Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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Verify proof that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism

I have to prove that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism for all $p$ Where $H_p(X)$ is the $p$th homology group of $X$. To me this seems to come down to ...
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47 views

The cohomology of the fiber of the Hopf fibration using the Eilenberg-Moore Spectral Sequence

In John McCleary's book about Spectral Sequences he computes on p.248 the cohomology of the fiber of the Hopf fibration $S^3 \to S^7 \to S^4$ using the Eilenberg-Moore Spectral Sequence. He deduces ...
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Two term free resolution of an abelian group.

This is probably a very easy question but I think I am missing some background regarding free abelian groups to answer it for myself. In Hatcher's Algebraic Topology, the idea of a free resolution is ...
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33 views

How does Tor behave with respect to restriction of scalars?

Let $A$ be an augmented commutative $k$-algebra, where $k$ is a commutative ring. Let $\varepsilon:A\to k$ be the augmentation and $\eta:k\to A$ be the unit. Let $M$ be a right $A$-module. Is it true ...
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65 views

What is the relationship between the module structure of cohomology groups and chain complexes in Ext groups?

Let $X$ be a Noetherian scheme and $\mathcal{G} \in \text{Coh}(X)$. Hartshorne III.6.3(c) states $$ \text{Ext}^i(\mathcal{O}_X, \mathcal{G}) \cong H^i(X, \mathcal{G}) $$ as modules. What does the ...
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145 views

When does tensor product have a (exact) left adjoint?

Let $A$ be a commutative Noetherian ring, and let $F$ be a flat $A$-module. We can assume $A$ is local, so $F$ is projective. Question 1. When does $F\otimes_A-$ preserve injective objects? ...
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65 views

$\text{Ext}(H, \mathbb{Z})$ is isomorphic to the torsion subgroup of $H$ if $H$ is finitely generated

From Hatcher, page 196, before corollary 3.3. We are first given these three properties of the Ext functor: $\text{Ext}(H \oplus H', G) \cong \text{Ext}(H, G)\oplus\text{Ext}(H′, G)$. ...
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39 views

Category of Morphisms Between Modules

Let $A$ be a connected finite dimensional basic $k$-algebra with $k$ an algebraically closed field, and denote by $mod(A)$ the category of finite dimensional left $A$-modules. We define the category ...
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1answer
20 views

Isomorphisms Between some terms of a Long Exact Sequence

Suppose we have two long exact sequences of finite dimensional $k$-vector spaces: $$ 0 \to A_1 \to A_2 \to A_3 \to \cdots $$ $$ 0 \to B_1 \to B_2 \to B_3 \to \cdots $$ And assume that $A_6 ...
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1answer
17 views

Relation between long exact sequence and associated graded

I'm reading these notes by Hutchings on spectral sequences. In the first section, he motivates spectral sequences with the long exact sequence in relative homology. Given a chain complex $C_*$ and a ...
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46 views

Derived equivalences and complexes of injective modules

I have a question about derived equivalences: Let $k$ be a field and $A$ and $B$ two finite dimensional algebras over $k$. Let $F : D^-(A) \to D^-(B)$ be an equivalence of triangulated categories. ...
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22 views

$E^{\bullet} \rightarrow \text{cone}^{\bullet}(u)[-1]$ is a quasi isomorphism

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $E^\bullet$ be a cochain complex with objects in $\mathcal{A}.$ Let $i^n : E^n \hookrightarrow I^n$. Put $F^n = I^n \oplus ...
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31 views

Hochschild cohomology of a boolean ring

I can't find any papers studying the Hochschild cohomology ring $H^*(B,B)$, where $B$ is a boolean ring, so I was wondering if this is known.
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93 views

Higher Ext's vanish over a PID

Let $R$ be a PID and $M$, $N$ be $R$-modules. I am trying to show that $$\forall n\ge 2~: \operatorname{Ext}_{R}^{n}(M,N)=0.$$ For example $\forall n\ge 2~: \operatorname{Ext}_{\mathbb ...
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69 views

Why is the sequence exact?

Bredon states: For $A \subset X$, the sequence $$ 0 \rightarrow \Delta_{*}(A) \otimes G \rightarrow \Delta_{*}(X) \otimes G \rightarrow \Delta_{*}(X,A) \otimes G \rightarrow 0$$ is exact ...
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147 views

Is this “snake lemma” true in derived category?

Suppose there is a diagram of cochain complexes $$ \begin{array}{c} 0 & \to & X_1\phantom\alpha & \to & Y_1\phantom\beta & \to & Z_1\phantom\gamma & \to & 0 \\ & ...
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56 views

Injective version of Horseshoe lemma

Do we have an injective version of Horseshoe lemma? I am trying to construct injective resolution of a chain complex. If there exits an injective version of Horseshoe lemma, then I guess I know ...
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88 views

A homological perspective on the Hodge-theorem

Let $M$ be a smooth oriented manifold of dimension $n$. Let $(\mathcal{A} ^*(M),d)$ be the chain complex of differential forms on $M$. Endowing $M$ with an inner product gives us a hodge star operator ...
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60 views

Example of Category, covariant functors

This example was given by my professor: -The category $F(Q, \mathrm{Vec}_K)$ of covariant functors from $Q$ to $VecK$, with morphisms being natural transformations is a well-defined category. Here ...
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15 views

Setup of Salamander Lemma

On ncatlab, in explaining the basics of the Salamander lemma, they state the following. For $A_{\square}$, they write the denominator as $\operatorname{im}(\partial^{hor}_{in}) \oplus ...
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1answer
58 views

Cup product is the zero homomorphism?

Let $A$ and $B$ be subspaces of a space $X$. Let $X = A \cup B$, where $A$ and $B$ are contractible and $A \cap B \neq \emptyset$. How do I see that the cup product$$\tilde{H}^p(X) \otimes ...
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27 views

Projective Resolution of Simple Modules over Upper Triangular Matrices

Let $R$ be the ring of $n\times n$ upper triangular matrices over a field $F$. I've found that the simple modules $L_i$, $i=1,\ldots, n$ are all $1$-dimensional over $F$, where the module action is ...
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14 views

Cokernels in Co-Cartesian Diagram are isomorphic.

We assume the category is abelian. Let the below be a co-cartesian diagram. A co-cartesian diagram is just the dual of a cartesian diagram. \begin{array}{ccc} A& \xrightarrow{f} & B ...
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25 views

Another Stable Category

Let $A$ be a finite dimensional $k$-algebra. For $M,N \in mod(A)$, define: $$ \mathcal{P}_m(M,N) = \{ f\in Hom_A(M,N) | \exists \ P\in mod(A) \ with \ pd(P)=m \ and \ f \ factors \ through \ P \} ...
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30 views

Projective Dimension of a Direct Sum

Let $A$ be a finite dimesional $k$-algebra, and let $M$ ad $N$ be finite dimensional $A$-modules of finite projective dimension $p > 0$. Does the module $M \oplus N$ has projective dimension $p$? ...
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53 views

Computing $\text{Tor}$ for modules over a PID

This is essentially an exercise in Sze-Tsen Hu's "Introduction to Homological Algebra", page 143. Let $R$ be a PID and consider two $R$-modules $X$ and $Y$. Let $S$ denote the subset of the Cartesian ...
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1answer
69 views

Verifying that $T_0 = F, T_n = 0$ is a universal $\delta$-functor [closed]

I am trying to prove the following: An Introduction to Homological Algebra - C. A. Weibel (1994) Exercise 2.1.2) If $F~:A\to B$ is an exact functor, show that ${{T}_{0}}=F$ and ${{T}_{n}}=0$ for ...
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1answer
65 views

$X$ has a basepoint $* \in A \cap B$, commuting diagram involving relative cup product.

This is a followup to my question here. Let $A$ and $B$ be subspaces of a space $X$, and let $X$ have a basepoint $* \in A \cap B$. How I deduce the following commutative diagram?$$\require{AMScd} ...
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21 views

homology of a specific complex

Consider the complex $\lbrace 1, 2, 3, 4, 12, 23, 34, 41, 13, 123 \rbrace$. Visually, it's a square with a diagonal edge, with one hole and one face. In computing the homologies, I ended up getting ...
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24 views

Verification of Ext groups and projective resolution for S3 over F3

So I've been looking at Ext groups of irreducible representations of $S_3$ over $\mathbb{F_3}$. Specifically, I'm doing a project where I'll be looking at extensions themselves, so am really only ...
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2answers
36 views

Property of minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm stuck on the proof of corollary 2.5.4 : If $M$ is a module for an Artinian ring $\Lambda$ and $S$ is a simple ...
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1answer
51 views

$M$ is finitely generated as an $A$-module iff $M/A_{>0}M$ is finitely generated as an $A$-module?

Let $A$ be a nonnegative graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. How do I see that $M$ is finitely generated as an $A$-module if ...
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1answer
59 views

Chain map induces map of chain complexes and induced product, intuition behind isomorphism preserving products?

I know there is an isomorphism$$H^*(K(\pi, 1), A) \cong \text{Ext}_{\mathbb{Z}[\pi]}^*(\mathbb{Z}, A).$$When $A$ is a commutative ring, the $\text{Ext}$ groups have algebraically defined products, ...
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38 views

$X$ space satisfying hypotheses used to construct a universal cover, $A$ abelian, $C^*(X, A) \cong \text{Hom}_{\mathbb{Z}[\pi]}(C_*(\tilde{X}), A)$?

Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\tilde{X}$ and let $A$ be an abelian group. What is the most elementary way to see that$$C^*(X, A) \cong ...
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1answer
52 views

Why is the cohomological $\delta$-functor for $H^\ast(K(\pi,1),–)$ effaceable?

Let $\pi$ be an abelian group and $\mathbf Z[\pi]$ its group ring. One obtains a cohomological $\delta$-functor from the short exact sequence of $\mathbf Z[\pi]$-modules $$0\rightarrow A\rightarrow ...
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15 views

Homology of Eilenberg-MacLane $K(\pi,1)$ in terms of group homology and Tor [duplicate]

Let $\pi$ be a group and $K(\pi,1)$ be the associated Eilenberg-MacLane space (it has a contractible universal cover). I have proved the following isomorphism for the the chain complex with local ...
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relation between the homology of a $\mathbb{ Z}\oplus \mathbb{Z}$-filtered chain complex and the homology of the induced bi-graded complex

Let $\mathbb{ Z}\oplus \mathbb{Z}$ be equipped with the order defined by $$(a,b) \leq (c,d)\; \; iff \; \; a \leq c \; \; and \; \; b \leq c $$ Let $C$ be a $\mathbb{ Z}\oplus \mathbb{Z}$-filtered ...
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27 views

Calculating $\hom(\mathbb{Z}_p,\mathbb{Z})$.

I am working on some homework and I am trying to compute $\hom(\mathbb{Z}_p,\mathbb{Z})$. I found a guide that said it would be the set $\{n\in\mathbb{Z}:np=0\}$. But the only element in $\mathbb{Z}$ ...
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51 views

$Tor^*_{\mathcal O_{\mathbb P^3}}(\mathcal O_{L_1}, \mathcal O_{L_2})$ for two lines $L_i$ in the projective space

I need to calculate $Tor^*_{\mathcal O_{\mathbb P^3}}(\mathcal O_{L_1}, \mathcal O_{L_2})$, where $L_1$ and $L_2$ are lines on $\mathbb P^3$. If they are intersecting at a point, I believe that they ...
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30 views

B/A a ring extension. B is faithfully flat over A if and only if B/A is a flat A-module.

I came across this problem in Enochs' and Jenda's Relative Homological Algebra. Suppose $A$ is a subring of $B$. Show that $B$ is faithfully flat as an $A$-module if and only if $B/A$ is a flat ...
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1answer
30 views

If dual of quasi-isomorphism quasi-isomorphism?

Let $L,M$ be complex of finitely generated free $\mathbb{Z}$ modules, $f\colon L\to M$ is a quasi-isomorphism, $g\colon M^\vee\to L^\vee$ is a quasi-isomorphism. Is it still true for $\mathbb{Z}/n$ ...
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118 views

are two definitions of a *minimal* chain complex equivalent?

In Eisenbud, a complex $C_\ast$ over a local ring $(R,\mathfrak{m})$ is minimal when $C_\ast\!\otimes_R\!\frac{R}{\mathfrak{m}}$ has zero boundaries. My more intuitive definition, for a chain complex ...
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54 views

Natural versus Unatural splitting, the difference is…

I am working on an assignment currently and come across the Universa Coefficient Theorem for Homology/Cohomology. I am still getting somewhat used to the terminology and concepts in questions but I ...
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1answer
38 views

Projective resolution of $R/rR$

R ring, and $R/rR$ as a $R$-module, what is the best projective module projecting to $R/rR$ to get a projective resolution of $R/rR$? Is $R$ a free $R$-module? If it is, how should I view it as free? ...
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42 views

Difference between $\mathbb{Z} G$-module and $G$-module

I am studying Group Cohomology, but I am a little confused about $\mathbb{Z} G$-module and $G$-module. Some text uses the $G$-module for group cohomology, but I thought group cohomology of $G$ is ...
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1answer
52 views

Example of Minimal Projective Resolutions

I am reading "Elements of the representation theory of associative algebras"'s book of Skowronski, Simson and Assem. I want to compute the global dimension of the example 2.5 c), of chapter 3, page ...
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1answer
101 views

Isomorphism involving Eilenberg-Maclane space, Tors.

Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. Does there exist an isomorphism between $H_*(K(\pi, 1); ...
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1answer
21 views

Simplicial Sets Notation

I just started reading about simplicial sets from Manin and Gelfand's "Methods of Homological Algebra," and I have run into some notation with which I am not familiar in their discussion of n ...
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1answer
37 views

How to prove Hom$_{k}(M,N)^{G}\cong$Hom$_{kG}(M,N)$

$M,N$ are $kG$-module, $G$ group and Hom$_{k}(M,N)^{G}$ is the invariants of Hom$_{k}(M,N)$ which I believe is $\{ f\in \text{Hom}_k (M,N) | gf=g \forall g \in G \}$? And the isomorphism here is ...
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1answer
50 views

How to compute Tor$_1$ ($\mathbb{Q} /\mathbb{Z}$, $\mathbb{Q} /\mathbb{Z}$).

How to compute Tor$_1$ ($\mathbb{Q} /\mathbb{Z}$, $\mathbb{Q} /\mathbb{Z}$). I am having trouble finding a projective resolution of $\mathbb{Q} /\mathbb{Z}$.