The chain-complexes tag has no usage guidance.

**28**

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### Motivating the category of chain complexes

Let $R$ be a commutative ring. For awhile I have been trying to motivate to myself more fully the definition of and various structures on the category $\text{Ch}(R)$ of chain complexes of $R$-modules ...

**25**

votes

**2**answers

486 views

### Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
An object $X$ in $\mathcal C$ gives a "point":
$$X$$
A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...

**20**

votes

**0**answers

528 views

### Are there “chain complexes” and “homology groups” taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A ...

**17**

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**6**answers

2k views

### Why chain homotopy when there is no topology in the background?

Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow ...

**16**

votes

**1**answer

504 views

### Is there a notion of a chain complex with corners?

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes.
On the topological side, although in the PL ...

**13**

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**3**answers

1k views

### how to make the category of chain complexes into an $\infty$-category

I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.
Has anyone ever written down ...

**13**

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**1**answer

537 views

### Lift chain complex from $\mathbb{F}_2$ to $\mathbb{Z}$

We start with a finite dimensional chain complex over $\mathbb{F}_2$, equipped with a basis. That is, we have finitely many finite dimensional $\mathbb{F}_2$-vector spaces $C_0,\dots,C_k$ with bases ...

**12**

votes

**1**answer

346 views

### Are totally degenerate chains null-homologous?

Let $X$ be a CW complex.
Suppose $\gamma\in C_n(X)$ is a cycle which is a sum of maps $\sigma:\Delta^n\to X$ which factor as $\Delta^n\to\mathbb R^{n-1}\to X$. Does it follow that $\gamma$ is ...

**12**

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**0**answers

447 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 ...

**11**

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**1**answer

315 views

### Can we use unparameterized chains to calculate singular homology?

Most models of singular chains on a topological space $X$ use maps from some particular collection of "nice" objects, such as the standard simplices $\Delta^n$, the standard cubes $[0,1]^n$, etc.
...

**10**

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**3**answers

854 views

### What are normalized singular chains good for?

One of the common definitions of homology using the singular chains, i.e. maps from the simplex into your space. The free abelian group on these can be made into a chain complex and one can take the ...

**9**

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**0**answers

168 views

### Is it possible to take “limits up to homotopy”?

Before I begin, an apology: it's been a while since I've done much analysis, so I might be misusing (or just missing) terminology.
I have a chain complex $(V_\bullet,\partial)$ of topological vector ...

**8**

votes

**3**answers

836 views

### Module category equivalent to graded module category?

Main Question
Let $R$ be a graded ring, graded by the nonnegative integers. Denote by $\mathrm{gr}R-\mathrm{Mod}$ the category of $\mathbb{Z}$-graded left $R$-modules with morphisms that preserve ...

**8**

votes

**1**answer

765 views

### Explicit description of the “simplicial tensor product” of chain complexes

Recall that there is an equivalence of categories (Dold-Kan) $$N:\mathrm{s}\mathbf{Ab}\simeq \operatorname{Ch}_{\geq 0}(\mathbf{Ab}):\Gamma$$ between simplicial abelian groups and (connective) chain ...

**8**

votes

**1**answer

432 views

### Left adjoint of totalization?

There is a functor from bicomplexes to chain complexes sending a bicomplex to its associated total chain complex. Does this functor have a left adjoint, and if so, what is it?

**7**

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**2**answers

1k 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 ...

**7**

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**1**answer

544 views

### Is there a good computer package for working with complexes over non-commutative rings?

I'm interested in doing computations with certain non-commutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find ...

**7**

votes

**1**answer

315 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 ...

**6**

votes

**0**answers

179 views

### Is there an obstruction which classifies “quasi-isomorphism but not chain equivalence”?

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...

**5**

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**2**answers

513 views

### Resolutions of unbounded complexes and homotopy (co)limits.

I want to understand once and for all what the resolution of an unbounded complex is. I've been trying to read 'Homotopy limits in triangulated categories' by Marcel Bokstedt and Amnon Neeman and ...

**5**

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**1**answer

91 views

### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...

**5**

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**0**answers

323 views

### Sign problem for the shift functor on DG modules

Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...

**4**

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**1**answer

163 views

### Transgression in terms of k-invariant for chain complexes

I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups (in field coefficients $k$) are non-zero only in dimension zero and for a fixed $n>0$. Let $M$ ...

**4**

votes

**2**answers

362 views

### Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes

It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the ...

**3**

votes

**2**answers

451 views

### how to make the category of chain complexes into an $(\infty,1)$-category

Related to this question, I would like to know, if there is an explicit presentation
of the $(\infty,1)$-category of a model category of abelian chain complexes, but this time in terms of simplicial ...

**3**

votes

**1**answer

191 views

### Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers?
In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...

**3**

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**0**answers

229 views

### Cisinski-Deglise model structure on chain complexes

Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and Déglise define the ...

**3**

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**0**answers

205 views

### Reference Request: The Categorification of $\mathbb{Z}$ as cochain complexes of vector spaces

Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain ...

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**2**answers

898 views

### Proving homotopy invariance of cellular homology by constructing a chain homotopy

I'm trying to follow an argument in Lück's "Algebraische Topologie: Homologie und Mannigfaltigkeiten" (to which there apparently doesn't exist an english translation). The aim is to check homotopy ...

**2**

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**1**answer

128 views

### Is a inverse limit of indecomposable again indecomposable?

In truth, I do not need in the general case.
Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$.
If $\mathbb{N}$ is the ...

**2**

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**1**answer

67 views

### Is the class of acyclic complexes deconstructible?

Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of ...

**2**

votes

**1**answer

305 views

### Brutal truncation of indecomposable complexes

Let $P^{\bullet} = (P^i, d^i)_{i\leq 0}$ be an indecomposable object in category of complexes bounded above, where each $P_i$ is a finitely generated projective module over an finite dimensional ...

**2**

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**1**answer

207 views

### Endomomorphisms of Chain Complexes of vector spaces and determinants

Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : ...

**2**

votes

**1**answer

118 views

### pure sub-complexes of exact subcomplexes

In https://www.google.com/#q=tensor+product+of+complexes%2Benochs a new tensor product of complexes is defined which characterizes flatness in the category of complexes of $R$-modules. That is, a ...

**2**

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**0**answers

438 views

### Why chain complexes? [duplicate]

Possible Duplicate:
Motivating the category of chain complexes
Chain complexes (of, say, abelian groups) are fundamental in homological algebra and algebraic geometry. For example, using the ...

**1**

vote

**1**answer

325 views

### Are chain complexes over a field always injective?

Question: Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and ...

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vote

**2**answers

163 views

### Terminology: complex of sheaves with cohomology sheaves concentrated in degree zero

What is the proper terminology for a complex of sheaves $\mathcal F^\bullet$ whose homology sheaves $\mathcal H^i\mathcal F^\bullet$ vanish for $i\ne 0$?

**1**

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**1**answer

185 views

### Question on resolutions for arbitrary chain complexes.

Consider a $\mathbb{Z}$-graded chain complex $A^{\bullet}$, I know that a bounded below complex is one such that $A^i = 0$ for $i$ sufficiently small, and a bounded above complex is one such that $A^i ...

**1**

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**0**answers

271 views

### How to prove that any perfect complex on an affine scheme is strictly perfect?

Let $(X,\mathcal{A})$ be a ringed space. A complex $\mathcal{S}^{\bullet}$ of $\mathcal{A}$-modules is $\textit{perfect}$ if for any point $x\in X$, there exists an open neighborhood $U$ of $x$ and a ...

**1**

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**0**answers

108 views

### when does a “triangulated” functor factor over the homotopy category?

The setup is as follows:
We have the category $C$ of chain complexes over some additive/abelian category and want to pass to the category $K$ of chain complexes modulo homotopy.
So we have an ...

**1**

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**0**answers

115 views

### Dold-Kan preserves weak equivalences and fibrations

It's well known that Dold-Kan correspondence is an isomorphism between simplicial vector spaces and non-negative chain complexes of vector spaces. Moreover, weak equivalences and fibrations are ...

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185 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 = ...

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74 views

### Extending Reedy dimension to augmented chain complexes of abelian groups

Recall that a normal continuously-graded finite interval is given by a pair $a=([a],f)$ consisting of:
1.) A finite totally ordered set $[a]=[a_0< \dots < a_n]$
2.) A grading function $f:U[a] ...

**0**

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**1**answer

104 views

### Perfect $Q[G]$-complex

Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex.
Suppose that $M\otimes_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to ...

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126 views

### Torsion in cohomology

Suppose to have a short exact sequence of chain complexes of $\mathbb{Z}$-modules:
$$0\to A^\bullet\to B^\bullet\to C^\bullet\to 0$$
such that $A^k,B^k,C^k$ are non zero for $k=0,1,2$.
Moreover, ...

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77 views

### Quick question on chain maps and maps induced by truncations.

Let $A^\bullet$ be the complex:
$\cdots \rightarrow A^{n-2} \xrightarrow{d^{n-2}} A^{n-1} \xrightarrow{d^{n-1}} A^{n} \xrightarrow{d^{n}} A^{n+1} \xrightarrow{d^{n+1}} A^{n+2} \xrightarrow{d^{n+2}} ...

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204 views

### Homomophism from Koszul complex to the original ring

In an article, I encounter an isomorphism relation as follows:
Let S be a comm. ring, x an element in S. K[x,S] be corresponding Koszul complex. The article says "this is a classical isomorphism":
...

**-1**

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**1**answer

312 views

### How does a chain map induce another chain map on an isomorphic chain complex?

I have 2 ways of defining a chain complex on a manifold, one of which is the cellular chain complex, $C^{CW}_*$. I know that a cellular map $f: X^n \rightarrow Y^n $ such that $ f(X^n) \subset Y^n $ ...