All Questions
Tagged with at.algebraic-topology homological-algebra
388 questions
3
votes
1
answer
575
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 ...
- 9,019
11
votes
1
answer
502
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 ...
- 37.8k
22
votes
0
answers
866
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 ...
- 985
3
votes
0
answers
300
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 ...
- 31
4
votes
0
answers
357
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 ...
- 141
4
votes
0
answers
170
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 ...
- 1,974
0
votes
1
answer
425
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 $M_n$ ...
- 4,826
6
votes
0
answers
1k
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 ...
- 461
3
votes
0
answers
170
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 ...
- 583
1
vote
0
answers
357
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 ...
- 187
11
votes
0
answers
1k
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 ...
- 787
3
votes
1
answer
641
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 ...
- 38.6k
3
votes
1
answer
551
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 ...
- 175
4
votes
4
answers
284
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 ...
- 4,920
9
votes
2
answers
1k
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 ...
- 10.5k
6
votes
1
answer
1k
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 ...
- 1,243
9
votes
1
answer
370
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 "...
- 743
10
votes
3
answers
2k
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 ...
- 4,920
8
votes
2
answers
2k
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 ...
- 1,589
7
votes
4
answers
797
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 ...
- 1,936
8
votes
1
answer
2k
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 \...
- 241
6
votes
1
answer
537
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 ...
- 16.9k
4
votes
1
answer
438
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 ...
- 16.9k
3
votes
1
answer
149
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 ...
- 16.9k
19
votes
5
answers
2k
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 ...
- 5,534
7
votes
0
answers
540
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}^* \...
- 543
7
votes
2
answers
3k
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 ...
- 583
10
votes
1
answer
855
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 ...
- 6,210
15
votes
2
answers
2k
views
Cosheaf homology and a theorem of Beilinson (in a paper on Mixed Tate Motives)
I'm trying to understand the proof of Theorem 4.1 in the paper Multiple Polylogarithms and Mixed Tate Motives by AB Goncharov (http://arxiv.org/pdf/math/0103059v4.pdf). In it, the author uses cosheaf ...
- 15.4k
0
votes
1
answer
364
views
Existence of a chain map lifting the identity; Alexander-Whitney/Eilenberg-Zilber maps
Some preliminary definitions:
Let $\Pi = \langle \alpha | \alpha^2 = 1\rangle$ be the cyclic group of order $2$ and let $\mathbb{Z}\Pi$ denote the group ring of $\Pi$ over $\mathbb{Z}$. Embed $\Pi$ ...
- 13
1
vote
0
answers
241
views
Two-point desuspension for augmented chain complexes?
Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = \...
- 19.6k
1
vote
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question
Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
- 360
6
votes
1
answer
1k
views
Solid rings and Tor
A solid ring is a ring $R$ such that the multiplication
$R\otimes_{\mathbb{Z}} R \to R$ is an isomorphism.
These were classified by Bousfield and Kan; they are
subrings of $\mathbb{Q}$,
$\mathbb{Z}/...
- 12.5k
2
votes
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves
Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
- 360
1
vote
2
answers
1k
views
Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
- 360
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
23
votes
2
answers
3k
views
Calculating Mayer-Vietoris efficiently
This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere....
- 156k
11
votes
1
answer
793
views
Is the pairing between contours and functions perfect (modulo the kernel given by Stokes' theorem)?
Let $s: \mathbb C^n \to \mathbb C$ be a homogeneous degree-$d$ polynomial which is nonsingular (in the sense that the hypersurface it defines in $\mathbb{CP}^{n-1}$ is smooth; equivalently the ...
- 54.6k
5
votes
1
answer
304
views
flat maps of monoids which are not localizations
It is well known that a localization $S^{-1}R$ of a commutative ring $R$ is flat as a $R$-module.
Rather, I am looking for extensions of rings which share certain properties of localizations, like ...
- 6,210
23
votes
3
answers
3k
views
Homology theory constructed in a homotopy-invariant way
Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
- 3,033
3
votes
2
answers
250
views
A model structure on the category of "dualizing maps"
Let $C$ be the category with objects being maps $h:M\to A$ where $A$ is a commutative graded $\mathbb{Q}$-algebra (cdga), $M$ is a differential graded (dg) $A$-module and $h$ is an $A$-dg-module map, ...
- 23.5k
6
votes
1
answer
974
views
Constructible sheaves and dg-modules
Let $M$ be a smooth manifold, $A_M$ the de Rham algebra of $M$, $D_{A_M}$ the derived category of the category of differential graded (dg) $A_M$-modules and $D^+_c(M)$ the bounded below constructible ...
- 23.5k
3
votes
4
answers
874
views
Extension of Tate's result regarding Tor
In a 1957 paper (Link), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R then there is a graded commutative DGA $X$ over $R$ with $H_i X=0$ except $H_0 X= R/I$ (I guess R should ...
- 3,726
5
votes
0
answers
477
views
Alternative approaches to the universal coefficient theorem
Let $A$ be a chain complex of free $R$-modules over a PID $R$, and let's assume $A$ has finite cohomological type, by which I mean $H^\ast(A)$ is finitely generated in each dimension and $0$ for large ...
- 5,534
6
votes
0
answers
723
views
On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
- 21.8k
8
votes
1
answer
353
views
Does there exist a model of chains on oriented manifolds with both a strict intersection pairing and strict functoriality for closed embeddings?
Let $M$ be a smooth oriented $n$-dimensional manifold. My favorite model of $\operatorname{Chains}_\bullet(M) \otimes \mathbb R$ is the space of smooth compactly-supported de Rham forms on $M$, ...
- 54.6k
3
votes
3
answers
757
views
Are these Two Definitions of Quadratic Form (Algebraic, Topological) Related to Each Other?
Hi, All:
I am trying to see if there is a nice relation between two different definitions of quadratic form q; a topological definition $q_T$, and an algebraic definition $q_A$, and, if there is, how ...
- 105
2
votes
1
answer
295
views
Sg: How to Show this Sequence is Exact?
Hi,All:
I am seeing a result in which the following sequence, in the context of the genus-g surface Sg, is described as being exact:
1-->Tg-->$M^{(2)}g$-->$Sp^{(2)}(2g,\mathbb Z)$-->1
Where :
i)Tg ...
- 105
3
votes
1
answer
817
views
dg-lie structure on $HH^*$ and Koszul duality
This is shamelessly close to my other question: A Question on Koszul duality and $B(\infty)$ structures on $HH^*$. Maybe this one will get a better response. Rather than rewrite that one, I am going ...
- 3,758
9
votes
1
answer
542
views
Signs and functoriality of tensor products
Let $C,C',D,D'$ be chain complexes of $R$-modules (let's say with upper indexing, so perhaps I should call them cochain complexes, though they're not duals of anything). Let $f\in Hom^\ast(C,C')$ and $...
- 5,534