Questions tagged [chain-complexes]
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47
questions with no upvoted or accepted answers
23
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0
answers
661
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 \...
13
votes
0
answers
664
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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 ...
11
votes
0
answers
200
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 ...
7
votes
0
answers
582
views
Understanding the higher stack of perfect complexes
One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:
We fix a function $b: \mathbb{Z} \rightarrow
\mathbb{N}$ which is zero ...
7
votes
0
answers
630
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 ...
6
votes
0
answers
133
views
Does homotopy equivalence between deformations of cochain complexes imply gauge equivalence?
Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator $d+\epsilon: C^{\bullet}\otimes_k m\to C^{\...
6
votes
0
answers
461
views
When is the dual of a limit the same as the colimit of the duals?
We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that ...
6
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0
answers
179
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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, ...
5
votes
0
answers
123
views
The interaction between differentials on a graded ring and chain-homotopy equivalences
I am wondering about the following question:
Given a differential graded algebra $A$, how many other differentials can we put on the underlying graded ring of $A$, which are also chain-homotopy ...
5
votes
0
answers
679
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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0
answers
122
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$K$-group of category of bounded chain complexes of Projective modules with finite length homologies
For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
4
votes
0
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186
views
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 $\mathcal{O}$ ...
4
votes
0
answers
105
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 ...
3
votes
0
answers
103
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Multiplication map by a ring element on an object vs. all its suspensions in singularity category
Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
3
votes
0
answers
200
views
What do the indecomposable objects of the homotopy category of chain complexes look like?
I am trying to understand indecomposable objects in the homotopy category of chain complexes. Let $\mathcal{A}$ be an abelian category. Denote by $C(\mathcal{A})$ the category whose objects are chain ...
3
votes
0
answers
121
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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 ...
3
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0
answers
63
views
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 ...
3
votes
0
answers
141
views
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 ...
3
votes
0
answers
318
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 $\mathcal{A}$ ...
3
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0
answers
241
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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 ...
2
votes
0
answers
82
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Multiplicative structure on Čech–Alexander complexes
I have the following basic question on Čech–Alexander complexes.
Let $R$ be a ring and $A$ be an $R$-algebra. To this datum one can attach a cosimplicial ring which assigns to an object $[n]=\{0,1,\...
2
votes
0
answers
110
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Hypercube of chain complexes as functor from (Δ^1 )^n to ∞-category of chain complexes
A hypercube of chain complexes consists of $\mathbb{Z}$-graded vector spaces $C_\epsilon$ for $\epsilon\in\{0,1\}^n$ and maps $D_{\epsilon,\epsilon^\prime}:C_{\epsilon}\to C_{\epsilon^\prime}$ for $\...
2
votes
0
answers
104
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Homotopy equivalence of chain complexes from subcomplexes and quotient complexes
Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
2
votes
0
answers
89
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Generalizations of elliptic chain complexes
I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....
2
votes
0
answers
303
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Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy
$\newcommand\Ab{\mathrm{Ab}}\newcommand\ab{\mathrm{ab}}\DeclareMathOperator\Ch{Ch}\DeclareMathOperator\Kom{Kom}\newcommand\ho{\mathrm{ho}}$Let $\Ab$ be the category of finitely generated abelian ...
2
votes
0
answers
79
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Chain complex of the Salvetti complex of an Artin group
Let $A_\Gamma$ be an Artin group. The Salvetti complex $Sal(A_{\Gamma})$ can be briefly defined as the $2$-presentation complex associated to the usual presentation of the Artin group after attaching ...
2
votes
0
answers
177
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Is the category of cochain complexes with terms in an additive category a 2-category?
$\def\hom{\operatorname{Hom}}
\def\bbZ{\mathbb{Z}}$This question is a follow-up to this other one. There the OP asks whether "the category of chain complexes" (can be interpreted in several ...
2
votes
0
answers
363
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Existence of quasi-isomorphisms between complexes with same homology
Consider an abelian category $\mathcal{A}$ (or more specifically, $R$-Mod). Suppose $C_1$ and $C_2$ are chain complexes with componentwise isomorphic homology. What conditions must be imposed upon $\...
2
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0
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201
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Do the polyhedral homologies of a polyhedron coincide with the polyhedral homologies of its subdivision?
Definition. A convex polytope is a compact finite intersection of hyperplanes in $\mathbb{R}^n$
Definition. The polycomplex is the following data set:
a set of convex polytopes, closed under ...
2
votes
0
answers
263
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References for Homotopy transfer problem
I am trying to read Algebra+homotopy=operad by Bruno Vallette.
Consider the following set up :
chain complexes $(A,d_A),(H,d_H)$,
a degree $1$ morphism of chain complexes $h:(A,d_A)\rightarrow (A,d_A)...
2
votes
0
answers
54
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contracting homotopies of Koszul resolution of $\mathbb{C}[x_1, \ldots, x_n]$ and $\mathbb{C}_{q}[x_1, \ldots, x_n]$
Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$.
By Koszul resolution I mean
$$\ldots \to A \...
2
votes
0
answers
673
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How does "chain complex functor" from $Top$ take mapping cones to mapping cones?
I am wondering how the singular chain complex functor from the category of topological spaces to the category of chain complexes of abelian groups takes a mapping cone to a mapping cone in the sense ...
2
votes
0
answers
67
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Order relation between cohomology groups
We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex
$$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
2
votes
0
answers
835
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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 ...
1
vote
0
answers
76
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Singular chain complex of balanced products
Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring)
$$f:C_*(V) \...
1
vote
0
answers
119
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Lift up characteristic class to chain complex
In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...
1
vote
0
answers
56
views
Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective complexes themselves?
Let $R$ be an associative ring with unit and $I$ be a complex of $R$-modules. We call $I$ is h-injective if for any acyclic complex $T$ of $R$-modules, the mapping complex $\text{Hom}_R(T,I)$ is ...
1
vote
0
answers
196
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Sheaves and isomorphisms with chain complex of singular chains (Sheaf Theory, Bredon)
Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$.
How to show that the homomorphism of ...
1
vote
0
answers
99
views
Are mapping cones in the bounded homotopy category of chain complexes isomorphic?
Let $A$ be an additive category. Suppose we have distinguished triangles
$$X \rightarrow Y \rightarrow Z \rightarrow X[1]$$
and
$$X \rightarrow Y \rightarrow Z' \rightarrow X[1]$$
in the bounded ...
1
vote
0
answers
170
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Dold-Kan preserves weak equivalences and fibrations
It's well known that Dold-Kan correspondence is an isomorphism between simplicial vector spaces and non-negative chain complexes of vector spaces. Moreover, weak equivalences and fibrations are ...
1
vote
0
answers
236
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Two-point desuspension for augmented chain complexes?
Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = \...
1
vote
0
answers
80
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Extending Reedy dimension to augmented chain complexes of abelian groups
Recall that a normal continuously-graded finite interval is given by a pair $a=([a],f)$ consisting of:
1.) A finite totally ordered set $[a]=[a_0< \dots < a_n]$
2.) A grading function $f:U[a] \...
0
votes
0
answers
40
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Is any deformation of an acyclic complex gauge equivalent to a trivial one?
This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator ...
0
votes
0
answers
201
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Determinant of chain complexes
Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...
0
votes
0
answers
215
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Torsion in cohomology
Suppose to have a short exact sequence of chain complexes of $\mathbb{Z}$-modules:
$$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, ...
0
votes
0
answers
89
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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}} \...
0
votes
0
answers
211
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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":
$...