Questions tagged [kt.k-theory-and-homology]
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
926
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Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2
Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...
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Roadmap for Quillen 1
Question
Suppose you grasped and enjoyed reading Quillen's "Higher Algebraic K-theory I". Now, if you could go back in time to when you started studying algebraic topology and create a reading list / ...
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For an additive category $\mathcal{A}$, how does one show $K_0(\mathcal{A})\cong K_0(\mathcal{K}^b(\mathcal{A}))$?
This is an exercise in §3.13 Beilinson's notes on homological algebra. He doesn't specify but I'm pretty sure $K_0(\mathcal{A})$ is defined as the free group on the isomorphism classes of $\mathcal{A}$...
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Question on Cuntz' proof of Bott periodicity
I am reading the presentation of Cuntz' proof of Bott periodicity for $C^*$-algebras in Wegge-Olsen (Thm. 11.2.1). Here one considers the short exact sequence of $C^*$-algebras
$$0 \longrightarrow \...
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Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
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Dixmier trace, Wodzicki residue and topological index
There are a well-known facts about Dixmier trace and Wodzicki residue. Let $P$ be an elliptic pseudodifferential operator of degree $−n$ on a compact Riemannian manifold $(M,g)$, than its Dixmier ...
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What are the generator for $K_1(C(\mathbb{T})\otimes\mathbb{K})$?
I've been working computing several K-groups associated to some $C^*$-algebras involved with my master's thesis, however I've just got stucked finding some generators for $K_1(C(\mathbb{T})\otimes\...
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Examples of noetherian local rings $R$ such that $K'_0(R)$ is not isomorphic to $\mathbb Z$
Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'_0(R) = K_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) ...
- 1,229
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Criteria for a map of rings to induce an equivalence on K-theory?
Algebraic $K$-theory is Morita invariant, but surely it does not detect Morita equivalence. What are some examples of rings (or ring spectra) $R$ and $S$ that are not Morita equivalent, but ...
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Functoriality of Hochschild cohomology for Drinfeld quotients
Let $C$ be a dg category and $C \to D$ a Drinfeld localization. Is there an induced pushforward map on $\operatorname{HH}^*(C) \to \operatorname{HH}^*(D)$, where $\operatorname{HH}^*$ denotes the ...
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Looking for an invariant similar to algebraic K-theory
I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties:
a) It attach to each small ...
- 40.2k
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Dennis trace map for stable $\infty$-category, naively
I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of ...
- 40.2k
4
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Split cofibrations up to quasi-isomorphism
$R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules).
Suppose that $A_{\bullet}\rightarrow B_{\bullet}\rightarrow B_{\bullet}/A_{\bullet}$ a short ...
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Descent properties for rational topological cyclic homology
Descent properties can be extremely useful for studying $\operatorname{TC}$ (topological cyclic homology), since it is a sheaf in many well behaved topologies.
I was wondering what is known about $\...
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Left and right topological K-theory of Banach algebras
Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette.
The definition is based on ...
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Exact subcategory with trivial Grothendieck group: what are the consequences and examples
Let $C$ be (a full) exact subcategory of the category of $R$-modules. We suppose that $C$ is essentially small. If the Grothendieck group $K_{0}(C)=0$, what can be said about the higher groups $K_{n}(...
- 153
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Motivation for Karoubi envelope/ idempotent completion
This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered ...
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Application of higher categories in algebra
Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher ...
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Modern context for hypercohomology spectra
In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) Thomason develops an elaborate theory of hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of ...
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When is a virtual bundle an actual bundle?
As far as I understand, one can make a monoid from the space of vector bundles on a (compact) manifold $M$, with respect to the direct sum operation $\oplus$. In order to make this into a group, one ...
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Relations between rational algebraic K-theory and Chow groups
A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic
$ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...
- 6,158
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Vector bundles on $\mathbb{A}^n / G$
Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...
- 2,260
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$KO_*$ groups of $\mathbb{R}P^\infty$, "Snaiths" theorem for $KO$
I posted this question some days ago at math.stackexchange, but didn't receive an answer.
I have two questions:
I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$?
The ...
- 301
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3
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K-theory of free $G$-sets and the classifying space, and generalization
$\newcommand\Sets{\mathrm{Sets}}\DeclareMathOperator\Nerve{Nerve}$Let $G$ be a finite group, $\mathcal{G}^0$ be the category of finite free
$G$-sets and isomorphisms between them. Then $\mathcal{G}^0$...
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Confusion regarding a definition of cycles
For a projective variety $X$ over $\mathbb{C}$, let us denote by $CH_k(X)$ the Chow group of $k$-cycles of $X$, modulo rational equivalence. Also, let $CH_k(X)_{hom}$ denote $k$-cycles modulo ...
- 737
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Hauptvermutung for non-manifolds
The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement.
People are mostly interested ...
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Recover the field from its Milnor K-groups
For every field $F$, consider $K_n^M(F)$ the $n$-th Milnor K-group of $F$ for each $n \in \Bbb N$, and form the Milnor K-ring $K^M(F)=\oplus_{n \geq 0}K^M_n(F)$. For instance, $K_1(F)=F^{\times}$.
...
- 6,158
4
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143
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Alternative definitions of Weibel's homotopy K-theory
Consider a sort of $\mathbb{A}^1$-homotopy-stable algebraic $K$-theory for rings constructed as follows.
For $K_0$ we take a symmetrization subject to natural direct sum operation of $\mathbb{A}^1$-...
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141
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A kind of isomorphicity of vector bundles
Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))...
- 312
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Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
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Chromatic Homotopy Theory and Physics
Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
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Computing K-theory for cellular vector bundles
One of the most computationally convenient properties of singular cohomology $X \mapsto H^\bullet(X;\mathbb{Z})$ is the fact that one can extract it from a good cover $\{U_i\}$ of $X$ via Cech ...
- 15.4k
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Cohomology theory with only one Adams operation?
Let $E$ be a multiplicative cohomology theory. Fix a prime p. Call a ring map $\psi^{p}:E\rightarrow E$ an Adams operation if it lifts the Frobenius map $E/p\rightarrow E/p$.
It is of course well-...
- 1,131
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202
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Bott periodicity homeomorphisms for spaces of Clifford extensions
I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules).
Let $W = \mathbb{R}^{\...
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Simpicial resolution in Cotangent Complex
I am reading Cotangent complex from Loday's book Cyclic Homology. My doubt is related to the simplicial resolution of $k$- algebra $A$ which is used in the definition of cotangent complex.. Let me ...
- 609
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Why does K-theory need schemes to be Noetherian?
The definition of K-theory of a scheme $X$ is defined as
$G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$.
But usually the schemes are required to be (at least locally) Noetherian, and ...
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Homotopy type of a $4$-manifold with finite fundamental group (paper by S. Bauer)
I'm studying Stefan Bauer's paper
The homotopy type of a 4-manifold with finite fundamental group. In: tom Dieck T. (eds) Algebraic Topology and Transformation Groups. Lecture Notes in Mathematics,...
- 6,000
8
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Pinwheel Tilings and C* algebras, K-theory
I was reading that spaces of tilings can be related to C*-algebras and K-theory. Here is an example of the pinwheel tiling. [1]
They construct a space called $\mathcal{A}\mathbb{T}_{pin}$ and show ...
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Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?
Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the ...
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Is the deformation along flow lines a simple homotopy equivalence?
Let $(M,g)$ be a compact, smooth $n$-manifold with boundary $\partial M$ and let $f: M \to [a,b]$ be a Morse function, whose critical points are interior and which satisfies $f^{-1}(b) = \partial M$.
...
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When and why are Adams operations "non-negative"?
We can think of the unary operations in a lambda-ring as integer linear combinations of Young diagrams; for example the operation $\lambda^n$ corresponds to the Young diagram with $n$ rows and one ...
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643
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$d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$
Cross posted from here after no responses and a bounty being placed on the question.
Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-...
- 835
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Chern character of coherent sheaf on singular variety
Does there exist a definition of Chern character (or Chern classes) for a coherent sheaf $\mathscr{F}$ on a singular variety $X$? In this case I might not be able to find a projective resolution for $\...
- 607
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91
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elementary matrices over a regular ring
Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...
- 3,432
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225
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Equivariant Venice Lemma
In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as
Theorem: For ...
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226
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K-theoretic derivation of Bézout theorem
In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says:
"When the ground field $k = \mathbb C$, Bézout’...
2
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1
answer
221
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$C^*$-algebras appearance in study of Lie groupoids and differentiable stacks
I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...
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5
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1
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350
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K-theory for a (geometric) stack
There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...
- 5,931
2
votes
0
answers
765
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Applications of Jordan-Holder theorem in an abelian category
The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length.
This theorem holds ...
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"a sign that one should be computing K-theory"
Allen Knutson said here in comments below the question that
I generally regard torsion in (co)homology as a sign that one should be computing K-theory instead, which has less of it.
I know one ...
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