# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
I have editted this question from the previous version which did not obtain much attention. Suppose I have two diagrams of chain complexes: $A^* \rightarrow C^* \leftarrow B^*$ $\tilde{A}^* ... 2answers 783 views ### On the difference between a projective chain complex and a level-wise projective chain complex Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module,$Ch_R$. A map$f:M\to N$in$Ch_R$is a weak equivalence if it ... 1answer 326 views ### finite complex with non-finitely generated homology with local coefficients I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex$X$such that some homology group with local coefficients$H_n(X,{\mathbb Z}[\pi_1 X])$is not a ... 1answer 480 views ### Cosheaf homology and a theorem of Beilinson (in a paper on Mixed Tate Motives) I'm trying to understand the proof of Theorem 4.1 in the paper Multiple Polylogarithms and Mixed Tate Motives by AB Goncharov (http://arxiv.org/pdf/math/0103059v4.pdf). In it, the author uses cosheaf ... 1answer 215 views ### Existence of a chain map lifting the identity; Alexander-Whitney/Eilenberg-Zilber maps Some preliminary definitions: Let$\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$\Pi$... 0answers 163 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 = ...
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 ...