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3
votes
2answers
56 views

$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von Neumann ...
1
vote
1answer
255 views

research articles in topology/geometry [closed]

There is a saying "Do you read the masters?" I want to read some basic papers in Topology/geometry... I can not clearly state what is basic as of now... My back ground includes course in ...
1
vote
1answer
150 views

Milnor-Witt K-theory for finite fields

Are $K^{MW}_*(\mathbb{F_q})$ and $K^{MW}_n(\mathbb{F_q})$ already known? Where can I read about it?
6
votes
1answer
388 views

K-Theory and Tame Symbol

This might be a little bit specific but here it goes. While reading a paper (Brauer-Manin pairing...) by Yamazaki, I encountered this definition. Let $V$ be a variety and $y$ be a one dimensional ...
11
votes
1answer
412 views

Plugging $1-x$ into Schur polynomials

I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x_1,\ldots,x_k)\mapsto ...
9
votes
0answers
100 views

Finiteness of torsion in $\mathcal{K}_2$-cohomology

Let $F$ be a number field, $C$ be a smooth projective curve over $F$ and $\mathfrak{C}$ be a proper regular model. I am interested in $\mathcal{K}_2$-cohomology, i.e., Zariski cohomology of the sheaf ...
2
votes
0answers
168 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 ...
9
votes
0answers
281 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 ...
5
votes
1answer
294 views

Waldhausen and Segal's delooping machinery

I was recently thinking about the proof of a theorem where Waldhausen compared the Segal's delooping machinery with his, in the case when the cofibration is splittable (sec.1.8 in 'Algebraic ...
9
votes
0answers
176 views

Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ? I ...
6
votes
2answers
327 views

Pullback along Frobenius morphism

Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...
6
votes
2answers
419 views

K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ...
11
votes
0answers
370 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, ...
14
votes
1answer
472 views

Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
6
votes
0answers
99 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 ...
11
votes
1answer
856 views

When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims. On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams ...
7
votes
2answers
191 views

An extension of $K$-theory to topological $^*$-algebras

What I have in mind is the following: a (sequence of) functor(s) $K_\bullet$ on the category of topological $^*$-algebras (with values in the category of commutative groups) that satisfies (among ...
4
votes
2answers
152 views

Differential structures and K-homology groups

What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...
15
votes
1answer
504 views

Is SL(n,Z[x]) generated by transvections?

Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?
14
votes
2answers
429 views

What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?

Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.) In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...
2
votes
0answers
146 views

Proof of Merkurjev's Theorem in “The Algebraic and Geometric Theory of Quadratic Forms”

I just have a little question about the above mentioned proof. I'm thinking for days, but I'm still not getting it. For those who have the book (or want to look it up via google books etc.), it's the ...
10
votes
2answers
353 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 $Z := \sum\limits_in_iZ_i$ ...
7
votes
0answers
123 views

Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...
4
votes
0answers
222 views

Pull-push formula?

There are many contexts in which the push-pull formula $f_*(f^*(\alpha)\cdot \beta) = \alpha \cdot f_*(\beta)$ holds. I am interesting mostly in the case of algebraic K-theory and Chow rings (under ...
2
votes
0answers
82 views

Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...
1
vote
1answer
127 views

When the restriction of derived equivalence to a summand is a derived equivalence as well

I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...
3
votes
2answers
135 views

generalization of result on K_1 of $SL(n,R)$

Let R be a "nice" ring with 1 (e.g. Euclidean domain). Then the subgroup E(n,R) generated by the elements $I+te_{i,j}$ is equal to $SL(n,R)$. My question is as follows: Instead of $SL(n,R)$ I look ...
11
votes
4answers
2k views

“Must read” papers in algebraic K-theory?

I'm mainly interested (graduate student) in surgery theory and geometric topology. If I have a chance to suggest "must read" papers in geometric topology for beginner, I'm very glad to suggest ...
5
votes
2answers
437 views

Is the algebraic Grothendieck group of a weighted projective space finitely generated ?

This is to be confronted with Joseph Gubeladze' paper : "Toric varieties with huge Grothendieck group" !
12
votes
2answers
637 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 ...
19
votes
1answer
484 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 ...
6
votes
0answers
443 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 ...
1
vote
1answer
87 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
376 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 ...
9
votes
0answers
95 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: $$ ...
5
votes
1answer
206 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} ...
12
votes
1answer
253 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
1k 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
143 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
159 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
445 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 ...
2
votes
1answer
234 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 ...
1
vote
1answer
203 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
151 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
755 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 ...
11
votes
4answers
1k 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
100 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
143 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
219 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
370 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 ...