# Tagged Questions

**13**

votes

**2**answers

657 views

### Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...

**2**

votes

**1**answer

172 views

### How do small changes in a filtered complex affect the associated spectral sequence

I have recently been learning about spectral sequences, not in the context of any problem, but mostly out of curiosity. There are two questions that have occurred to me which I've been unable to ...

**10**

votes

**2**answers

520 views

### (co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...

**12**

votes

**1**answer

270 views

### What is the geometric fixed points of an (equivariant) Eilenberg Maclane Spectrum?

The following was posted to math.stackexchange to no avail: http://math.stackexchange.com/questions/908756/an-exercise-in-homology-computation-what-is-the-geometric-fixed-points-of-an-e
The question ...

**4**

votes

**1**answer

152 views

### AdicCompletion$\dashv$Torsion adjunction on spectra?

It seems to me that in slight paraphrase the central result of the article
Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386)
(theorems 6.11 and ...

**1**

vote

**2**answers

205 views

### If an abelian category $\mathcal{A}$ has enough injectives then so does $\mathrm{Ch}^{\geq 0}(\mathcal{A})$

Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if
...

**2**

votes

**0**answers

161 views

### On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...

**1**

vote

**0**answers

86 views

### generalisation of the universal coefficient spectral sequence

Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective ...

**0**

votes

**0**answers

84 views

### Dimension of category of sheaves [duplicate]

Let $k$ be a field. Consider the category $Shv(\mathbb{R}^n)$ of sheaves of $k$-vector spaces on $\mathbb{R}^n$. What is the cohomological dimension $d$ of $Shv(\mathbb{R}^n)$? I know that $d \in ...

**2**

votes

**1**answer

158 views

### Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...

**1**

vote

**0**answers

113 views

### Reference needed: Homology of the blow-up

Given an algebraic variety $X$ over the complex numbers.
Let $V$ be a subvariety of $X$ and $\pi_V \colon X' \rightarrow X$
be the blow-up of $X$ at $V$.
It is posible in general to compute the ...

**1**

vote

**0**answers

259 views

### Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...

**5**

votes

**1**answer

304 views

### Spectral Sequences reference

What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep.
I'm ...

**1**

vote

**0**answers

105 views

### (Co)homology of classifying space of spin group $BSpin(n)$

In the answer for question: Homology of classifying space of spin group BSpin(n),
it was shown that $H_i(BSpin(\infty),Z)$ is $0,0,0,Z$, for $i=1,2,3,4$.
What is $H_i(BSpin(\infty),Z)$ or ...

**1**

vote

**0**answers

81 views

### Homology of the fixed points of the singular complex of a G-space

I posted the following to stackexchange a while ago [1], without any answers. Maybe the question is too unmotivated, but it seems very natural to me.
Suppose $X$ is a topological space and $G$ a ...

**9**

votes

**1**answer

396 views

### Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles?
Can you give me a text or an article?
For example:
Consider a vector bundle $E$ with fiber $V$ and base manifold $M$.
Consider ...

**3**

votes

**1**answer

96 views

### Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...

**2**

votes

**1**answer

445 views

### A question about the universal coefficients theorems

This seemingly simple question stands unanswered on math.stackexchange.com for a couple of days (http://math.stackexchange.com/questions/680211/a-question-about-the-universal-coefficient-theorem), so ...

**11**

votes

**0**answers

369 views

### Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...

**7**

votes

**2**answers

283 views

### Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex

Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence
$$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...

**21**

votes

**4**answers

1k views

### origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange).
I have been "brought up" as an algebraic ...

**1**

vote

**0**answers

224 views

### Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...

**3**

votes

**1**answer

270 views

### What is the “higher version” of chain homotopy in singular homology?

In basic algebraic topology, we know the following well-known chain homotopy theorem:
Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...

**7**

votes

**1**answer

243 views

### Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?

Consider a diagram D: I→ChR of real connective chain complexes.
In the example I have in mind all chain complexes are concentrated in some fixed degree n.
There is a canonical map lim D → holim D ...

**11**

votes

**0**answers

300 views

### Bar construction vs. twisted tensor product

One may study the cohomology of a space $E$ expressed as a homotopy pullback of $X$ and $Y$ over $Z$ using either the Eilenberg-Moore spectral sequence or the Serre spectral sequence for the fibration ...

**3**

votes

**0**answers

156 views

### Hochschild homology of a tensor algebra modulo a two-sided ideal

Let $V$ is a module over a field $k$, and $A=T(V)$ the tensor algebra generated by $V$. The Hochschild homology $HH_*(A)$ has been determined by Loday and Quillen in their paper "Cyclic homology and ...

**4**

votes

**0**answers

167 views

### What is known about this short exact sequence in Lie algebra cohomology?

In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact
$ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...

**4**

votes

**0**answers

161 views

### Homotopy unites of a differential graded algebra

I apologize in advance if the question is too basic.
Let $A$ be a differential graded algebra and $H^{0}A$ the 0-cohomology of $A$ (which is an ordinary ring) and $A^0$ is the 0-level of $ A$ . An ...

**0**

votes

**1**answer

327 views

### A generalization of cochain complex: quasi-cochain complex

It appears that we can generalize cochain complex to quasi-cochain complex, that still allow us to define cohomology.
Definition:
A quasi-cochain complex is a sequence of commutative monoids $M_n$ ...

**3**

votes

**0**answers

193 views

### The normalised cochain complex, totalisation and cosimplicial simplicial $R$-modules

Short Version
Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by ...

**3**

votes

**0**answers

101 views

### minimal model of $A_\infty$ structure

Hi all,
I am reading about minimal model of $A_\infty$ structure. So far, I find two different ways of the construction, given by Kadeishvili and Kontsevich, Soibelman.
1) The construction of these ...

**0**

votes

**0**answers

193 views

### Homology of the dg-nerve vs Hochschild homology of the dg-category

Lurie in Higher Algebra, section 1.3 associates a quasi-category to a dg-category A via the so called dg-nerve construction, extending the classical nerve. I have a feeling the homology of the ...

**6**

votes

**0**answers

317 views

### Lyndon-Hochschild-Serre spectral sequence and cup products

First here is my setup:
Let $W$ be some group, and $C$ a normal subgroup of finite index, and let $W/C=G$. Now let $L$ be a a $G$-module on which $C$ acts trivially, so in particular we get on action ...

**3**

votes

**1**answer

225 views

### Resolutions chain homotopic to projective ones

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module ...

**1**

vote

**1**answer

248 views

### Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...

**4**

votes

**4**answers

198 views

### Stratifications and Cohomology Computations

I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...

**8**

votes

**2**answers

533 views

### H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...

**4**

votes

**1**answer

335 views

### Mayer-Vietoris sequence in homology with local coefficients

Background. I'm trying to compute some homology groups using a Mayer-Vietoris argument, but I really need local coefficients.
Question 1. What does the Mayer-Vietoris sequence look like when using ...

**7**

votes

**1**answer

245 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 ...

**8**

votes

**3**answers

776 views

### Serre Spectral Sequence of Representations

Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...

**5**

votes

**2**answers

1k views

### Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...

**6**

votes

**3**answers

421 views

### A lost lemma about periodicity in a grid of long exact sequences?

This is a question about finding references and hopefully a larger
context for a lemma in homological algebra I proved recently.
The motivation is to understand properties of characteristic
classes of ...

**4**

votes

**1**answer

356 views

### Homology of classifying space of spin group BSpin(n)

While dealing with $BO(n)$, $BSO(n)$ and $BSpin(n)$ with the universal coefficient theorem and Künneth formula, I came to have the following question:
The universal coefficient says $H^n(X;M)\cong ...

**6**

votes

**1**answer

330 views

### Exceptional collections of objects in topological triangulated categories?

People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in ...

**3**

votes

**1**answer

230 views

### If a t-truncation of the unit object in a stable homotopy category is a ring object up to homotopy, can it be lifted to a ring spectrum? What about the Postnikov t-truncations of the sphere spectrum?

Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to ...

**3**

votes

**1**answer

127 views

### Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?

It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder ...

**8**

votes

**4**answers

767 views

### References for Eilenberg-Zilber shuffle product

Most of the treatments I can find in the literature for the Eilenberg-Zilber shuffle product approach it from the point of view of simplicial sets (including the original Eilenberg-MacLane paper). I ...

**6**

votes

**0**answers

263 views

### homotopy pullbacks of tensor product chain complexes (towards Kunneth formula in diff cohomology)

I have editted this question from the previous version which did not obtain much attention.
Suppose I have two diagrams of chain complexes:
$A^* \rightarrow C^* \leftarrow B^*$
$\tilde{A}^* ...

**5**

votes

**2**answers

816 views

### On the difference between a projective chain complex and a level-wise projective chain complex

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it ...

**10**

votes

**1**answer

338 views

### finite complex with non-finitely generated homology with local coefficients

I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex $X$ such that some homology group with local coefficients $H_n(X,{\mathbb Z}[\pi_1 X])$ is not a ...