All Questions
Tagged with chain-complexes homological-algebra
59 questions
9
votes
3
answers
2k
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-...
- 37.8k
4
votes
1
answer
267
views
A particular morphism being zero in the singularity category
Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
- 35.2k
3
votes
1
answer
152
views
Homotopy coherent transformation and totalization
Let $C$ be the category of chain complexes over a field $F$ and $C^\prime$ be the subcategory of chain complexes with zero differentials. If $X:I\to C$ is a functor, there is an induced "homology&...
- 12.9k
3
votes
1
answer
129
views
Thick subcategory containment in bounded derived category vs. singularity category
Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...
- 15.8k
2
votes
1
answer
294
views
Projective objects in chain complexes of an abelian category: Further question
Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes
I am wondering why a level-wise projective chain complex $P$ which is split ...
- 30.3k
3
votes
0
answers
106
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\...
- 412
2
votes
0
answers
147
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 ...
- 673
0
votes
0
answers
41
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 ...
- 9,009
6
votes
0
answers
138
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^{\...
- 9,009
12
votes
3
answers
2k
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 ...
- 141
8
votes
1
answer
399
views
Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors?
Let $A$ be an abelian category (you can assume additional conditions for its goodness). Let $\mathrm{Seq}(A) = \mathrm{Func}(\mathbb{Z}, A)$, where $\mathbb{Z}$ is the standard order category on ...
- 3,411
2
votes
0
answers
310
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 ...
- 12.9k
2
votes
0
answers
545
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 $\...
- 175
2
votes
0
answers
221
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 ...
- 6,962
3
votes
0
answers
238
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 ...
- 41
0
votes
0
answers
253
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 ...
CommunityBot
- 1
4
votes
1
answer
558
views
derived tensor product and finite projective dimension
Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules.
Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ ...
- 361
2
votes
1
answer
124
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 ...
- 4,709
23
votes
3
answers
4k
views
how to make the category of chain complexes into an $\infty$-category
I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.
Has anyone ever written down ...
- 56.9k
2
votes
0
answers
272
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)...
- 6,023
2
votes
0
answers
64
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 \...
- 161
4
votes
0
answers
135
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 ...
- 1,209
5
votes
1
answer
2k
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 ...
- 498
5
votes
2
answers
683
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$ ...
- 30.4k
4
votes
1
answer
193
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 ...
- 35.2k
5
votes
0
answers
129
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 ...
- 179
5
votes
1
answer
540
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\...
- 30.4k
1
vote
0
answers
77
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) \...
- 1,623
3
votes
0
answers
126
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 ...
CommunityBot
- 1
4
votes
0
answers
190
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 $\mathcal{O}$ ...
- 1,623
3
votes
0
answers
68
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 ...
- 789
1
vote
0
answers
57
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 ...
- 9,009
2
votes
1
answer
342
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 $A$ ...
- 39.3k
1
vote
1
answer
197
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 ...
- 35.2k
1
vote
0
answers
100
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 ...
- 482
3
votes
0
answers
325
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 $\mathcal{A}$ ...
- 163
3
votes
2
answers
913
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 ...
- 118k
25
votes
2
answers
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 $\mathcal C$ ...
- 18.4k
7
votes
0
answers
668
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 ...
CommunityBot
- 1
11
votes
1
answer
502
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 ...
CommunityBot
- 1
51
votes
8
answers
7k
views
Motivating the category of chain complexes
Let $R$ be a commutative ring. For awhile I have been trying to motivate to myself more fully the definition of and various structures on the category $\text{Ch}(R)$ of chain complexes of $R$-modules (...
CommunityBot
- 1
2
votes
1
answer
108
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 $\mathcal{S}$...
- 35.2k
2
votes
0
answers
866
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 ...
- 9,009
2
votes
1
answer
522
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 ...
- 45
2
votes
1
answer
173
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 ...
- 35.2k
13
votes
0
answers
680
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 ...
- 18.7k
5
votes
0
answers
723
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 ...
- 345
11
votes
0
answers
204
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 ...
- 54.6k
3
votes
1
answer
282
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 ...
- 23k
0
votes
0
answers
90
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}} \...
- 456