Questions tagged [chain-complexes]
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118 questions
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Properties of a generalization (regularization) of the Euler characteristic?
Intro: This question is about a version of the Euler characteristic for infinite dimensional chain complexes. I have no idea if this is a pre-existing concept, that's essentially what my query is ...
- 1,231
4
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0
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190
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Dyer–Lashof operations for more than 2 inputs
Let $\mathcal{O}$ be a topological operad and $X$ an algebra over it. Let the base ring be $\mathbb{Z}_2$. If $C_*$ denotes the singular chain complex over $\mathbb{Z}_2$, the action of $\mathcal{O}$ ...
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1
answer
223
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The table reduction morphism of operads from Barratt-Eccles to Surjection
The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma_n\}_{n>0}$ in groups. Berger-Fresse defined here an ...
- 517
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67
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Hermitian structure for complexes of vector bundles
Does it exist a different notion of Hermitian metric for complexes of vector bundles, besides the obvious data of a metric for each vector bundle?
Same question for connections. In particular is there ...
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Are Chain Complexes Related to the Tangent Bundle Construction?
For a scheme $X$ over $\text{Spec}(K)$, we can consider maps $\text{Sch}(\text{Spec}(K[d] / d^2), X)$, which we can think of as the tangent bundle over $X$. A map $\text{Spec}(K) \rightarrow S$ picks ...
user30211
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127
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Lift up characteristic class to chain complex
In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...
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1
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92
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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
$$
C:=\mathrm{Hom} (\...
- 2,789
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57
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Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective complexes themselves?
Let $R$ be an associative ring with unit and $I$ be a complex of $R$-modules. We call $I$ is h-injective if for any acyclic complex $T$ of $R$-modules, the mapping complex $\text{Hom}_R(T,I)$ is ...
- 9,009
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105
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Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?
Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
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How does "chain complex functor" from $Top$ take mapping cones to mapping cones?
I am wondering how the singular chain complex functor from the category of topological spaces to the category of chain complexes of abelian groups takes a mapping cone to a mapping cone in the sense ...
- 751
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67
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Order relation between cohomology groups
We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex
$$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
- 173
4
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1
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814
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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 ...
- 527
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1
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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-...
- 21
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(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, ...
- 42.4k
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1
answer
342
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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 $A$ ...
- 841
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1
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197
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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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205
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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 ...
- 11
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100
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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 ...
- 482
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answers
325
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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 $\mathcal{A}$ ...
- 163
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2
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662
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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 ...
- 9
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2
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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 $\mathcal C$ ...
- 18.7k
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668
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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 ...
- 15.5k
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1
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128
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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 $...
- 1,613
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1
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363
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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 $M$ ...
- 235
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votes
1
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288
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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 ...
- 2,871
2
votes
1
answer
108
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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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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 ...
- 9,009
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913
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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 ...
- 547
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0
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222
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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, ...
- 41
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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-...
- 37.8k
2
votes
1
answer
173
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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 ...
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1
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Smith Normal Form for block matrices over the integers
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 $kr \times ks$ made of square blocks of size $k$ such that ...
- 41
2
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1
answer
359
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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 (...
- 103
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votes
1
answer
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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 $...
- 518
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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 ...
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1
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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.
...
- 18.7k
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1
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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-...
- 18.7k
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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 ...
- 18.7k
11
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1
answer
502
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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 ...
- 37.8k
2
votes
1
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522
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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 ...
- 45
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1
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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 ...
- 141
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204
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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 ...
- 54.6k
1
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0
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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 ...
- 1,271
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676
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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 \...
- 15.5k
3
votes
2
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634
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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 ...
- 624
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2
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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 $i\ne 0$?
- 18.7k
3
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1
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282
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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 ...
- 360
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0
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90
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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}} \...
- 456
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1
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391
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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} ...
- 945
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votes
2
answers
1k
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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'...
- 417