All Questions
Tagged with algebraic-k-theory homotopy-theory
47 questions
2
votes
0
answers
114
views
The induced map between push outs in an exact infinity category
Let $(\mathcal{C} , \mathcal{M} , \mathcal{E})$ be an exact $\infty$-category. (I am following the definition in Higher Segal Spaces $I$ by Dyckerhoff and Kapranov). Assume that $F$ is a cofiberation ...
- 167
3
votes
0
answers
107
views
rational homology of SO(2,1) over number fields
Let $\mathrm{SO}(2,1)$ be the special orthogonal group defined by the quadratic from $q(x,y,z)=x^2+y^2-z^2$.
This is a connected non-simpy connected algebraic group.
Now, let $F$ be a number field, ...
7
votes
1
answer
843
views
Algebraic K-theory and Witt groups
Let $S$ be a ring with involution (with 2 invertible). Suppose that the non connective algebraic K-theory $K(S)$ is 0 (i.e. $K_{n}(S)=0$, for all $n$).
Can we say something about the (higher) Witt ...
- 855
5
votes
3
answers
443
views
Group completion of a monoid (braid groups)
Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups.
For a discrete group $G$, we let $BG$ to be the classifying space of $G$.
After reading this question, I was ...
- 140
12
votes
1
answer
429
views
Plus construction on Simplicial Sets?
I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here.
Write $\mathsf{sSet}$ for the category of simplicial sets and $...
- 174
8
votes
0
answers
123
views
The homotopy inverse on Quillen's $S^{-1}S$ construction
Suppose $S$ is a symmetric monoidal groupoid. Take Quillen and Grayson's $S^{-1}S$-construction, which is a symmetric monoidal category with objects given by pairs $(m,n)$ and maps given by ...
- 2,303
4
votes
1
answer
172
views
The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$
$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
- 43
8
votes
1
answer
600
views
Why isn't the anchor map in Lurie's "Rotation Invariance in Algebraic K-Theory" zero?
I think this is a silly question, but I'm quite confused. In Lurie's "Rotation Invariance In Algebraic K-Theory" Notation 3.2.4. he defines a filtered spectrum $\mathbb{A}$ given by
$$\...
- 455
2
votes
0
answers
103
views
Generalisations of Volodin's construction of algebraic K-theory
In a previous question I asked about uses of Volodin's construction of the algebraic K-theory of rings. Some of these are striking and it made me wonder whether those proofs can be extended. This ...
- 637
2
votes
0
answers
178
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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, ...
- 6,018
2
votes
0
answers
210
views
pseudo-abelian category / Karoubian category in K-theory
A pseudo-abelian category or Karoubian category $\mathcal{C}$ is a preaditive
category such that every idempotent morphism
$i: A \to A$ in $\mathcal{C}$ has a kernel and consequently a
cokernel as ...
- 6,018
11
votes
2
answers
864
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 ...
- 63.9k
12
votes
1
answer
489
views
Homological stability and Waldhausen A-theory
$\DeclareMathOperator{\Diff}{Diff}$
From the work of Galatius - Randall-Williams and Berglund - Madsen we have homological stability (with respect to g) of $B\Diff_\partial (W_{g,1})$ and rational ...
- 5,839
7
votes
0
answers
223
views
Does the Whitehead torsion of a homotopy equivalence depend on the CW structure?
In the (old) literature I've seen referenced the question of whether simple homotopy equivalence is a topological property, i.e. whether it depends only on the underlying space, rather than the ...
- 5,839
9
votes
1
answer
1k
views
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 / ...
- 650
7
votes
0
answers
266
views
Homotopy invariant analogues of localizing invariants
Given a localizing invariant, $E$, valued in spectra, by following the recipe prescribed in 3.13 of https://arxiv.org/abs/1808.05559, we can define a homotopy-invariant version of $E$ on $H\mathbb{Z}$-...
- 532
5
votes
1
answer
579
views
Topological Hochschild homology using equivariant orthogonal spectra
In the Hesselholt-Madsen paper "On the K-theory of finite algebras over Witt vectors of perfect fields", the authors develop some results concerning the Topological Hochschild homology (THH) of ...
- 254
3
votes
1
answer
185
views
Spherical objects and K-theory
My question goes as follows: given a ring $R$ (with $1\neq 0$). Define $\mathbf{Perf}_{R}$ the the category of Prefect complexes over $R$. I want to prove that the Waldhausen $K$-theory of the ...
- 511
5
votes
0
answers
415
views
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 ...
- 254
5
votes
0
answers
311
views
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,...
- 6,018
10
votes
1
answer
604
views
Descent properties of topological Hochschild homology
Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent?
Adaptations of the arguments appearing in ...
- 532
11
votes
1
answer
432
views
$K_3(\mathbb{Z})$ and $\pi ^S_3$
This is an afterthought on this MO question, and also on Gannon's book mentioned there, about $K_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the ...
- 38k
3
votes
0
answers
260
views
Homotopy equivalence of $K$-theory and $G$-theory
Let $X$ be regular variety then it is known that $Q(Vect(X))\cong Q(\mathcal{M}_X)$. Where $Q$ is the Quillen's q-construction and $\mathcal{M}_X$ is the category of coherent sheaves on $X$. You can ...
- 5,901
4
votes
0
answers
252
views
Effects of the first algebraic K-theory on the higher algebraic K-theory
Is there any counterexamples known to the following statement? ($A$ a regular noetherian integral domain of finite Krull dimension)
If $A^{\times}$ is finitely generated then $K_n(A)$ is finitely ...
- 5,901
6
votes
0
answers
220
views
Adams operation on Q-construction of fields
Let $F$ be a field that we want to compute its rational algebraic $K$-theory using the Quillen's $Q$-construction. Let $QF$ be the $Q$ construction of the category of finite dimensional vector spaces ...
- 5,901
3
votes
0
answers
127
views
Adams operation on the rational homology
The Adams operation acts on the algebraic $K$-theory of $R$ but the action as far as I know doesn't come from a endo-functor on the category of projective modules over $R$. For the $K_0$ there is an ...
- 5,901
14
votes
1
answer
1k
views
Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?
It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.
This statement is ...
- 2,227
22
votes
1
answer
2k
views
Is algebraic $K$-theory a motivic spectrum?
I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy ...
- 63.9k
11
votes
0
answers
264
views
Direct proof of the equivalence of symmetric monoidal $K$-theory and exact sequence $K$-theory?
When all exact sequences split in $C$, we have $\Omega B C \simeq K(C):=\Omega Q(C)$. Heuristically, this is because the space of upper-triangular matrices is contractible. Can this be made precise? I ...
- 63.9k
12
votes
2
answers
1k
views
$K$-theory backwards
Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed?
That is, ...
- 63.9k
10
votes
1
answer
553
views
Waldhausen $K$-theory before group completion
$K$-theory is often billed as the "universal way to split exact sequences". But it seems we're too anxious to group-complete things to actually take the slogan at face value.
Consider the following $\...
- 63.9k
6
votes
1
answer
397
views
Do topological commutative monoids model all 0-connective spectra (after group completion)?
Of course, before group completion, topological commutative monoids do not model all connected $E_\infty$ spaces -- among the grouplike ones, they model only products of Eilenberg-Mac Lane spaces. But ...
- 63.9k
10
votes
1
answer
373
views
Do symmetric monoidal groupoids model all connective spectra?
Thomason showed that symmetric monoidal categories model all connective spectra. But can it be done with groupoids? In order for this to be the case, the group completion prices process must be very ...
- 63.9k
7
votes
1
answer
807
views
Etale and Algebraic K-theory with rational coefficients
We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
- 71
2
votes
0
answers
240
views
Algebraic K theory, Karoubi completion and splitting
Suppose $\mathcal{C}\subset \mathcal{C}'$ is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, $\mathbb{C}$), such that $\mathcal{C}$ is faithfully embedded in ...
- 7,952
9
votes
0
answers
745
views
When does algebraic K theory behave like a cohomology theory
Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (...
- 7,952
12
votes
0
answers
551
views
Goodwillie's notes from MSRI Lecture Series
Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...
6
votes
0
answers
144
views
$K_0$ an $KH_0$ of a normal crossing variety
Let $k$ be a field (say, algebraically closed to fix the ideas) and let $X$ be a strict (aka simple) normal crossing variety over $k$, so that $X$ is union of regular varieties with intersection that ...
- 61
5
votes
1
answer
300
views
Map between homotopy groups of O, related to J-homomorphism and K-theory of Z
Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi_{8s+1}(SO) = \mathbb Z/2$ in $\pi_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres.
Now regard $\pi_{8s+1}^s = \pi_{8s+1} ((B\...
- 11.9k
12
votes
1
answer
768
views
The multiplication on $THH$ of finite fields
Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...
- 303
14
votes
1
answer
800
views
Is there a category whose isomorphisms are precisely the simple homotopy equivalences?
If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...
- 18.7k
19
votes
2
answers
702
views
Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...
user1437
7
votes
1
answer
412
views
Can any suspension spectrum be realized as Waldhausen K-theory?
If we consider the category of finite, pointed sets and declare cofibrations to be inclusions and weak equivalences to be bijections, we get a Waldhausen category whose $K$-theory spectrum is the ...
- 13.5k
19
votes
4
answers
3k
views
Algebraic K-theory and Homotopy Sheaves
Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory ...
- 365
24
votes
3
answers
4k
views
Plus construction considerations.
In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = GL(R)/E(...
- 1,105
34
votes
1
answer
2k
views
Is every ''group-completion'' map an acyclic map?
I start with a longer discussion which will result in a precise version of the question. I am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...
- 20.9k
6
votes
0
answers
312
views
homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence
Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
- 5,575