Questions tagged [algebraic-k-theory]
The algebraic-k-theory tag has no usage guidance.
189 questions with no upvoted or accepted answers
38
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Homology of $\mathrm{PGL}_2(F)$
Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
- 21.3k
23
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0
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647
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Is this a model for $K$-theory of a triangulated category?
The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
- 18.4k
20
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0
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890
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Epsilon factors - a la Beilinson - What is it?
I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or "...
- 5,562
16
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0
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603
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K-theory and homology of groups
It is known that for any ring $R$,
$$K_{1}(R)=H_{1}(GL_{\infty}(R), \mathbb{Z})$$
$$ K_{2}(R)= H_{2}(E_{\infty}(R),\mathbb{Z})$$
$$ K_{3}(R)= H_{3}(St_{\infty}(R),\mathbb{Z})$$
where $GL_{\infty}= ...
- 1,613
15
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0
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402
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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 ...
- 42.4k
12
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0
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410
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Can Quillen-Lichtenbaum recover Borel's computation?
Borel famously used analysis on symmetric spaces to compute the rationalised algebraic $K$-theory groups of rings of integers $\mathcal{O}_F$ in number fields, e.g. $K_i(\mathbb{Z}) \otimes \mathbb{Q}...
- 8,167
12
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0
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586
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Global version of Gabber's rigidity theorem
I had a question regarding Gabber's rigidity.
Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), ...
- 5,901
12
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0
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551
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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, ...
12
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0
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1k
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Are there analogues of Beilinson's conjectures for motives with coefficients?
There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
- 9,408
11
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0
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623
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Does Merkurjev's argument help Voevodsky's program?
In the talk
Unimath - its present and its future, July 10, 2017. Video and slides of a talk, Isaac Newton Institute for Mathematical Sciences, Cambridge. (abstract)
Voevodsky mentioned that he was ...
- 35.5k
11
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0
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265
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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 ...
- 331
11
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264
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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
11
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340
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$K$-theory spectrum of the category of finite groups
(I asked some people this question in person and got the answer "no", but wanted to see if the Internet had more to say)$
\newcommand{\FinGrp}{\mathbf{FinGrp}}
$
Way back in my first group theory ...
- 1,838
11
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0
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779
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Rosenberg's proof of Bass-Heller-Swan
I'm reading the proof the Bass-Heller-Swan Theorem in Rosenberg's book Algebraic K-Theory and Applications (Theorem 3.2.22), which asserts
$$K_1(R[t,t^{-1}]) \cong K_0(R) \oplus K_1(R) \oplus NK^+_1(R)...
- 63.1k
10
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0
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470
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Using the universal property of K-theory
A paper of Blumberg, Gepner and Tabuada gives a universal property of K-theory: from their abstract "connective algebraic K-theory is the universal
additive invariant, i.e., the universal functor with ...
- 2,040
10
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0
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155
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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 ...
- 17.4k
9
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0
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446
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$K$-theory of $D$-modules
I have to admit I don't know much about topics appearing in this question, I just see very rough connections between these objects:
According to this page 23, a different $t$-structure on $D^b(\text{...
- 5,901
9
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0
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373
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Examples for a conjecture of Beilinson
Beilinson has conjectured that for a regular, complete, geometrically irreducible curve $C$ of genus $g$ over a number field $k$, $rank(K_2(C))=g[k:\mathbb{Q}]$. As far as I know it is not known in ...
- 5,901
9
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0
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388
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Finite generation of rational algebraic k-theory
Parshin's conjecture states that higher algebraic k-theory of smooth projective varieties over finite fields are rationally trivial. This has been shown for curves. Quillen showed that the K-groups in ...
- 5,901
9
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0
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717
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Andrei Suslin's works
Andrei Suslin, a well known mathematician, died 10 July 2018. (https://en.wikipedia.org/wiki/Deaths_in_2018) I believe it may be appropriate to give an overview of his work on this site. Personally, ...
- 6,891
9
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631
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Relative Chow groups
Most cohomologies have the notion of cohomology with support on a closed subspace, and also cohomology with compact support. In general, for any morphism $f\colon Y\to Z$ the inverse image fits into a ...
- 2,871
9
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0
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426
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Is the class of commutative generalized Euclidean rings stable under quotient and localization?
Let $R$ be an associative ring with identity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. Let ...
- 7,893
9
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0
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1k
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Motivic cohomology of a point
I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that $H^{n,n}_{\mathrm{mot}}(\...
- 596
9
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0
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745
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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
9
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0
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260
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Is the generation of rings by their units a question in K-theory?
Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...
- 13.3k
9
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0
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417
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Geometrizing the Third Cohomology of a Complex Lie Group
If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural $\...
- 23k
8
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0
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123
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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
8
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0
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382
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On the existence of a norm map for Milnor K-theory for a finite extension $A \to B$ which is free of finite rank
I have one technical question on norm maps on Milnor K-theory.
When $K \subset L$ is a finite extension of fields, we know (by Bass-Tate and Kato) that there exists a norm map $N_{L/K} : K^M_n (L) \...
- 937
8
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0
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307
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How bad can $SK_1$ of a commutative ring be?
For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...
- 3,186
7
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0
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223
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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
7
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0
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266
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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
7
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0
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764
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Serre presentations over $\mathbb{Z}$
Given a Cartan matrix $A=(a_{ij})_{i,j\in I}$, a classical result of J.-P. Serre asserts that the complex semisimple Lie algebra $\mathfrak g=\mathfrak g(A)$ corresponding to $A$ admits a presentation ...
- 563
7
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0
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279
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Adequate equivalence relations and algebraic $K$-theory
I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...
- 5,901
7
votes
0
answers
832
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Difference between algebraic and etale K-theory
Due to the Quillen-Lichtenbaum conjecture (now proven by Rost, Voevodsky, and Weibel), the map $K_\ast(X,\mathbb{Z}/n)\rightarrow K_\ast^{et}(X,\mathbb{Z}/n)$ from algebraic K-theory to etale K-theory ...
- 889
7
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0
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385
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K-theory of the infinite dimensional projective space
What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be "...
- 16.9k
7
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0
answers
181
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Conceptual proof of a theorem of Bloch on $K_2$ of Artinian $\mathbb Q$-algebras
Recall the following theorem of S. Bloch from his paper ($K_2$ of Artinian $\mathbb Q$-algebras, with applications to algebraic cycles, 1975):
For any local $\mathbb Q$-algebra $B$ and an augmented $B$...
- 528
7
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0
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279
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Is there a derived geometric interpretation of morse functions?
Given a smooth affine scheme $X = \mathbb{V}(g)$ over a field of characteristic 0, let $f:X \to \mathbb{A}^1$ be a morphism of schemes. Then, the critical locus is given by $\pi_*(dg \cap df)$ for $\...
- 1,716
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206
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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 ...
- 2,934
7
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0
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374
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Split exact categories arising naturally
If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is ...
- 770
7
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0
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191
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Torsion in Whitehead group
Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?
- 468
7
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1
answer
518
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When is $GL_m(R)$ generated by elementary and diagonal matrices?
Let $D$ be a division ring and $R=D[t_1,\ldots,t_n]$ the polynomial ring in $n$ variables. Now let $GL_m(R),\,E_m(R)$ be the usual general linear group and its subgroup generated by the elementary ...
- 348
6
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0
answers
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 ...
6
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0
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158
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Questions about the $K$-theory of the algebraic standard Podleś sphere
Given $\theta \in \mathbb{R}$ irrational, the $K$-theory of the smooth noncommutative $2$-torus $C^\infty_\theta(\mathbb{T}^2)$ is well understood in relation to that of the corresponding $\mathrm{C}^\...
- 2,385
6
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0
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368
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Subgroup of algebraic $K$-theory generated by split vector bundles
Is there any description of a subgroup of the algebraic $K$-groups of a curve that its generators lie in the subcategory that its objects are direct sums of $\mathcal{O}(n)$'s (for possibly different $...
- 5,901
6
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0
answers
180
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Abelian localisation for K theory?
Let $X$ be a scheme acted upon by $\mathbf{G}_m$ and $K(X)=K_0(\text{Perf}X)$ the Thomason-Trobaugh K theory. Is there a localisation theorem in this context? By this I mean something like
$$\text{id}...
- 5,701
6
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0
answers
100
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$K_0$ of configuration of hyperplanes
Let $\ell_n$ where $n\geq 3$ be the configuration of $n$ lines in a plane, such that $n-1$ of them pass through a single point and the last one does not and it intersects rest of the $n-1$ lines. I'm ...
- 5,901
6
votes
0
answers
98
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Elliptic deformation of the second Chern class
Second Chern class
$$c_2 \in H^4(BGL,\mathbb{Q}(2))$$
admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ...
- 4,316
6
votes
0
answers
220
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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
6
votes
0
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233
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Group $C^*$ vs group von-Neumann algebras
Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
- 9,853
6
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0
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236
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Fundamental class in equivariant K-theory
I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory.
The setup I'm interested in is the following: suppose $V$ is a vector space equipped with ...
- 371