**1**

vote

**0**answers

126 views

### Cotorsion theory and its relative homology

Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $,
$ \text{Ext}_{F(R)}^i(M, N)\cong ...

**3**

votes

**1**answer

114 views

### ring with prescribed K group

One can construct topological spaces with prescribed homotopy groups or, say, homology groups.
But is it possible to construct a ring with any given $K_0$ group? What about $K_1$ group et.c.?
I know ...

**0**

votes

**1**answer

189 views

### About regular local rings and Socles

Let R be a regular local ring with $ \text{dim} R = d $. If $ 0\rightarrow R\rightarrow I_0\rightarrow ...\rightarrow I_d\rightarrow 0 $. Then why for $ 0\leq i\leq d-1 $, the socle of $ I_i $ is ...

**0**

votes

**1**answer

237 views

### Length of a module

Let R be a commutative ring, M an R-module of finite length and let N be an Injective R-module with zero socle. Then why $ \text{Hom}_R(M, N) $ is zero?

**0**

votes

**0**answers

179 views

### A question about higher K-theory

Suppose $\mathcal{A,B,C}$ are additive categories, $\mathcal B$ is a subcategory of $\mathcal C$. Now let $F,G: \mathcal A\rightarrow\mathcal B$ be two additive functors. Suppose $F,G$ are naturally ...

**6**

votes

**1**answer

279 views

### Coefficients of real k-theory with coefficients

Question: Calculate the group $ \pi_{8k+2}(KO \wedge M\mathbb Z/l\mathbb Z) $.
Here $KO$ denotes the real k-theory spectrum and $M\mathbb Z/l\mathbb Z $ denotes a Moore Spectrum associated to the ...

**14**

votes

**2**answers

465 views

### Is there an effective way to calculate K-theory using Morse functions?

Let $M$ be a compact manifold and let $f$ be a Morse function with exactly one critical point at each critical level. Then one can recover a CW-complex with the homotopy type of $M$ from just the ...

**18**

votes

**0**answers

617 views

### Computational complexity of topological K-theory

I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...

**5**

votes

**1**answer

300 views

### Taking direct sums in $K$-theory in Kirchberg-Phillips classification

A theorem by Kirchberg and Phillips states that two unital separable nuclear simple purely infinite $C^*$-algebras (so called Kirchberg algebras) satisfying the Universal Coefficient Theorem are ...

**1**

vote

**0**answers

82 views

### Homomorphism of algebra of “Clifford-valued continuous functions”

I am interested in the general question of
When is a map between algebra of "Clifford-valued continuous functions" homomorphism?
As a starter, I would like to first understand the case for ...

**8**

votes

**1**answer

599 views

### Motivic cohomology and cohomology of Milnor K-theory sheaf

Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}_p$).
...

**7**

votes

**1**answer

271 views

### Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the ...

**12**

votes

**3**answers

847 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) = ...

**1**

vote

**0**answers

74 views

### linear independence of values of the polylogarithm at different roots of unity

I am interested in the real and imaginary part of the complex polylogarithm
$$L_{k+1}(\zeta):=Re(\frac 1 {i^k}\sum_{m=1}^\infty \frac{\zeta^m}{m^{k+1}}),$$
where $\zeta$ is a primitive $n$-th root ...

**4**

votes

**2**answers

191 views

### On the descent homomorphsim of Kasparov equivariant KK theory

Hello,
I have recently read about the construction of the descent map in Kasparov KK theory, which, for a group $G$ and two $G$-equivariant $C^*$ algebra $A$ and $B$ send $KK_i^G(A,B)$ to $KK_i(A ...

**4**

votes

**0**answers

221 views

### A canonical way to kill a subset of cohomology in a dg-algebra: via $A_\infty$-algebras? References?

Let $A$ be a differential graded algebra, $S\subset H^*(A)$. I would like to 'kill $S$ in a canonical way'. Is it possible to do it as follows: consider the $A_\infty$-algebra structure on ...

**4**

votes

**1**answer

288 views

### References for geometric K-homology

Can anyone give me some good references to read geometric K-homology. I know bit of Kasparov's KK theory and analytic K-homology.

**5**

votes

**1**answer

275 views

### A theory of bifurcation of braids ?

I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional ...

**2**

votes

**1**answer

146 views

### For which local $R$ its K-theory mod l is isomorphic to the one of its residue field?

It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong ...

**15**

votes

**1**answer

563 views

### Explicit path in the unitary group of a $C^*$-algebra

For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...

**8**

votes

**2**answers

332 views

### Injectivity of the Baum-Connes assembly map for locally compact groups

Skandalis, Tu and Yu in "The coarse Baum-Connes conjecture and groupoids" proved that:
Let $\Gamma$ be a countable group with a proper left-invariant metric $d$. If $\Gamma$ admits a uniform ...

**1**

vote

**1**answer

129 views

### Does the global dimension gldim R equal the projective dimension of R as bimodule over its enveloping algebra?

I know that generally the answer is no, for example the weyl algebra。
But is this true for commutative algebra？ or we may restrict to affine commutative algebras。
Maybe ，it is a classical result. ...

**5**

votes

**1**answer

368 views

### Terminology question for real K-theory

This is a terminology question. Answers will help me satisfy a referee but I'm also genuinely interested. Consider the following two things that you could define for a topological space X:
(1) The ...

**0**

votes

**2**answers

338 views

### The First Homology Group of Configuration Space and Knot Theory

Let $\pi_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the fundamental group functor and let $H_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the first homology group functor. We can then define ...

**4**

votes

**1**answer

309 views

### Disconnectedness of Hilbert schemes of projective schemes

Let $Y$ be a projective scheme. The naive definition of a Hilbert scheme of subschemes $X$ of $Y$ would require us to projectively embed $Y$, then ask that $X$ have a fixed Hilbert polynomial $p$.
...

**3**

votes

**0**answers

104 views

### Extending cohomology classes to compactifications of Kuga varieties

I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon,
available at ...

**7**

votes

**2**answers

476 views

### Baum-Connes-like “conjecture” for $l^p$-spaces

Let $G$ be a (discrete) group. For the Baum-Connes conjecture, one looks at the reduced group $C^{\star}$-algebra: Look at the Hilbert space $l^2(G)$ and the representation of $G$ on this Hilbert ...

**2**

votes

**1**answer

189 views

### Ideal spanned by matrix units isomorphic to compact operators

Hello,
Assume we have $(n+1)$ isometries $S_1,...,S_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum_{i=1}^{n+1}S_iS_i^*=I, S_i^*S_j=0$ (i.e. $S_i$ are the generators of the ...

**0**

votes

**2**answers

120 views

### free complex with mod-p coefficients

How does one prove the following fact. I could not find anything in literature.
Let $\pi$ be a subgroup of the symmetric group $S_p$ and let $W$ be a free $\pi$-complex. Then for any space $X$ there ...

**2**

votes

**1**answer

135 views

### Can I bound the degree of a contracting homotopy in an exact filtered complex?

Suppose that I have given you a bigraded vector space $V = \bigoplus_{i,j} V_{i,j}$. The first grading is a "homological" $\mathbb Z$-grading, and the second is an independent $\mathbb Z$-grading. ...

**34**

votes

**3**answers

977 views

### Lambda-operations on stable homotopy groups of spheres

The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable ...

**3**

votes

**2**answers

282 views

### Equivariant K-theory of $S^1$-action on $S^2$

Is there any references for the structure of the equivariant K-theory $K_{S^1}(S^2)$ where the action of $S^1$ on $S^2$ is defined to be rotation about the $z$-axis? What is the ring structore of ...

**2**

votes

**1**answer

300 views

### Kasparov's Dirac element and the index map

In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) $G$- Riemanian manifold $X$ as an element in the $K$-homology ...

**2**

votes

**0**answers

143 views

### Two questions on axiomatic homology

1) Given the Eilenberg-Steenrod axioms, there are several Mayer-Vietoris type sequences that can be deduced. The most general form seems to be
$$\rightarrow H_n(X \cap Y, A \cap B) \rightarrow ...

**24**

votes

**1**answer

862 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. A am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...

**3**

votes

**1**answer

274 views

### Euler characteristics and the difference bundle construction

I am reading on K theory in Lawson and Michelson (Spin Geometry). One has the "exact sequence spaces" $L(X,Y)$ and there is the theorem that there is a unique equivalence of functors $\chi$ between ...

**6**

votes

**0**answers

183 views

### K-theory of topological groupoids

For a topological groupoid $G$, there are two kinds of ``topological'' $K$-homology theories, $K_{\ast}^{G}(\underline{EG})$ the $K$-homology of $G$ with $G$-compact support, and $K_{\ast}(BG)$ the ...

**3**

votes

**0**answers

108 views

### How to deduce (8.1) in Lusztig's “Equivariant K-theory and representations of Hecke Algebras”

Let G be a connected complex algebraic group G and X=G/B, where B is the Borel subgroup. $ M=G\times C^{*}$, where $C^*$ acts on X trivially. Let $K_M(X)$ be the equivariant K-theory. Let $s\in S$ be ...

**2**

votes

**1**answer

197 views

### $K$-groups and dual graphs of special fibers

Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let ...

**5**

votes

**0**answers

86 views

### Characterization of the sequences in the equivalence classe of the zero element in higher extension groups

Hello,
I am looking for a characterization of the long exact sequences in the equivalence classe of the zero element (for the Baer sum) in $Ext^n(U,V)$ for $n>1$.
If $n=1$, then these are the ...

**15**

votes

**2**answers

2k views

### Who named it the Snake Lemma?

What is the history behind the colorful name of this result? Cartan-Eilenberg states it without any particular fanfare.

**6**

votes

**0**answers

687 views

### Is there any “deep” relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex ...

**4**

votes

**1**answer

271 views

### K theory of a simplicial monoidal category, Cofinality theorem

Let $X=(d\mapsto X_d)$ be a simplicial symmetric monoidal category. We define the $K$-theory space of $X$ to be $K(X)=|d\mapsto K(X_d)|$, the geometric realisation of the simplicial space $d\mapsto ...

**5**

votes

**2**answers

501 views

### When do the $\gamma$-filtration and codimension filtration of K-theory agree?

Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by
...

**15**

votes

**2**answers

901 views

### Who first noticed that the Hilbert symbol is a Steinberg symbol ?

Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula
$$
\prod_v(a,b)_v=1
$$
for the various local Hilbert symbols. For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ ...

**12**

votes

**2**answers

1k views

### Finiteness of stable homotopy groups of spheres

Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or ...

**12**

votes

**1**answer

839 views

### finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from algebraic ...

**4**

votes

**1**answer

439 views

### second fundamental form

Hi all,
Currently I'm reading a paper about the geometry of Grassmannians:
www.omup.jp/modules/papers/riemann/04Nagatomo.pdf
In there, the author regards the second fundamental form of the ...

**4**

votes

**2**answers

198 views

### Steinberg Group as a Lattice in a lie group

Given an integral domain $R$, the Steinberg group $St_n(R)$ is the group given by generators
$e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda)$,
$p\neq q$, $1\leq p,q \leq n$
Subject to ...

**9**

votes

**2**answers

377 views

### f.g. modules vs. f.g. projective modules

In algebraic K-theory one defines $K_0(R)$ as the result of application of the Grothendieck construction to the semigroup of isomorphism classes of left f.g. projective $R$-modules.
But we can also ...