Questions tagged [homological-algebra]
Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.
5,548 questions
5
votes
2
answers
213
views
Best approximation to an adjoint functor
I have the following question.
Suppose I have a functor $F\colon C\to D$ between two categories. I would like it to have an adjoint (say, right), but it doesn't. Is there a way to define a "best ...
- 185k
3
votes
1
answer
63
views
Projective resolutions in the wrong direction
Let $M$ be an abelian group, and consider a “projective resolution”
$$
0 \to M \to P_0 \to P_1 \to \cdots \to P_n \to 0.
$$
Note that this is not a projective resolution in the usual sense since it's ...
- 33k
1
vote
1
answer
76
views
Does condition $\operatorname{Ext}^i(E(-p), C) \cong 0$ imply that $C \cong 0$?
Let $X$ be a smooth projective variety over a field $k$, $E$ is a vector bundle on $X$ and $C$ is a complex of vector bundles on $X$ i.e. $C \in D^b(Coh (X))$. Assume that
$$
\operatorname{Ext}^i(E(-p)...
- 900
3
votes
2
answers
500
views
Showing that $k[G]$ is a self-injective module
Let $k$ be a field, $G$ a finite group, and $k[G]$ the group ring. I'm trying to show that $k[G]$ is self-injective, meaning that it is injective as a (left) module over itself.
One possible approach ...
- 18.4k
3
votes
1
answer
119
views
Doubt on a non canonical isomorphism in exact functors
Let $\mathcal T$ and $\mathcal S$ be triangulated categories (I'm using the definition of triangulated category given here, because that is the definition that my teacher uses, even if I have seen ...
0
votes
0
answers
29
views
Prerequisites for learning grid homology / knot Floer homology?
I would like to study grid homology (combinatorial version of knot Floer homology) as described by Ozsváth–Szabó, Rasmussen, and later Sarkar–Wang and Manolescu–Ozsváth–Sarkar. What are the minimal ...
- 109
1
vote
1
answer
49
views
Lurie's definition of K-injective complexes
Let $\mathcal A$ be an abelian category, $Ch(\mathcal A)$ its category of cochain complex, $K(\mathcal A)$ be the homotopy category of cochain complexes.
Usually, K-injective complex is defined as a ...
- 99.7k
9
votes
6
answers
1k
views
(Elementary) applications of group (co-)homology
I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology.
My background is, ...
- 2,014
5
votes
1
answer
112
views
The relation between the categories of chain and cochain complexes
Let $\sf A$ be an additive category.
Define the category $\mathrm C_\bullet(\mathsf A)$ as follows:
An object $A_\bullet$ in $\mathrm C_\bullet(\mathsf A)$ is a diagram (indexed by the category $\...
- 13.8k
0
votes
1
answer
81
views
Homotopy category is (?) equivalent to the terminal category
Let $\sf A$ be an additive category. Let $\mathrm{C}\sf A$ be the category of cochain complex over $\sf A$ and denote by $\mathrm{K}\sf A$ the homotopy category. I have made a reasoning whose ...
- 13.8k
48
votes
5
answers
10k
views
Proving the snake lemma without a diagram chase
Suppose we have two short exact sequences in an abelian category
$$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$
$$0 \to A' \mathrel{\overset{f'}{\to}} B' \...
- 11.4k
3
votes
2
answers
137
views
When can we define a derived functor (and on which side)?
Consider a ring and the its derived category $D_\cdot(R)$ (resp. $D^\cdot(R)$), which is equivalent to the homotopy category of chain (cochain) complexes of projective (injective) $R$-modules. So far, ...
- 48.8k
3
votes
0
answers
27
views
Geometric intuition for Massey’s chain homotopy for the cubical subdivision operator
In Singular Homology Theory by Massey, a chain homotopy is constructed between the identity and the cubical subdivision operator. Instead of simplices, Massey works with singular cubes, i.e., ...
- 161
2
votes
0
answers
84
views
Construction of simplicial homotopy equivalence of Path space and Constant
Definition of Path space of a simplicial object:
There is a functor $P: \Delta \to \Delta$ with $P[n] = [n+1]$ such that the natural map $\epsilon_0: [n] \to [n+1] = P[n]$ is a natural transformation $...
- 11k
1
vote
2
answers
5k
views
How to prove that $\mathbb Q$ is flat as a $\mathbb Z$-module? [duplicate]
$\DeclareMathOperator\Tor{Tor}$I know that $\Tor^{\mathbb Z}_1(\mathbb Z, N) = 0$ for any $\mathbb Z$-module, because free modules are flat. Then because $\Tor_1$ is local, we have $\Tor_1^{\mathbb Q}(...
- 4,234