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6
votes
0answers
424 views

Inverse Galois Problem…and parallelizable vector fields?

Usual approaches to the inverse Galois problem start with realizations of a group $G$ over a larger field, and then try to specialize to ${\Bbb Q}$. One could also start by building suitable objects ...
0
votes
0answers
174 views

Whitehead group [closed]

Whitehead group (WG) is known for some groups (e.g. free abelian group, cyclic group,Braid group etc). For example-1)The Whitehead group of the trivial group is trivial. 2)The Whitehead group of a ...
1
vote
1answer
78 views

Is there something like an “f-regular” $K$-theory?

Let $R$ be a ring and $f\in R$. Is there something like an $f$-regular $K$-theory group of $R$ based on the category of $f$-regular $R$-modules, i.e. modules that do not have any $f$-torsion? If ...
7
votes
1answer
368 views

Waldhausen Additivity in a More General Context

The following arose when I was thinking about a talk at the Midwest Topology Seminar: Background I want to consider a generalization of a Waldhausen-like structure on a category $C$ with 0-object ...
-1
votes
0answers
77 views

Isomorphism of tensor product

The following question is related with the definition of the homothety operators defined in algebraic $K$-theory, more specific, in the Witt-vector module structure of the Bass Nil-groups of a ring ...
8
votes
0answers
76 views

Gersten complexes in Quillen's and Milnor's K-theories

Consider a good enough scheme $X$ (e.g. an algebraic variety over a field). Let $X_i$ be the set of points of dimension $i$ in $X$. Then we have the Gersten complex in Quillen's K-theory: $$ ...
14
votes
0answers
237 views

Group with finite outer automorphism group and large center

Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$? A motivation is ...
5
votes
1answer
183 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} ...
11
votes
1answer
188 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 ...
10
votes
1answer
969 views

Explicit description of boundary map in algebraic K-theory

Recall that for a DVR A with fraction field F and residue field k, there is a "localization" fiber sequence in algebraic K-theory, $$K(k) \rightarrow K(A) \rightarrow K(F).$$ In Remark 5.17 of his ...
2
votes
0answers
125 views

Do we have the following “devissage commutative diagram” in K-theory?

Let $X$ be a non-reduced Noetherian scheme. We define $K^0(X)$ to be the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ to be the Grothendieck group of the derived category ...
2
votes
1answer
133 views

Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...
18
votes
1answer
413 views

Can topological cyclic homology compute Picard groups?

Let $K$ be a number field, and $\mathcal{O}_K$ its ring of integers. Then there is an isomorphism $$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K)$$ where $Pic(\mathcal{O}_K)$ is the ...
0
votes
0answers
79 views

$K_1(R)$ and splitting

Let $R$ be a commutative ring with unit. Under what conditions does the following exact sequence split? $1\to E(R)\to Gl(R)\to K_1(R)\to 1$.
2
votes
1answer
213 views

Map from algebraic K-theory to topological K-theory

Suppose that $A$ is a Banach algebra with unit. We can consider $GL(A)$ as a topological group in either the discrete topology or the topology that it inherits from the norm topology of $A$, and the ...
3
votes
0answers
135 views

Equivalence of various definitions of arithmetic Chow groups

If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ ...
1
vote
1answer
191 views

What is $K_2(\mathbb{Z}[x,x^{-1}])$?

The question is as in the title: is $K_2(\mathbb{Z}[x,x^{-1}])$ known?
5
votes
0answers
129 views

Gersten Conjecture for Milnor K-theory

The Gersten conjecture for Milnor K-theory, saying that the Gersten complex $$0\rightarrow \mathcal{K}^M_X\rightarrow \oplus_{x\in X^0}i_{x*}(K^M_n(x))\rightarrow \oplus_{x\in ...
10
votes
2answers
682 views

Symplectic K-theory

For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...
9
votes
4answers
992 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 ...
0
votes
0answers
91 views

Central extensions of SL2(R) by U(1) ?

Can somebody please tell me what are the central extensions of SL2(R) by U(1), that is, what is $H^2(SL2(R), U(1)) $ ? Thank you
1
vote
0answers
137 views

map in K-theory

I have a very stupid question on a map I have seen in K-theory. The situation is as follows: $X$ is a smooth variety over a field $k$ and $\iota: Z \to X$ is the inclusion of a smooth closed ...
5
votes
0answers
199 views

Beyond Bloch-Kato conjecture [repost from math.stackexchange]

I have just asked this question on math.stackexchange, and would like to repost it here: The norm residue isomorphism theorem establishes that the norm residue map between Milnor K-theory of a field ...
-1
votes
1answer
347 views

Coherent sheaves on Proj

Roughly speaking , the question is : when a f.g. graded module induces a trivial coherent sheave on Proj ? More precisely, let S be a (complex) graded polynomial algebra, where the variables have ...
14
votes
1answer
604 views

Link: Serre's intersection formula <-> Bloch-Quillen Thm / When only intersecting divisors, is there 'shorter' approach of proof known?

In very short: When proving the equivalence of intersection theory constructed through (Milnor) K-sheaves and their product vs. defining the product via Serre's local multiplicity formula + moving, I ...
1
vote
0answers
148 views

The definition of $SK_1$ for an arbitrary ring

Let $R$ be a unitary associative ring. If $R$ is commutative, then one defines $SK_1(R)$ as the quotient $$SK_1(R)=SL(R)/E(R)$$ (Definition 2.8 of ...
4
votes
0answers
119 views

residue and regulator

Let $C$ be a curve defined over $\mathbb{Q}$. The regulator is a map $$ reg: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}). $$ Here $K_2(C)_{\mathbb{Q}}$ is the K-group tensor ...
13
votes
1answer
467 views

References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...
3
votes
1answer
106 views

Stable rank of finite rings

Has any finite ring (not necessarily commutative) always stable rank 2 ? How do you prove that or does it follow from something ? May be this question is trivial but I'm not familiar with K-theory.
11
votes
1answer
482 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 ...
6
votes
1answer
371 views

K-theory of the h-cobordism category

I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group ...
5
votes
1answer
283 views

Is there a simple proof that Milnor $K_2$ of a number field is torsion?

This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...
3
votes
0answers
76 views

The 4-th generator of $K_1$ group for 3-dimensional NC tori algebra

An $n$-dimensional NC torus algebra $A_\theta^{(n)}$ is defined for any antisymmetric $n\times n$ matrix $\theta$ of real numbers as the universal $C^*$-algebra, generated by unitaries ...
0
votes
0answers
93 views

Reciprocity laws in different dimensions

Let $M/L/Qp$ be a finite galois abelian extension of local fields and define $\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to ...
1
vote
0answers
77 views

Relation between 1-dimensional and 2-dimensional reciprocity maps

Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define $\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, ...
1
vote
0answers
161 views

Surjectivity of the algebraic K-functor

Let $R \to S$ be a surjective morphism of commutative rings. For a fixed integer $q$, is there any known condition under which the resulting morphism of the K-groups, $K_q(R) \to K_q(S)$ is ...
1
vote
1answer
88 views

Equivalence of definitions of the Milnor $K$-groups

In Kurihara's paper: "The exponential homomorphisms for the Milnor $K$-groups and an explicit reciprocity law" he difines, in the first page, the $q$-th Milnor K-group for the ring $R$ as ...
7
votes
1answer
187 views

Transfers on Bloch groups and scissors congruence groups

I have a couple of questions concerning existence and description of transfers for Bloch groups and scissors congruence groups/pre-Bloch groups. To fix notation and recall definitions: From the ...
3
votes
1answer
220 views

Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
9
votes
2answers
436 views

Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that $$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$ i.e. the connected ...
15
votes
1answer
229 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" ...
6
votes
2answers
749 views

an elementary question on K-theory

Sorry for asking such an elementary question. 1) What is Quillen's $K_1$ of a (nice) scheme $X$? If $X=Spec(k)$, I guess one gets $k^\times$, is that correct? What about the case of a curve $C$ ...
4
votes
0answers
114 views

The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?

There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...
1
vote
1answer
318 views

Reference request: Grothendieck´s period conjecture?

I would like to know if Grothendieck published something about this conjecture? Is there some book (or expository article) about this conjecture? Is there any connection between this conjecture and ...
5
votes
1answer
224 views

Definition of a cylinder functor in Waldhausen's K-theory

In Waldhausen's Algebraic K-Theory of Spaces, he defines a cylinder functor on a category $\mathcal C$ with cofibrations and weak equivalences (henceforth called a Waldhausen category) as the ...
6
votes
0answers
86 views

Bloch group, hyperbolic manifolds and rigidity

I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to ...
8
votes
2answers
474 views

is there a p-adic Borel theorem?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. ...
6
votes
1answer
337 views

Detection of stable homotopy by K-theory spectra

This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from ...
2
votes
0answers
170 views

What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?

Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures? What is the ...
5
votes
0answers
142 views

Strengthening of Suslin's rigidity argument?

To fix the situation, let $k$ be an algebraically closed field, and let $C$ be a smooth projective curve over $k$. Suslin's rigidity argument implies in particular that any class in ...