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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9
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839 views

Hom of finitely generated modules over a noetherian ring

This is an exercise from Rotman, An Introduction to Homological Algebra, which I've been thinking now and then for a few days and I haven't solved it yet. I've decided to ask here because it is ...
7
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3answers
699 views

“The Yoneda embedding reflects exactness” is a direct consequence of Yoneda?

Let $A,B,C$ be objects of a category of modules over a ring. It is not hard to see that the Yoneda embedding "reflects exactness" (as Weibel puts it, on p. 28), i.e. if ...
4
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1answer
360 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 ...
4
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1answer
947 views

Relative homology of a retract

Show that if $A$ is a retract of $X$ then for all $n \ge 0$ $$H_n(X) \simeq H_n(A) \oplus H_n(X,A)$$ So we have a retraction $r:X \to A$, which is surjective. Consider the long exact sequence ...
11
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1answer
434 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
9
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1answer
282 views

Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq ...
6
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1answer
129 views

Is $R/N(R)$ a faithfully flat $R$-module?

I'm studying recently faithfully flat modules and I'd like to know the following: Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the subset of ...
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1answer
392 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
15
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1answer
1k views

Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what ...
15
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3answers
1k views

How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
16
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1answer
929 views

When is the derived category abelian?

I read in the book Methods of homological algebra of Gelfand and Manin that the derived category of an abelian category $A$ is never abelian. Now to me this seems to be wrong, because if $A=0$ then ...
16
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2answers
2k views

Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...
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2answers
610 views

Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
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2answers
1k views

Short Exact Sequences & Rank Nullity

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher) If $0 \stackrel{id}{\to} A ...
6
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0answers
165 views

Description of $\mathrm{Ext}^1(R/I,R/J)$

Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$? What do I mean by a nice description? For example ...
4
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4answers
85 views

Proving that P/PJ is a projective right module over R/J

If P is a projective right module over a ring R and J is a two sided ideal of R. Prove that P/PJ is a projective right module over R/J . My idea was trying to proof that " $M$ is an $R$-module ...
5
votes
1answer
183 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
2
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1answer
49 views

$i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism.

Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. ...
2
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1answer
87 views

The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
5
votes
2answers
285 views

Does finite projective dimension localize?

Let $R$ be a commutative (but not necessarily Noetherian) ring with unity. Let $M$ be an $R$-module. Suppose that, for all $\mathfrak p \in\text {Spec}(R),$ $\text{pd}_{R_{\mathfrak p}}M_{\mathfrak ...
3
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2answers
96 views

Hochschild cohomology of a formal quantization of an associative algebra

Let $A$ be a commutative associative $k$-algebra and let $A[[\hbar]]$ be the formal deformation of $A$. I would like to know if there is a relation between the Hochschild co-homologies ...
3
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1answer
283 views

About presentation of module

Let $R$ be a ring and $R[\mathbb{Z}]$ be the group ring obtained from ring $R$ and group $\mathbb{Z}=<s>$. Suppose that $M$ be a $R[\mathbb{Z}]$-module and it is isomorphic to $R^n$ as ...
2
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1answer
276 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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2answers
98 views

What is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$?

In the category of $\mathbb{Z}$-modules, what is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$? I was hoping to find a divisible group containing $\mathbb{Z}/n\mathbb{Z}$ such that it is ...
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1answer
171 views

Every chain complex is quasi-isomorphic to a $\mathcal J$-complex

I found this in "Algebra & Topology" by Schapira, but I'm not able to prove it: Suppose $\mathcal J$ is a cogenerating family in an abelian category $\mathbf A$. Then for any positive complex ...
30
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2answers
1k views

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 ...
28
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3answers
616 views

How did we know to invent homological algebra?

Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought. (Original question ...
19
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4answers
995 views

Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module

Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module? The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
17
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8answers
1k views

Reference for spectral sequences

What are good expositions of spectral sequences, which include a thorough introduction to the topic as well as the most important examples of applications - maybe with an emphasis an topological ...
12
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1answer
507 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 ...
18
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2answers
2k views

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
29
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1answer
676 views

Applying Freyd-Mitchell's embedding theorem on large categories

One commonly reads that the Freyd-Mitchell's embedding theorem allows proof by diagram chasing in any abelian category. This is not immediately clear, since only small abelian categories can be ...
16
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2answers
536 views

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: ...
12
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1answer
225 views

Can it happen that the image of a functor is not a category?

On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a ...
11
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2answers
467 views

Computation of $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$

Can you please give some examples of computation of the derived functors $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$ for some simple cases, say $R=\mathbb{Z}$ or $R=\mathbb{Z}[G]$ for some ...
7
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1answer
262 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 ...
8
votes
1answer
143 views

Generators of a certain ideal

Crossposted on MathOverflow. The MathOverflow version of the question has been rewritten. For the sake of completeness, I pasted it here in a condensed form. I also deleted the old version. Let $K$ ...
7
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0answers
298 views

Condition for a ring on projective and free modules problem

Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
6
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1answer
269 views

The Zig Zag Lemma in Cohomology

I´m reading the Zig Zag lemma in Cohomology and i want to prove the exactness of cohomology sequence at $ H^k(A)$ and $H^k(B)$ : A short exact sequence of cochain complexes $ 0 \to A \ ...
6
votes
1answer
228 views

Does maximal Cohen-Macaulay modules localize?

Let $A$ be a Noetherian local ring and $M$ a finitely generated $A$-module such that $$\operatorname{depth}M= \dim M=\dim A.$$ I can prove that $$\operatorname{depth}M_{\mathfrak{p}}= \dim ...
5
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2answers
2k views

Minimal free resolution

I'm studying on the book "Cohen-Macaulay rings" of Bruns-Herzog (Here's a link and an image of the page in question for those unable to use Google Books.) At page 17 it talks about minimal free ...
4
votes
1answer
352 views

Additive functor over a short split exact sequence.

$0\to A'\stackrel{f}{\longrightarrow} A \stackrel{g}{\longrightarrow} A''\to 0$ is a short split exact sequence, where $A'$, $A$, $A''$ are $R$-modules, and $T$ is an additive functor from ...
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5answers
328 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
9
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1answer
128 views

How *should* we have known to invent homological algebra?

Previously I asked How did we know to invent homological algebra?, because I was under the misapprehension that concrete examples of long exact sequences had been a major motivation for developing ...
9
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1answer
431 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} ...
8
votes
2answers
211 views

What is the kernel of the summation map from the direct sum to the sum?

Let $R$ be a ring, and let $I_1,\ldots,I_n$ be ideals in $R$ (or submodules of some $R$-module). Consider the sequence $$ \bigoplus_{1\leq j < k\leq n} I_j\cap ...
7
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1answer
408 views

The construction of the localization of a category

I was reading the construction of the localization of a category in the book "Methods of homological algebra" of Manin and Gelfand. Let me remind you the definition of the localization of a category: ...
7
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2answers
274 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 ...
6
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2answers
245 views

Submodules of a free module over a commutative ring

Let $R$ be a commutative unital ring, $I$ a set, and $R^{(I)}$ the free module on $I$. Can there be a submodule $R^{(J)}\cong M\leq R^{(I)}$ with $|J|\!>\!|I|$? Can $R^{(I)}$ be generated (as a ...
6
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2answers
482 views

$\mathbb{Z}/2\mathbb{Z}$ coefficients in homology

I don't see the point in using homology and cohomology with coefficients in the field $\mathbb{Z}/2\mathbb{Z}$. Can you provide some examples for why this is useful?