Questions tagged [chain-complexes]

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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 \...
Vidit Nanda's user avatar
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13 votes
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664 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 ...
John Pardon's user avatar
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11 votes
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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 ...
Theo Johnson-Freyd's user avatar
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 ...
Martin Hurtado's user avatar
7 votes
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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 ...
Vidit Nanda's user avatar
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6 votes
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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^{\...
Zhaoting Wei's user avatar
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6 votes
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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 ...
Daniel Robert-Nicoud's user avatar
6 votes
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179 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, ...
Simon Henry's user avatar
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5 votes
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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 ...
Mo Behzad Kang's user avatar
5 votes
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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 ...
Sondre's user avatar
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122 views

$K$-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 ...
user521337's user avatar
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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 $\mathcal{O}$ ...
FKranhold's user avatar
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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 ...
Zhaoting Wei's user avatar
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3 votes
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103 views

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\...
uno's user avatar
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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 ...
death_cube_k's user avatar
3 votes
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121 views

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 ...
Julian Chaidez's user avatar
3 votes
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 ...
BinAcker's user avatar
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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 ...
Ronald J. Zallman's user avatar
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 $\mathcal{A}$ ...
kesa's user avatar
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241 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 ...
Zhaoting Wei's user avatar
  • 8,657
2 votes
0 answers
82 views

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,\...
Stabilo's user avatar
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2 votes
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110 views

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 $\...
Faniel's user avatar
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2 votes
0 answers
104 views

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 ...
Faniel's user avatar
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2 votes
0 answers
89 views

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....
Arturo's user avatar
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2 votes
0 answers
303 views

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 ...
Student's user avatar
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2 votes
0 answers
79 views

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 ...
Marcos's user avatar
  • 577
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0 answers
177 views

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 ...
Elías Guisado Villalgordo's user avatar
2 votes
0 answers
363 views

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 $\...
Sam K's user avatar
  • 113
2 votes
0 answers
201 views

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 ...
Arshak Aivazian's user avatar
2 votes
0 answers
263 views

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)...
Praphulla Koushik's user avatar
2 votes
0 answers
54 views

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 \...
Invincible's user avatar
2 votes
0 answers
673 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 ...
ARA's user avatar
  • 739
2 votes
0 answers
67 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\...
King Khan's user avatar
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2 votes
0 answers
835 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 ...
Zhaoting Wei's user avatar
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1 vote
0 answers
76 views

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) \...
FKranhold's user avatar
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1 vote
0 answers
119 views

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. ...
userabc's user avatar
  • 667
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 ...
Zhaoting Wei's user avatar
  • 8,657
1 vote
0 answers
196 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 ...
User2211's user avatar
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 ...
Iteraf's user avatar
  • 482
1 vote
0 answers
170 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 ...
Ma Ming's user avatar
  • 1,261
1 vote
0 answers
236 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 = \...
Harry Gindi's user avatar
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1 vote
0 answers
80 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] \...
dcomplex's user avatar
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0 answers
40 views

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 ...
Zhaoting Wei's user avatar
  • 8,657
0 votes
0 answers
201 views

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 ...
Ronald J. Zallman's user avatar
0 votes
0 answers
215 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, ...
user53075's user avatar
0 votes
0 answers
89 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}} \...
Mario Carrasco's user avatar
0 votes
0 answers
211 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": $...
AlgRev's user avatar
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