# Questions tagged [homological-algebra]

(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

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### Every surface of sufficiently large genus separates

Let $M^3$ be a smooth closed orientable manifold. Does there exist a non negative integer $g_0$ such that every closed orientable embedded surface $\Sigma \subset M$ of genus $g \geq g_0$ represents ...
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### How to define cohomology of algebraic structures?

I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain \begin{align*} \cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
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### On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak m G$. Let $X$ be a finitely ...
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1 vote
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### Induced exact sequence on symmetric bilinear forms on abelian groups

Let $F=\operatorname {Hom}(S^2(\,\_\,),\mathbb{Q}/\mathbb{Z}):\mathsf{Ab} \to \mathsf{Ab}$ be the functor that sends an abelian group to the group of symmetric bilinear forms on this group. As far as ...
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### Restriction vs. multiplication by $n$ in Tate cohomology

$\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Cor}{Cor}$ This question was asked in MSE. It got no answers or comments, and so I post it here. Let $H$ be a subgroup of a finite group $G$, and ...
1 vote
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### Iterating exact triangles (particularly in Floer homology)

There are several different Floer-homological invariants of 3-manifolds (and knots). The most prominent of these are Heegaard Floer homology, monopole Floer homology, and instanton Floer homology. It ...
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### Generalizations of global Euler characteristic formula

Let $K$ be a number field, $S$ a finite set of primes of $K$ including the archimedean primes and $G_{K,S}$ be the Galois group of the maximal extension of $K$ unramified outside $S$. Assume ...
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### Mayer-Vietoris sequence in group cohomology for arbitrary pushout squares of groups?

Suppose that we have (not necessarily injective) group homomorphisms $H \to G_1$ and $H \to G_2$, and we construct the pushout (i.e. amalgamated free product) $G_1 \sqcup_H G_2$. Suppose that we have ...
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1 vote
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### About a recent paper of Rickard on finitistic dimension

Apologizes if this is a basic question, but I am new to the area of finite dimensional algebras. I am reading the paper "Unbounded derived categories and the finitistic dimension conjecture" ...
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### Finite lattices that are Koszul

Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$. It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
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### When does $FP_n(R)$ imply $F_n$?
It is known that if a group $G$ is of type $F_2$ (finitely presented) and of type $FP_n(\mathbb{Z})$, then $G$ is of type $F_n$. However, is this true also for other rings which are not $\mathbb{Z}$? ...