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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.

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Necessary and sufficient condition of injectivity for a module based on exact sequences

Show that Q is injective if and only if, whenever $$0\to A\stackrel{f}{\to} B\stackrel{g}{\to} C\to 0$$ is exact, then $$0\to \mathrm{Hom}_R(C,Q)\stackrel{g^*}{\to} \mathrm{Hom}_R(B,Q)\stackrel{f^*}{\...
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Connections between Ext and vertex

I haven't gotten a chance to search through the literature yet, but I was wondering if anyone knows anything about or has any thoughts about this idea: So, if we have a module $M$, then we can easily ...
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group cohomology equivalent to topological singular cohomology

Let $G=<\sigma>$ be a cyclic group of order $n$. For any $\mathbb{Z}[G]$ module $M$ it is known that the group cohomology $$ H^i(G, M) = \begin{cases} M^G &\text{ if } i = 0 \\ M^G/NM &...
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$\text{Tor}_0^R(M,N)\simeq M\otimes_R N$ [duplicate]

I'm trying to understand why $\text{Tor}_0^R(M,N)\simeq M\otimes_R N$. Let $\dots\to F_1\to F_0\to M\to 0$ be a free resolution of $M$. Then $$ F_1\to F_0\to M\to 0$$ is exact. Hence $$ F_1\otimes_R ...
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Cech cohomology of coefficient in a presheaf and in its associated sheaf

I am reading Jean-Luc Brylinski's book on loop spaces. In his book, he claims that: Suppose that for any presheaf $F$ of Abelian group such that its associated sheaf $aF$ is $0$, the groups $\check{H}...
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A spectral sequence with only one index in Atiyah's paper?

I would like to read Atiyah's paper Characters and cohomology of finite groups; but when I started reading the introduction, Atiyah mentions that he will prove that there is a "spectral sequence $\{E^...
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Homology - long exact sequence from short exact sequences of cofibrations

I am currently self-studying a course in algebraic topology (http://ium.mccme.ru/f18/f18-topology-2.html, it's in Russian) and I've been stuck at one problem for quite some time. Let $\mathcal{HoCW}$ ...
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Asymmetry in the notion of exactness

$\newcommand{\im}{\operatorname{im}}$We all know that, in an abelian category, a sequence $A\xrightarrow{f} B\xrightarrow{g} C$ is called exact if $\im f=\ker g$. The one inclusion $\im f\subseteq\ker ...
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A question about dual map on modules

Let $R$ be a commutative ring (noetherian if needed), and $P$ a finitely generated projective $R$-module (of constant rank $n$). Let $P^\vee:={\rm Hom}_R(P, R)$ be the dual. For $\varphi\in{\rm Aut}_R(...
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Differential on tensor product of $A_\infty$ modules

Let $A$ be an $A_\infty$-algebra over a commutative ring $k$, and $M$ is a left $A$-module and $N$ is a right $A$ module. By modules I mean $A_\infty$-modules. Then we can define $A_\infty$ tensor ...
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Commutative rings as co-limit of Noetherian rings?

Question 1: Does there exist a small category $\mathcal J$ such that for every commutative ring $A$, there is a functor $F :\mathcal J \to \mathcal CRing$ such that $ F$ takes every object to a ...
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Definitions of Group Cohomology

I am trying to understand group cohomology, and I have a very basic question. So as I understand it, let $\Gamma$ be a group, and $V$ be a $\Gamma$-module (which is essentially another abelian group ...
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Commutative ring as a direct limit of Noetherian rings

Does there exist a directed set $\mathcal J$ such that every commutative ring with unity is a direct limit of a family of (commutative) Noetherian rings indexed by $\mathcal J$ ?
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$\infty$-category of chain complexes via Dold-Kan correspondence

Suppose we are given the category of abelian groups $\mathsf{Ab}$. Then I am aware of two procedures how to turn the category of chain complexes $\operatorname{Ch}_{\geq 0}(\mathsf{Ab})$ that are ...
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Group homomorphism induces cohomology homomorphism

Let $\phi:G\to H$ a group homomorphism. I'm interested in knowing what can be said about $H^*(X;G)$ and $H^*(X;H)$ (singular cohomology) in terms of $\phi$. One can define $\phi^*:C^n(X;G)\to C^n(X;...
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Defining the cycle functor $Z_* : \mathbf{Ch_R} → \mathbf{GrMod_R}$

In example (viii) of section 1.3.2 of Category Theory in Context, the $n$-cycle functor is defined on objects as \begin{align} Z_n: \mathbf{Ch_R}&\to\mathbf{Mod_R} \\ C_\bullet&\mapsto \ker(...
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Why does $Ext_{R}^{n}(I,B)\neq 0$, $I$ is injective with $pd(I)=n$, & $B$ the image of a projective $P$, imply $Ext_{R}^{n}(I,P)\neq 0$?

I've see a proof that say that as $Ext_{R}^{n}(I,B)\neq 0$, where $I$ is an injective module with projective dimension $n$, and as $B$ is the image of a projective module $P$ $Ext_{R}^{n}(I,P)\neq 0$ ...
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image of generator in filtered colimit in grothendieck category

Suppose $\mathscr{A}$ is a grothendieck abelian category with generator $R$, is it true that $$\varinjlim \mathrm{Hom}(R,M_i) =\mathrm{Hom}(R,\varinjlim M_i)$$ if $M_i$ is a filtered system of objects....
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Is $ \operatorname{Hom}_R(R,M)$ an $R$-module?

Let $R$ be a ring and $M$ be a left $R$-module,then I have to show that $ \operatorname{Hom}_R(R,M) \cong M$. But for that I will have to make $ \operatorname{Hom}_R(R,M)$ an $R$-module. I tried $r.\...
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Why does spectral sequence $E^\infty$ need AB4*

So in Weibel, he states Warning: In an unbounded spectral sequence, we will tacitly assume that $B^{\infty}$, $Z^{\infty}$, and $E^{\infty}$ exist! The reader who is willing to only work in the ...
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Does trivial cohomology of spectra imply trivial homology?

It is known that for any spectrum $X$, $H\mathbb{Z}^*(X)=0$ implies that $H\mathbb{Z} \wedge X =0$. Also, for the case $HF_p,$ if we consider $ HF_p^*(X) =0.$ This gives $[HF _ p \wedge X , \sum^i ...
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Injective Dimension and Base Change

I know that if $R \rightarrow S$ is a ring map and $A$ is an $S$ module, $pd_R(A) \le pd_R(S)+pd_S(A)$. Is this true when projective dimension is replaced by injective dimension? i.e Is $id_R(A) \le ...
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Theorem on Injective Dimension in Weibel's “An Introduction to Homological Algebra”

This fact is stated in Weibel's book without proof: If $A$ is a finitely generated module over a commutative Noetherian local ring $(R,\mathfrak{m},k)$, then $id(A) \le d$ is equivalent to $\...
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Definition of $A_\infty$-module

Let $A$ be a $A_\infty$-algebra over a commutative ring $k$, suppose $V$ is a complex of $k$-modules. The "usual" definition of the structure of $A_\infty$-module on $V$ the sequence of map $$ s_n : A^...
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Three questions about Properties of Ext functor.

I have three questions about the Ext functor's properties. (i) $Ext(H \oplus H',G) = Ext(H,G) \oplus Ext(H',G)$ (ii) $Ext(H,G) = 0$ if $H$ is free (iii) $Ext(\mathbb{Z_n}, G) = G/nG$ There is a ...
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On the dual group of the group of units of commutative ring

All rings below are commutative with unity. For a ring $R$, let $U(R)$ denote its group of units, which is in particular abelian. Now, let $R$ be a ring. Consider the dual group of $U(R)$ namely $\...
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Calculated Hochschild (co)homology [duplicate]

I am looking for examples of algebras that it is known how to calculate its Hochschild homology and/or cohomology. I am restricting, for now, to bound quiver algebras. I have found that it has been ...
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Exercise 4.4.1 in Weibel's 'An Introduction to Homological Algebra'.

I can solve this question on the assumption that the $x_i$s are not zero-divisors since $\dim(R/(x)) = \dim(R)-1$ if $x$ is not a zero-divisor. My question is, how do I prove that they are not zero ...
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Homology of infinite dimensional real projective space given by Tor-functor

Let $Z$ be the trivial $Z[Z/2]$-module (i.e. $Z/2$ acts trivially). How can one show that for all $n\geq0$ $Tor_n^{Z[Z/2]}(Z,Z) = H_n (RP^{\infty},Z)$ without calculating Tor and the homology of the ...
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About proving a cokernel is not representable

I have been confused by an example from Dolgachev: Derived Categories, which aims to show that the cokernel of a morphism between two presheaves $h_A$ and $h_B$ may not be representable, even when $\...
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Question on a proof about spectral sequences from exact couples

I am going through Proposition 2.9 in User's guide in spectral sequences (2nd edition) by McCleary. This is a proof on defining spectral sequences using the language of exact couples. Towards the end ...
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Given filtered complex $K=\oplus_n K^n$ with gradation $n$ and filtration $\{K_p\}$ of $K$, $K_p\cap K^n$ is filtration of $K^n$?

Let $K$ be a filtered graded complex s.t. $K=\oplus_{n\in Z}K^n$ and $\{K_p\}$ filtration of $K$. Define $K_p^n=K_p\cap K^n$. $\textbf{Q:}$ Why $\{K_p^n\}_{p\in Z}$ forms filtration of $K^n$? The ...
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Projective resolution of short exact sequence: definition in Jacobson

Let $0\rightarrow M'\rightarrow M \rightarrow M''\rightarrow 0$ be a short-exact sequence of modules (over a ring $R$). By a projective resolution of this sequence, we mean (according to Jacobson ...
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about the cohomology of a tensor module.

Let $F$ be a field and let $C_*$ be a graded vector space over $F$ such that $C_i = 0$ for $i<0$ and $i>N$ for some integer $N$. Consider $R=F[t]$ as a graded ring (with the usual grading) and $...
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Associated graded object

I have a question, how do I prove the following proposition? A and B are filtered chain complex , $f: A \rightarrow B $ is a filtered chain map if indicated mapping on the associated graded object ...
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Homology of a chain complex with unknow boundary map

Question. Let $(C_\bullet,\partial)$ be a $\mathbb{Z}/2\mathbb{Z}$-chain complex generated by $a$, $b_1,b_2,b_3$, $c_1,c_2$ with gradings: $$|a|=2,|b_1|=|b_2|=|b_3|=1,|c_1|=|c_2|=0.$$ Which ...
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If $V$ is free, show that $f$ is surjective.

Let $R$ be a commutative ring with $1$. Let $V$ and $W$ be $R$-modules. a) Exhibit a canonical $R$-linear map $f: V^* \otimes V \to R $ b) If $V$ is free, show that $f$ is surjective. Now for the ...
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A kind of horseshoe lemma on the injective resolution

We know that in an appropriate setting, with some exact sequence $0 \to A \to B \to C \to 0$ of $R$-modules with suitable $R$ (commutative ring with unity), we have injective resolution of $A,C$. Then,...
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Is quasi-isomorpism of $A_{\infty}$ algebras invertible

Let $A_1$ and $A_2$ be two $A_{\infty}$-algebras over a field $k$. Suppose $f=\{f_i\} : A_1 \to A_2$ is $A_{\infty}$ quasi-isomorphism of $A_{\infty}$-algebras that is a map of $A_{\infty}$-algebras ...
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how to construct free resolution of the $ℤ_{4} $ module $ℤ_{2}$? [duplicate]

can somebody please explain how to construct free resolution of the $ℤ_{4} $ module $ℤ_{2}$??
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How to prove that for $K$ $\mathbb{Z}$-free & $Q$ projective, $K\otimes_{\mathbb{Z}}Q$ is a projective?

Let $K$ be $\mathbb{Z}$-free and $Q$ a projective $\mathbb{Z}G$ module then $K\otimes_{\mathbb{Z}}Q$ is a projective $\mathbb{Z}G$-module. I believe that this follows from the adjoint isomorphism ...
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Is the kernel of a map from a $\mathbb{Z}$-free module to an abelian group $\mathbb{Z}$-free? [duplicate]

Is the kernel of a map from a $\mathbb{Z}$-free module to an abelian group $\mathbb{Z}$-free? It seems like the answer should be yes as the kernel is contained inside the free $\mathbb{Z}$-module but ...
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1answer
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Why is this a filtered category? Localization of rings.

I was reading this post I am trying to show that $S^{-1}R=\operatorname{colim}F(s)$, where $S$ is a multiplicative closed set in a commutative ring $R$ and $F$ is a functor from a filtered category ...
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Homology groups of the set with Mayer-Veitoris

i need some help please my professor asked for homology groups of $S^2\cup d$ which $d={\{ (0,0,t) \mid -1 \leq t \leq 1}\} $ . We just find homology groups with some points and Mayor-...
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Existance of certain spectral sequences implies the existence of a long exact sequence of derived functors associated to a short exact sequence

I saw this question asked in some form somewhere, but couldn't quite prove the statement that I wanted to, so I am hoping that someone can help. Essentially I want to prove that a spectral sequence is ...
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56 views

What is $Ext^1(\overline{\mathbb{Q}}^\times, \overline{\mathbb{Q}})$ in abelian groups?

I want to find a way to describe all the extensions of $\overline{\mathbb{Q}}^\times$ by $\overline{\mathbb{Q}}$, i.e., all the abelian groups $A$ (and the maps $\alpha$ and $\beta$) that fit into the ...
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51 views

When can I have “short” projective resolution?

When we use universal coefficient theorem, we only need to compute Tor$(A,B)$. But suddenly, later on, like in more advanced homological algebra. We start to construct Tor$_n$, Ext$_n$... Can't we ...
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Equivalent form of the Whitehead problem

Let $M$ be an $R$-module and consider the following statement. $M$ is projective whenever the obvious group map $\tau: \text{Hom}_R(M, R) \to \text{Hom}_R(M, {R}/{I})$ is surjective for any ideal ...
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(Solved) Exact functors commute with homology

I want to show that for any exact functor $F\colon {}_R\mathrm{Mod}\rightarrow {}_S\mathrm{Mod}$ there is a natural isomorphism $$F\circ H_n \cong H_n \circ \mathbf{Ch}(F)$$ where $\mathbf{Ch}(F)$ ...
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79 views

$ \operatorname{Ext}_{k[x]}^n(k,k)$ for a field $k$ [closed]

Consider the polynomial ring $k[x]$ for a field $k$ and the $k[x]$-module $k$, letting $x$ act trivially on $k$. What is $ \operatorname{Ext}_{k[x]}^n(k,k)$ for $n\geq0$?