudo-abelian Banach category as equivalence classes of triples $(E,F,\alpha)$, where $E,F \in \mathcal{...
- 233
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249
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Phantoms and Geometry
Let $\mathcal{D}(X)$ be the bounded derived category of coherent sheaves on a smooth projective variety $X$. An autoequivalence $\Phi: \mathcal{D}(X) \to \mathcal{D}(X)$ is called phantom if it ...
- 71
2
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0
answers
162
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Equivariant Künneth formula for partial flag variety
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$. Let $P$ be a parabolic subgroup of $G$, $\mathscr{F}:=G/P$ the partial flag variety associated to $P$. For a $G$-variety $X$, ...
- 653
2
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1
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399
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${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 2,964
4
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1
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418
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Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
- 41
6
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1
answer
186
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Plus construction of the product spaces
I am newly learning plus construction in topology. My question is how to prove the following:
The plus construction of the product of two CW complexes is homotopically equivalent to the product of ...
- 613
5
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0
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255
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Necessity of Banach category for K-theory
In Karoubi's book on K-Theory (K-Theory, An Introduction), he introduces the groups $K^{p,q}(C)$ for a pseudo-abelian Banach category $C$.
My question is whether the requirement that $C$ be a Banach ...
- 233
1
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0
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111
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Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
- 11
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1
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271
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$K$-theory and its dual
I am reading a paper which uses some $K$-homology which is the homology theory dual to $K$-theory can be defined using the homotopy theoretic formulation:
$$
K_\ast(X)\cong\pi_\ast(K\wedge X).
$$
...
- 81
3
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answers
122
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Canonical basis in equivariant K-theory of the Springer resolution
In Definition 15.0.2 of the notes from a course by Bezrukavnikov there is a characterization of canonical basis in K-theory of a Springer fiber which is due to Lusztig. This characterization is in ...
- 2,964
2
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59
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Proof of K-theory version of Poincaré duality for closed spin$^c$ manifold
Let $X$ be a oriented Riemannian $n$-manifold. It seems to be well known that if $X$ is closed and spin$^c$ then the Poincaré duality reads
$$ K^*(X) \cong K_{*+n}(X)$$
between the complex K-theory ...
- 63
4
votes
1
answer
216
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On Connes' fomulae of pairing between cyclic cohomology and K-theory
The following proposition comes from Connes' paper in IHES. See the link Non-commutative differential geometry.
On page 109, Proposition 15. of Part II, he claims that
(1) The following equality ...
- 41
3
votes
1
answer
265
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Base change in Chriss-Ginzburg
Below is a fragment of the book by Chriss and Ginzburg. Proposition 5.3.15(b) is stated in $K$-theory. My question is, does the same conclusion (and proof?) of proposition 5.3.15(b) (i.e. base change) ...
- 2,964
13
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1
answer
558
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Intuitive reason for periods of 2 and 8 in Bott periodicity?
Is there a reasonably simple explanation for why Bott periodicity for $U$ and $O$ have periods 2 and 8, respectively? For example, in the $h$-cobordism theorem the requirement that $n \geq 5$ has the ...
- 131
1
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240
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Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
- 356
3
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2
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342
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Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism
I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map.
More precisely, I'...
- 2,907
5
votes
1
answer
144
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Equivariant KR-theory of representation sphere
I would like to say my question first.
Let $G$ be a compact Lie group acting on a good space $X$ in a good way. Let $V$ be a $G$-representation whose real dimension may be less than 8, and let $S^V$ ...
- 1,040
6
votes
1
answer
426
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Nilpotency of generalized cohomology
$\newcommand\pt{\mathrm{pt}}$Let $(X,\pt)$ be a connected, pointed, finite CW complex and let $h$ be a generalized cohomology theory. Let $\smash{\tilde{h}}^*(X)$ denote the kernel of restriction $h^*(...
- 959
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43
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How to define a family of Hilbert $A-B$-bimodules $ \pi \ : \ M \to X $, parametrized by a $C^*$-algebra $X$?
Let $A$ and $B$ two $ C^* $ - algebras.
I would like to define a functor $ X \to \mathrm{Bimod}_{A,B} (X) $ which associate to any object $X$, the set of isomorphism classes of a family of Hilbert $A-...
- 595
10
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0
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225
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Third homology of simply connected Chevalley–Demazure group schemes
I’ve been studying the group of $\mathbb{Z}$-points of the simply connected Chevalley–Demazure group scheme of type $E_7$, denoted $G_{\text{sc}}(E_7,\mathbb{Z})$; see Vavilov and Plotkin - Chevalley ...
- 545
3
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2
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161
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Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
It is known that $K_* K^* $-theory is a common generalization both of $K$-homology and $K$-theory as an additive bivariant functor on separable C*-algebras.
Is it possible to construct a $ H_* H^* $-...
- 595
4
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1
answer
187
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Poincaré duality, spin structure and Connes' reconstruction theorem
In the paper "ON THE SPECTRAL CHARACTERIZATION OF MANIFOLDS", A.Connes proves two "reconstruction" theorems. Informally, they are like :
Theorem 1 : Commutative spectral triple $(A,...
- 63
5
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1
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195
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What is the most general notion of exactness for functors between triangulated categories?
For triangulated categories $T,T'$ I would like to define "weakly exact" functors as those that respect cones, that is, $F(Cone f)\cong Cone(F(f))$ for any $T$-morphism $f$, and I do not ...
- 16.9k
7
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1
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274
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Proof of the equivalence of spectra $(\mathbb{S}^{-1} \otimes \mathbb{S}^{-1})_{h \Sigma_2} \cong \Sigma^{-1} \mathbb{RP}_{-1}^{\infty}$
$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\Th}{Th}$I am trying to give a hands-on proof of the equivalence of spectra in the title. I am using the definitions $\mathbb{RP}^{\infty}_{-...
- 348
3
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2
answers
246
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Explicit description of transfer for $K_1$
Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence
$$
\ldots \rightarrow K_i(R/s) \rightarrow K_i(...
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0
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101
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A roof genus of high dimensional lens space
Let $p$ be a natural number, and for $i\in \{0,
..., p-1\}$,
denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$.
Let $a=(a_{1},\ldots a_{d}) $ ...
- 2,630
4
votes
1
answer
319
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K-theory of simplicial rings
Given a simplicial commutative ring $R$, you have the usual K-theory spectrum $K(R)$ defined as the group completion of the symmetric monoidal category of finitely-generated projective modules over $R$...
- 197
4
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0
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87
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Shearing maps on domain of assembly map in algebraic $K$-theory
Let $H \to G$ be an inclusion of abelian groups, and let $R$ be a ${\Bbb Z}[H]$-algebra. Assume that the assembly map ${\Bbb S}[BG] \otimes_{\Bbb S} K(R \otimes_{{\Bbb Z}[H]} {\Bbb Z}[G]) \to K((R \...
6
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162
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$K_0$ of arithmetic surfaces
In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated ...
1
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0
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124
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Computing the induced homomorphisms of derived functors using acyclic resolutions
Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
- 38.6k
7
votes
1
answer
101
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Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces
From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...
- 71
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86
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Projectivity of equivariant K-theory of toric variety
I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups.
In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
- 959
2
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0
answers
129
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Flag variety type Beilinson resolution
The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
- 653
2
votes
1
answer
99
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Stabilizing conjugacy classes of integer matrices
$\DeclareMathOperator{\Conj}{Conj} \DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\id}{id} \newcommand\Z{\mathbb{Z}}$
For an $n \times n$ integer matrix $A \in \GL_n(\Z)$, let $\Conj(A)$
be the ...
- 23
13
votes
1
answer
2k
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Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection
I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
- 155
4
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0
answers
110
views
category of vector bundles with connections and its K-theory
For the category of Hermitian vector bundles with unitary connections, an object is (of course) a Hermitian vector bundle with a Hermitian metric and a unitary connection $(E, g^E, \nabla^E)$. For ...
- 1,173
3
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0
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90
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When does homology preserve inverse limits of Eilenberg-MacLane spaces?
Let $... \to G_3 \to G_2 \to G_1$ be an inverse system of abelian groups and $G$ the limit of the system. By a theorem of Goerss the integral homology of the Eilenberg-MacLane space $K(G,n)$ for $n &...
- 499