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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A particular isomorphism between Hom and first Ext.

Let $R$ commutative ring and $I$ an ideal of $R$. How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ? This question is an exercise of the course ...
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192 views

Lifting isomorphisms between derived categories

Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. I will denote by $D(A)$ and $D(B)$ the derived categories of unbounded complexes over $A$ and $B$. Suppose ...
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431 views

Equivalent definition of exactness of functor?

I'll use the following definition: (Def) A functor $F$ is exact if and only if it maps short exact sequences to short exact sequences. Now I'd like to prove the following (not entirely sure it's ...
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362 views

When $\mathbb{Z}/pq\mathbb{Z}$ is not semisimple?

Prove that for any primes $p$, $q$, $p\neq q$, the ring $\mathbb{Z}_{pq}$ (the ring of integers modulo pq) is semisimple, and for $p=q$ the same ring is not semisimple. I was told that the easiest ...
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174 views

How are injective model structures cofibrantly generated?

I have a question about the injective model structure on functor categories. As background : If $\mathcal{M}$ is a combinatorial model category and $\mathcal{C}$ is a small category, then there are ...
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307 views

When is the pushout of a monic also monic?

Let $$\matrix{ A& \mathop{\longrightarrow}\limits^f &B\\ \Big\downarrow & & \Big\downarrow\\ C&\mathop{\longrightarrow}\limits_g &D }$$ Be a pushout diagram in a category ...
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78 views

Does the tensor product of two complexes with acyclic augmentation have acyclic augmentation?

More specifically, let $(K,\partial^K,\varepsilon^K)$ and $(L,\partial^L,\varepsilon^L)$ be augmented aclycic complexes of free abelian groups with augmentation module $\mathbb{Z}$; that is, ...
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185 views

Flabby sheaves and comparison of topologies

Let $A^p$ be a group of sheaves on a topological space $X$, let $F$ be the global sections functor $F(A^p) = A^p(X)$. I have to compute the cohomology of the complex $0\rightarrow A^1(X) \rightarrow ...
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145 views

The relation between betti numbers and Tor functor?

Let $M$ be finitely generated Module over the Polynomial ring $R=k[x_{1},..,x_{n}]$, then there is a free resolution of M of the form $$0\rightarrow F_{n}\rightarrow ...\rightarrow F_{0}\rightarrow ...
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451 views

Does the splitting lemma hold without the axiom of choice?

In part of the proof of the splitting lemma (a left-split short exact sequence of abelian groups is right-split) it seems necessary to invoke the axiom of choice. That is, if $0\to A\overset{f}{\to} ...
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222 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...
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155 views

Computing Betti numbers using Macaulay program ??

Let $k$ be a field and $R=k[x,y,z]$, let $M=R/\langle x^2,xy,yz^2,y^4\rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the betti numbers of this resolution?
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225 views

cohomology of a finite cyclic group

I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title. Let $G=\langle\sigma\rangle$ where ...
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1answer
294 views

Bruns-Herzog problem 3.1.25

This is problem 3.1.25 (page 97) in Cohen-Macaulay Rings by Bruns and Herzog. The direction I am interested in is the following. Let $R$ be a Gorenstein local ring and $M$ a finite $R$-module. If ...
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tensor product of two chain homotopic maps are again chain homotopic?

Let $C$,$C'$, $D$, $D'$ be chain complexes, $f$, $f'\colon C\to C'$ and $g$, $g'\colon D \to D'$ two pairs of homotopic chain maps.How to show $f \otimes g$ and $f' \otimes g' \colon C\otimes D\to ...
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293 views

Computing an example of Ext

Let $k$ be a field. I want to compute $\operatorname{Ext}_{k[x] / \langle x^2 \rangle}(k,k)$. However I have no idea how to do this? I cannot even think how to construct a projective resolution ...
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Wedge product of Hochschild Cohomology classes in characteristic 2

Let $A$ be a smooth commutative $k$-algebra, for $k$ a commutative ring. By the Hochschild-Kostant-Rosenberg theorem, we have that $HH^*_k(A)\cong \Lambda^* \mathrm{Der}_k(A,A)$, where ...
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285 views

Universal coefficient theorem of relative homology

In Hatcher, Corollary 3A.4 stated a universal coefficient theorem for relative homology, i.e. the following short exact sequence splits: $0 \rightarrow H_n(X,A) \otimes_\mathbb{Z} G \rightarrow ...
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375 views

Is the image of a tensor product equal to the tensor product of the images?

Let $S$ be a commutative ring with unity, and let $A,B,A',B'$ be $S$-modules. If $\phi:A\rightarrow A'$ and $\psi:B\rightarrow B'$ are $S$-module homomorphisms, is it true that ...
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198 views

(Non-)Formality of A-infinity algebra implies derived (non-)equivalence?

Take an unital differential graded (dg) $k$-algebra $A$, we can regard it as $A_\infty$-algebra with $m_1$ as differential and $m_2$ as algebra multiplication, and $m_n=0$ or all $n\geq 0$. Take a dg ...
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105 views

Other differentials for group cohomology other than the standard one.

In group cohomology, one defines $H^i(G;A)$ for $G$ a group and $A$ a $G$-module (an abelian group with a $G$-action) as the $i$-th right derived functor of the functor $$(-)^G: G-mod \rightarrow Ab, ...
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286 views

Koszul Complex Homology

I'm attempting to understand Eisenbud's proof that: If $x_1,x_2,\ldots,x_i$ is an $M$-sequence, then $H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M$. Here ...
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608 views

Confused about Hypercohomology terminology and meaning

check this: Given a sheaf complex $F^\bullet$, let's say I want to compute the hypercohomology of this complex, if we consider the bicomplex of sheaves $C^\bullet(F^\bullet) = (C^p(F^q))\quad ...
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1answer
130 views

equivalence of definition of the first cohomology group

I've found different definitions of the same cohomology group and I would like to prove that they are equivalent. For $G$ a group and $A$ a $G$-module, Weibel defines in "An introduction to ...
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748 views

History behind Exact Sequences.

I am very much interested in listening to the history behind the exact sequence. We know that the exact sequence is sequence of objects with morphisms such that image of one morphism equals to the ...
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118 views

How to prove $\mathrm{Im}(\mathrm{Ext}_R^1(g,A'))=\mathrm{Ker}(\mathrm{Ext}_R^1(f,A'))$

I'm reading MacLane's "Homology" and got stuck at the proof of the following fact. Theorem. Let $E:0\xrightarrow{}A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{}0$ be a short exact sequence of left ...
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666 views

Motivation behind the ingredients of First Cohomology group $H^1$

I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following : It is concerned with the formal definitions of crossed and principal ...
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What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
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Tensor-free proof that for finite modules over reduced Noetherian rings, locally free = projective

Is there an elegant tensor-free proof of the fact that over a reduced Noetherian ring $A$, every finitely-generated $A$-module which is locally free, is projective? EDIT: I would be content with the ...
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Constructing bijection between $\mathrm{Ext}_{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z},A)$ and $A/mA$

I'm reading Mac Lane's Homology and get stuck at the proof of proposition $1.1$ chapter $3$. This proposition states that there exist bijection $$ ...
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Is there an easy formula for $\operatorname{Tor}_i^{\mathbb{Z}/(p^n)}(\mathbb{Z}/(p),\mathbb{Z}/(p))$?

I've been following some old slides from the Spring 2010 Algebra Seminar at UWaterloo. I now know that $$ \operatorname{Ext}_{\mathbb{Z}/(p^n)}^i(\mathbb{Z}/(p),\mathbb{Z}/(p))\cong\mathbb{Z}/(p) $$ ...
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Is there a general formula for $\operatorname{Ext}_{\mathbb{Z}/(p^n)}^i(\mathbb{Z}/(p),\mathbb{Z}/(p))$?

The other day I was reading through some slides I found online about Ext and Tor. One of the examples gave a cursory derivation for a general formula $$ ...
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1answer
677 views

What are $E_\infty$-rings?

I've been working with DG-algebras for the last year, and was able to obtain using them some nice commutative homological algebra results. However, I keep hearing about a (more general???) concept of ...
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171 views

Why can't the projective dimension of $\bigwedge V$ be finite?

While studying the Koszul complex, I can't properly recall a certain fact. I remember if $\bigwedge V$ is the exterior algebra of a finite dimensional vector space, then $\bigwedge V$ has infinite ...
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318 views

Different definitions of projective objects

There are various characterizations for an $R$-module to be projective. Two of them can be generalized to any category: i) $P$ is an object such that given morphisms $\alpha: A \rightarrow B$ and ...
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Is the projective resolution of this artinian module finite?

Suppose $\Delta(n,k)$ is the algebra of upper triangular $n$ by $n$ matrices over a field $k$. Furthermore, let $M$ is an artinian module over $\Delta(n,k)$, and let $$ \cdots\to ...
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180 views

Showing there is an exact sequence

Consider the following commutative diagram with exact rows (of $R$-modules and $R$-linear maps): $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} ...
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1answer
132 views

Uniqueness of projective covers

I want to show that if projective covers exist then they are unique up to isomorphism. More precisely let $f: P \rightarrow M$ and $g: Q \rightarrow M$ be projective covers of an $R$-module $M$. ...
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193 views

About Gorenstein ring

Is it true that in a (non-local) Gorenstein ring, every maximal ideal has the same height? It seems a little strange, but I don't see any reason why it shoudn't.
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Additive functors preserve split exact sequences

How can I prove that additive functors preserve split exact sequences?
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Alternative construction of Direct Limit

The construction of the direct limit that I learned from Atiyah Macdonald is the following: Suppose we have a directed system $(M_i,\mu_{ij})$ of $A$ - modules and $A$ - module homomorphims over a ...
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Injective module and Noetherian ring

In the book Abstract Algebra of J.Antoine Grillet there is a theorem as follow: A ring R is left Noetherian if and only if every direct sum of injective left R-modules is injective The ...
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Can we think of a chain homotopy as a homotopy?

I'm taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I'm looking for a bit of intuition regarding chain homotopies. The definitions I'm using are: ...
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Equivalent definition of injective module

When I studied injective module there is a theorem which say that the two following statement are equivalent: Let $R$ be a ring, $I$ is a left ideal of $R$, $J$ is a left $R$-module. For every ...
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Homology/Cohomology of Free Product

I recently completed an exercise showing that $$ H_1(G*H,A) \cong H_1(G,A) \oplus H_1(H,A) $$ for $A$ a trivial $G*H$-module, and also proved a similar statement for cohomology. This is exercise ...
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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 $m=0$) ...
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139 views

Cohomology of the trivial action of $\mathbb{Z}_p$ on $\mathbb{Z}$

I'm wondering if the next exercise in 'An introduction to homological algebra' by Weibel is correct: Let $G$ be the profinite group $\widehat{\mathbb{Z}}_p$. Show that $$H^i(G;\mathbb{Z}) = ...
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273 views

Homology calculation

When playing around with a homological calculation I came across a short exact sequence of the form $$ 0 \to \mathbb Z^{2g} \to H \to \mathbb Z / 2 \to 0 $$ My background in algebra is not very ...
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How to view set of equivalence classes of extensions of M by N as an A-module

I know that for a commutative ring $A$ and $A$-modules $M$ and $N$, the set $E_A(M, N)$ of extensions of $M$ by $N$ can be equipped with the Baer sum which gives it an additive group structure. ...
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56 views

$C^n(G,A)$ with $G$ a profinite group and $A$ a discrete $G$-module as direct limit

Let $C^n(G,A)$ be the set of continuous functions $G^n \rightarrow A$ with $G$ a profinite group and $A$ a discrete $G$-module (these are the functions that are locally constant). I want to prove that ...