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Questions tagged [chain-complexes]

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0
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1answer
60 views

Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$

Let $M$ be a compact smooth manifold. And particularly I am interested in the case the torus $M=T^n$. Consider the de Rham complex $(\Omega^*(M), d)$ and the cochain complex $$ C:=\mathrm{Hom} (\...
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0answers
35 views

Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective complexes themselves?

Let $R$ be an associative ring with unit and $I$ be a complex of $R$-modules. We call $I$ is h-injective if for any acyclic complex $T$ of $R$-modules, the mapping complex $\text{Hom}_R(T,I)$ is ...
4
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0answers
88 views

Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?

Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
2
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0answers
274 views

How does “chain complex functor” from $Top$ take mapping cones to mapping cones?

I am wondering how the singular chain complex functor from the category of topological spaces to the category of chain complexes of abelian groups takes a mapping cone to a mapping cone in the sense ...
2
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0answers
63 views

Order relation between cohomology groups

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex $$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
2
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1answer
198 views

A-infinity modules

Using: https://arxiv.org/pdf/math/9910179.pdf as a reference... My question involves spelling out explicitly the comment in 4.2 - "Equivalently, the datum of an $A_\infty$-structure on a graded ...
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0answers
87 views

Cochain homotopy from topological right homotopy

Suppose I have a right homotopy between two maps $f,g:X\rightarrow Y$ given by a map $h:X\rightarrow Y^I.$ This should give a cochain homotopy $C^*(Y)\rightarrow C^{*-1}(X).$ How do I express this ...
2
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1answer
128 views

A model category structure on chain complexes

The wikipedia claims that there is a model category structure of the category of arbitrary chain-complexes of R-modules which is defined by: weak equivalences are chain homotopy equivalences of chain-...
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0answers
89 views

(Reference Request) Tensor product of chain complexes in terms of strict $\infty$-categories

(note: this question is essentially a reference request for the tensor product described at the end. the rest is context) It is well known that the category of chain complexes (in positive degree, ...
2
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1answer
228 views

A question on some lemmas in Orlov's “Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models” (Exts vanishing)

I'll write the two lemmas I have questions about, and then ask my questions. For reference, I'm using the following definition of Gorenstein: $\mathbf{Definition\ 1.15}$ A local noetherian ring $A$ ...
1
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1answer
129 views

Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory

Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex ...
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0answers
119 views

Sheaves and isomorphisms with chain complex of singular chains (Sheaf Theory, Bredon)

Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$. How to show that the homomorphism of ...
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0answers
77 views

Are mapping cones in the bounded homotopy category of chain complexes isomorphic?

Let $A$ be an additive category. Suppose we have distinguished triangles $$X \rightarrow Y \rightarrow Z \rightarrow X[1]$$ and $$X \rightarrow Y \rightarrow Z' \rightarrow X[1]$$ in the bounded ...
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0answers
159 views

Derivators - diagrams in homotopy category of chain complexes

$\require{AMScd}$ Let $\mathcal{A}$ be an additive category and $K(\mathcal{A})$ be the homotopy category of $\mathcal{A}$, i.e. the category of chain complexes $Ch(\mathcal{A})$ over $\mathcal{A}$ ...
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2answers
364 views

Relation between different definitions of homotopy

When I did a course in topology, we defined "homotopy" in the following way: "A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is ...
25
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2answers
1k 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 $\mathcal C$ ...
6
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0answers
297 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 ...
0
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1answer
117 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 $...
5
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1answer
260 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 $M$ ...
7
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1answer
159 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 ...
2
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1answer
79 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 $\mathcal{S}$...
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0answers
545 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 ...
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2answers
569 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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0answers
163 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, ...
6
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2answers
762 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 Dold-...
2
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1answer
148 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 ...
3
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1answer
331 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 ...
2
votes
1answer
252 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 (...
13
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1answer
663 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 $...
5
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0answers
447 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 ...
12
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1answer
336 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. ...
12
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1answer
379 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 null-...
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0answers
548 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 ...
8
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1answer
351 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 ...
2
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1answer
356 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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1answer
127 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 ...
11
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0answers
186 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 ...
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0answers
135 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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0answers
592 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 \...
3
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2answers
527 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 ...
1
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2answers
221 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 $i\ne 0$?
2
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1answer
213 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 ...
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0answers
85 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}} \...
2
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1answer
272 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} : C_{\ast}...
5
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2answers
775 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 can'...
16
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1answer
537 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 ...
3
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0answers
222 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 ...
7
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2answers
2k 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 ...
8
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1answer
589 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?
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0answers
204 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 = \...