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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.

A chain complex is a sequence of abelian groups, vector spaces, or modules, with linear maps connecting them which compose to zero.

Homological algebra is the study of chain complexes and their homology groups.

5114 questions
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Is a $p$-torsion-free $\mathbb{Z}_{(p)}G$-module with finite projective dimension projective?

Let $G$ be a finite group, $\mathbb{Z}_{(p)}$ be the ring of p-local integers (localization of $\mathbb{Z}$ at $p\mathbb{Z}$). Let $M$ be a $p$-torsion-free (i.e. $pm = 0$ implies $m=0$) $\mathbb{Z}_{(p)}G$-module with finite projective dimension.…
Cihan
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Injective dimension $n$ implies $Ext^n$ does not vanish with an injective

Let $M$ be a finitely generated module and suppose that the injective dimension of $M$ is $n$. I want to show that there exists an injective module $I$ such that $Ext^n(I,M)\neq 0$ (and if the projective dimension of $M$ is $n$, then there exists a…
koizumi
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A question on the standard chain complex

Suppose that $G$ is a group, and $\mathbb{Z}[G]$ the group ring. Then $\mathbb Z$ can be considered as a $\mathbb{Z}[G]$-module if every $g (\in G)$ acts trivally, and every $n (\in \mathbb Z)$ acts by multiplication on $\mathbb Z$. In order to find…
ShinyaSakai
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Homology of the complement of a real hypersurface

Consider a real algebraic set $Z(f) = \{x \in \Bbb{R}^n\,|\, f(x) = 0\} \subset \Bbb{R}^n$ (not necessarily irreducible). I'm thinking about wether the (Euclidean) closure of a connected component of the complement $\Bbb{R}^n\setminus Z(f)$ has…
Raclette
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Quasi-isomorphism from "almost acyclic" complex to its homology

The following is an exercise in the book Representation Theory of Finite Reductive Groups by Cabanes and Enguehard. Let $\mathcal{A}$ be an abelian category. Let $X$ be a complex of objects of $\mathcal{A}$. Assume $H^i (X ) = 0$ for all $i \neq…
Matthias Klupsch
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Ext groups for fraction field and a module annihilated by an element

Suppose $Q$ is the field of fractions for a domain $R$ and $A$ is an $R$-module such that $rA = 0$ for some $0 \ne r \in R$. Why is it the case that $\text{Ext}_R^n(Q,A) = 0$ for all $n \ge 0$? I have a feeling this isn't too hard, and maybe relies…
Mr. Chip
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Basic computation of exact sequence

Given a long exact sequence of vector spaces: $$...\longrightarrow V_1 \overset{f}{\longrightarrow}V_2\overset{g}{\longrightarrow}V_3\longrightarrow...$$ Given another vector space $W$, is the following sequence still exact?How to write down the map…
C Weid
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computation of a group homology

A few days ago, I asked a question about a group homology, and it was actually easy. I am continuing computing group homologies, but I am stuck on this: $H_*^{\textrm {grp}}(T, \mathbb{Z}) = \textrm{Tor} _* ^{\mathbb{Z}(T)}(\mathbb{Z}, \mathbb{Z})$…
S. Ha
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Extensions of G-modules

Let $G$ be a finite group of order $n$ and $\Lambda={\mathbb{Z}}[G]$ the group ring of $G$. Let $A$ be a finitely generated free abelian group on which $G$ acts. Let $B$ be a finitely generated $\Lambda$-module. Question. How can one prove that…
Mikhail Borovoi
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Constructing a cochain complex out of a chain complex

Let $(C,\partial)$ be a chain complex where $C_i$ is an $R$-module ($R$ is a given ring) , we can always construct a cochain complex out of the chain complex $(C,\partial)$ in the following way: We construct the $i$th $R$-module of the cochain…
palio
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About Project Module and Flat Module over a Non-commutative Ring

Given a non-commutative ring R, how to prove that a projective left module is a flat left module by using the natural isomorphism: $Hom_{\mathbb{Z}}(A\otimes_RB,G)\cong Hom_R(B,Hom_{\mathbb{Z}}(A,G))$, where A and B are right module and left module…
Karoo Yang
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Show that in any connected triangulation, a 0-cycle $\sum c_i[i]$ is a 0-Boundary $\iff$ $\sum c_i =0$.

Show that in any connected triangulation, a 0-cycle $\sum c_i[i]$ is a 0-Boundary $\iff$ $\sum c_i =0$. $(\Longrightarrow)$ Case 1: If the 0-cycle is the boundary of a 1-face, it must be of the form $[b]-[a]$ by the definition of the boundary…
user58289
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Exact sequence induced by two head-to-tail arrows

Let $f: A \to B$ and $g: B \to C$ be two arrows in an abelian category $\mathsf{A}$. Prove that they induce an exact sequence: $$0 \to kerf \to kergf \to kerg \to cokerf \to cokergf \to cokerg \to 0$$ This is exercise 8.4.6 in Category Theory for…
SalutaFungo
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Rotman Homological Algebra Proposition 4.76(ii)

Theorem Question I don't understand the normalization step. We start out with $S = R - \mathfrak p$, and it's clear that $\mathfrak p$ is the only prime ideal for $S^{-1} R$ because it is minimal prime. But then $S$ is changed, and yet we continue…
IsaacR24
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Generalized Associativity for tensor products

I'm following the steps suggested in Rotman's text (Problem 2.30). Let $R$ be a commutative ring and let $A_1, \ldots, A_n, M$ be $R$-modules with $n \ge 2$. Let $F$ be a free $R$-module with basis $\Pi A_i$ and define $N \subset F$ to be the…
IsaacR24
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