Questions tagged [chain-complexes]

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7
votes
0answers
453 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 ...
2
votes
1answer
117 views

Methods for finding complex for subobjects of homology

Let $\mathcal{C}$ be an abelian category and $$ C_\bullet:C_n \rightarrow C_{n-1}\rightarrow \ldots \rightarrow C_1\rightarrow C_0$$ a complex in $\mathcal{C}$. Suppose we have for each $i$ a ...
2
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0answers
198 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)...
2
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0answers
52 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 \...
7
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2answers
439 views

Trees in chain complexes

$\DeclareMathOperator{\Ch}{\mathit{Ch}}$Let $\Ch_\mathbb{Q}$ denote the model category of chain complexes over rational numbers. Let $T_\ast$ be a tree in $\Ch_{\mathbb{Q}}$ with $n$ vertices. How to ...
4
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0answers
94 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 ...
6
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0answers
274 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 ...
5
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1answer
383 views

Cohomology of derived tensor product of complexes and Künneth spectral sequence

Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
4
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2answers
279 views

Model categories and chain complexes

I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ ...
9
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2answers
771 views

Is the tensor product of chain complexes a Day convolution?

Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...
4
votes
1answer
146 views

Split cofibrations up to quasi-isomorphism

$R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules). Suppose that $A_{\bullet}\rightarrow B_{\bullet}\rightarrow B_{\bullet}/A_{\bullet}$ a short ...
5
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0answers
109 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
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1answer
440 views

Sign in May’s General algebraic approach to Steenrod operations

In the first section of J. P. May’s General algebraic approach to Steenrod operations, May defines for $\pi\subseteq\Sigma_r$ an integer $q\in\mathbb{Z}$ and a commutative ring $\Lambda$, the $\Lambda\...
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0answers
53 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) \...
3
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0answers
105 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 ...
4
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0answers
168 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 $\mathcal{O}$ ...
1
vote
1answer
141 views

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 ...
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0answers
48 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
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0answers
135 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 ...
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0answers
112 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. ...
0
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1answer
80 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
47 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
96 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
515 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
votes
1answer
403 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 ...
2
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1answer
290 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
151 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
votes
1answer
288 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
vote
1answer
153 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
152 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
88 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 ...
2
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0answers
247 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 $\mathcal{A}$ ...
0
votes
2answers
444 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
votes
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 $\mathcal C$ ...
7
votes
0answers
470 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
votes
1answer
123 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
votes
1answer
301 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
votes
1answer
201 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
votes
1answer
92 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 $\mathcal{S}$...
2
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0answers
727 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 ...
3
votes
2answers
747 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 ...
0
votes
0answers
191 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, ...
7
votes
2answers
1k 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
votes
1answer
158 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 ...
4
votes
1answer
459 views

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 ...
2
votes
1answer
297 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 (...
14
votes
1answer
709 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
votes
0answers
539 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
votes
1answer
346 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. ...