All Questions
Tagged with homological-algebra reference-request
271 questions
5
votes
2
answers
590
views
An explicit homotopy equivalence between the de Rham complex and the Cech-de Rham total complex
I'm currently in need an explicit formula in classical cohomology which I'm pretty sure is well known, but which I've been unable to find in the references I am aware of.
Let $X$ be a smooth ...
- 6,777
3
votes
1
answer
149
views
Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?
It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder ...
- 16.9k
3
votes
0
answers
244
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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 ...
- 9,019
13
votes
1
answer
694
views
Classification of long exact sequences
Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.
The category $\mathcal C$ is naturally additive as a subcategory of ...
- 3,184
3
votes
2
answers
543
views
Earliest/most standard reference for derived categories of hereditary algebras
Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C_{\bullet}$ in $\mathcal{D}$, we have $C_{\...
- 3,184
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 34.7k
14
votes
1
answer
625
views
$A_\infty$ structure on Ext-algebras well defined?
Let $M$ be an object in an $k$-linear abelian category with enough projectives. Then one can construct an $A_\infty$-structure on the Ext algebra
$$Ext^\bullet(M,M)$$
as follows: One chooses ...
- 30.3k
6
votes
2
answers
684
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Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"
Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...
- 1,738
6
votes
0
answers
723
views
On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
- 21.8k
3
votes
1
answer
172
views
Existence of a bounded finitely generated torsion-free resolution
I am looking for a reference for (or a proof of) the following fact:
Let $G$ be a profinite group.
Let $X^\bullet$ be a complex of discrete $G$-modules.
We assume that the cohomology $G$-modules of $...
- 14.2k
10
votes
1
answer
1k
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Is there a way to define a prime ideal object via diagrams in the category of rings?
I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...
CommunityBot
- 1
5
votes
0
answers
331
views
Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
8
votes
1
answer
1k
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Morphisms of Banach spaces
What is the standard name in English for bounded linear maps $f:E\to F$ between Banach spaces such that the kernel $\ker(f)$ has a complement, and $\text{im}(f)$ is closed, and has a complement?
...
- 4,586
0
votes
0
answers
301
views
Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
- 1,222
8
votes
2
answers
2k
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Modern reference for integral homology of a finitely generated abelian group
I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled "Alg&...
- 66.3k
11
votes
1
answer
3k
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Where can I easily look up / calculate (abelian) group cohomology?
For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...
- 22.6k
10
votes
1
answer
842
views
Is there a "correct" general setting for the principle: "tensoring any object with a projective object yields another projective"?
Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$. (See Exercise 2 on p. ...
CommunityBot
- 1
15
votes
1
answer
1k
views
Comodule exercises desired
This Question is inspired by a Quote of Moore's
"There are two ‘evil’ influences at work here:
1. we are toilet trained with algebras not coalgebras
2. some of us are addicted to manifolds and so ...
- 3,726
13
votes
1
answer
1k
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Can the Jacobi-Trudi identity be understood as a BGG resolution?
The thought process that led me to this question is that the identity
$$ \left(\prod_i \frac1{1-x_i}\right)\left(\prod_i {1-x_i}\right)=1$$
can be understood as expressing exactness of the Koszul ...
- 9,132
12
votes
0
answers
552
views
References for a certain generalization of Hochschild cohomology?
Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
- 21k
10
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0
answers
1k
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Complexes of representations with complementary central charges
This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
- 15.6k