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
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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 ...
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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
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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 ...
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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 ...
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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 $\...
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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 ...
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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., ...
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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 $...
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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^...
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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 ...
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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 ...
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Why self-injective path algebras with relations are covered by cycles?
In this question, it is stated that if a path algebra with relations $kQ/I$ (where I is an admissible ideal) is self-injective, then Q is covered by cycles, that is, each of its arrows can be included ...
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Borel-Moore homology of the nilcone of $\mathfrak{sl}_2$
I'm trying to compute the Borel-Moore homology (which, abusing notation, we write $H_i$ for the $i$-th Borel-Moore homology group with coefficients in $\mathbb{C}$ - which is usually denoted by $H_i^{...
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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 ...
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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 ...
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