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
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 ...
- 3,631
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)...
- 6,898
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 ...
- 745
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 $\...
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
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 ...
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 $...
1
vote
1
answer
105
views
Cohomology of a degree $5$ hypersurface $X\subseteq \mathbb{P}^{4}$ - Sanity Check
Consider a degree $5$ hypersurface $X\subseteq\mathbb{P}^{4}$. I know by computing the Euler characteristic that the hohomology $H^{2,1}(X)\cong\mathbb{C}^{101}$, but have confused myself on what ...
- 1,719
19
votes
2
answers
1k
views
Source of commutative diagram in a math meme
I am trying to find a reference, the source of this commutative diagram:
I have tried prompting an LLM, which erroneously thought it was the following (also quite complex) commutative diagram from ...
- 2,985
1
vote
0
answers
59
views
Can double complexes be relabelled without applying shift operators [closed]
Let $A^{p,q}$ be a bounded double complex concentrated in bidegrees
$$p \in \{0,1\}, \qquad q \in \{-1,0\}.$$
Thus the complex consists of two columns and two rows, and its total complex is
$$
Tot^n(A^...
- 43
3
votes
1
answer
228
views
What is the content of the splitting lemma?
a bit of an abstract question today.
I've seen the splitting lemma lots in homological algebra, but I feel that I don't have great intuition for what it means. Let me start by stating it properly for ...
- 431
1
vote
1
answer
142
views
Equivalence of $Supp(a) \subseteq D(J)$, $V(Ann(a)) \cap V(J) = \emptyset$, and $Ann(a) + J = A$
I'd like to proof that the following statements are equivalent for a ring $A$ (commutative I think), $a \in A$ and $J$ and ideal in $A$:
$$(i) Supp(a) \subseteq D(J)$$
$$(ii) V(Ann(a)) \cap V(J) = \...
- 672
3
votes
1
answer
86
views
Submodules of a finitely presented module over coherent ring annihilated by powers of a fixed element
Let $R$ be a commutative coherent ring of finite weak global dimension . Let $x\in R$. Let $M$ be a finitely presented module, i.e., a coherent module over $R$. For every integer $n>0$, consider ...
- 1,914
2
votes
0
answers
58
views
Question about strong homotopy Lie algebras regarding signs in the derivation of the Leibniz rule
In the paper "An introduction to the Batalin-Vilkovisky formalism" in the chapter on $L_{\infty}$-algebras on page 7, it states that one can define and $L_{\infty}$-algebra by defining a ...
- 39