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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71 views

Does Ext commute with surjective scalar extensions?

Let $A$ be a ring, $I\subset A$ an ideal, $M$, $N$ $A$-modules such that $IM=0$ and $IN=0$. Then the modules extend to $A/I$-modules, and we have ...
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1answer
179 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 ...
2
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1answer
190 views

Ext, Extensions and homomorphisms between them

Say that we have $R$-modules (let's assume over a commutative ring $R$). Consider extensions of a module $A$ by a module $C$, so that we have short exact sequences: $0 \rightarrow B \rightarrow^i E ...
6
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1answer
977 views

Hom and direct sums

Let $R$ be a ring (not necessarily commutative). Let $A$ be a left $R$-module. When does the functor $\text{Hom}(A,-)$ preserve direct sums - in the category of left $R$-modules? For example, this ...
3
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1answer
150 views

Isomorphism of First Ext groups

Let $A$ be a commutative ring with $1$ and $\mathcal m$ be a maximal ideal. One knows that then there is a canonical isomorphism $A_{\mathcal m}/{{\mathcal m}A_{\mathcal m}} \simeq A/{\mathcal m}$. ...
9
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1answer
285 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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3answers
1k views

Injective modules: examples and problem

In almost all textbooks on Homological Algebra, when they talk about injective modules, they do not give many examples, usually are $\mathbb{Q}/\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$. Is there a ...
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4answers
2k views

Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' ...
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2answers
310 views

Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?
3
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1answer
312 views

Homology and semidirect products

If $G=N\rtimes H$ what is the relation between the second integral homology groups (Schur multipliers) of $G,N$ and $H$.
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1answer
122 views

A simple second homology question

What is $H_2(\mathbb{Q},\mathbb{Z})$ where the action is trivial. Thanks in advance
6
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1answer
245 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 ...
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1answer
201 views

uniqueness of a direct limit

DEFINITIONS: $(I,\leq)$ is a preordered set when $I$ is a set and $\leq$ is a reflexive and transitive binary relation on $I$, i.e. $\forall i\!\in\!I\!: i\!\leq\!i$ and $\forall i,j,k\!\in\!I\!: ...
1
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1answer
279 views

In the existence of a short exact sequence, the projective dimension of $B$ is less than the larger of projective dimensions of $A$ and $C$

If there is an exact sequence of $R$-modules $0 \rightarrow A \stackrel{\alpha}{\longrightarrow} B \stackrel{\beta}{\longrightarrow} C \rightarrow 0$, then $\mathrm{pd}(B) \leq \mathrm{max}\{ ...
5
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0answers
183 views

Non-trivial conditions for $\mathrm{Ext}^2(A,B)=0$?

Edit: Since I had some trouble making my previous question precise without diving into details about the origin of the homological objects I'm interested in, let me ask a more open-ended question: ...
3
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1answer
144 views

Proving projective equivalence of Auslander Transpose

Let $$P_1\overset{\partial}{\rightarrow} P_0\rightarrow M\rightarrow 0$$ be an exact sequence of $A$-modules with $P_0$, $P_1$ finitely generated and projective. The transpose $T(M)$ is defined as ...
3
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1answer
64 views

About depth$(I,M)$ when $IM=M$

Suppose $A$ is a Noetherian ring, $I\subset A$ an ideal, and $M$ a finitely generated $A$-module. If $IM\neq M$, then the length of a maximal $M$-sequence inside $I$ is fixed by the number ...
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1answer
125 views

Question about derived functors

Let $F,G, H: Mod \to Mod$ be three left exact functors such that $R^iF(-)\cong R^iG(-)$ for all $i\in\mathbb{N}$. We consider the exact sequence $$\cdots\to R^iF(M)\to R^iG(M)\to R^iH(M)\to ...
9
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2answers
563 views

Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
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1answer
261 views

translation from French

A passage from Bourbaki's Algebre X reads, "... l'homothetie de rapport $a_1$ dans $\oplus_{i\geq0}I^iM/I^{i+1}M$ est injective,..." Here $M$ is an $A$-module and $I=(a_1,\ldots,a_n)\subset A$. ...
3
votes
1answer
409 views

how can we compute the homology of these groups without using topology?

I'd like to know the homology of a free group and a free abelian group of rank 2. I know that they could be computed topologically, but I'm searching a proof purely algebraic, could you help me ...
5
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1answer
420 views

Construction of the morphism from the zig-zag lemma

UPD: I'm not sure why i'm not getting any comments or votes, so I'm expanding a little bit below to make it easier to understand my question and make it more self-contained. For reference I'm using ...
2
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1answer
134 views

Acyclic resolutions

Hallo, I have to worry you one more time with these acyclicity problems, but as I am currently working on derived functors in a.g., I really need to understand derived functors in a very general ...
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1answer
208 views

Acyclic Objects and cohomologically finite functors

let's start with a left exact functor $F: A\longrightarrow B$ of abelian categories, where the derived functor $RF: D^{+}(A)\longrightarrow D^{+}(B)$ exists. Furthermore the class of F-acyclic objects ...
0
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1answer
253 views

derived functors and acyclics

I'm not sure how I can show the following: If F is a left exact functor from an abelian category A to an abelian category B, whose derived functor RF in the sense of derived categories exists, then ...
4
votes
1answer
417 views

Calculating Hom(A,B)

I have been studying modules and homological algebra as of late but somehow I have missed how to calculate Hom(A,B) for abelian groups, modules and Hom(A,_)/Hom(_,B) for exact sequences. I have no ...
3
votes
1answer
153 views

Does the minimal injective resolution have the smallest length?

Let $A$ be a Noetherian (not necessarily local) ring and $M$ a finitely generated $A$-moduel. Is the length of the minimal injective resolution of $M$ always equal to the injective dimension of $M$? ...
5
votes
2answers
759 views

Arbitrary products of quasi-coherent sheaves?

I have a short question: Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to ...
5
votes
1answer
163 views

Grothendieck spectral sequence

given functors $F,G$, left exact, with as good properties as you want we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}(F\circ G)$. I am looking for an analogous for a "mixed ...
2
votes
1answer
237 views

Confused about Weibel proof

In Weibel (Introduction to Homological Algebra)'s proof that left derived functors form a homological $\delta$-functor (Thm. 4.2.6), he does a lot of work that seems unnecessary to me. The relevant ...
4
votes
1answer
475 views

What are the relations between the Koszul complex and the minimal free resolution?

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and $F.$ the Koszul complex of a minimal system of generators of $\mathfrak{m}$. Let $G.$ be the minimal free resolution of $k$. In which cases they ...
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6answers
818 views

Why are projective objects important?

I belive we study them because in important categories they are close to free objects and even a retract of a free object in some algebraic instances (for example, direct summands in Mod_R, and ...
6
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3answers
3k views

Spectral Sequence proof of the five lemma

The five lemma is an extremely useful result in algebraic topology and homological algebra (and maybe elsewhere). The proof is not hard - it is essentially a diagram chase. Exercise 1.1 in McCleary's ...
2
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0answers
214 views

kernel of cokernel is cokernel of kernel [duplicate]

Possible Duplicate: Equivalent conditions for a preabelian category to be abelian Let $\mathcal{C}$ be an abelian category, and consider an arrow $f:A\rightarrow B$. In a number of sources ...
2
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1answer
430 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
10
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1answer
396 views

Derived functor of a derived functor

Given $F$ is a covariant additive functor from left R-module to a left S-module, show that $\mathscr{L}_n(\mathscr{L_m}(F))=0$ if $m>0$ (where $\mathscr{L}$ refers to the derived functor). I am ...
16
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1answer
2k 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 ...
6
votes
2answers
572 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?
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0answers
714 views

Does finite projective resolution imply finite free resolution?

Suppose that $R$ is a ring (commutative, if it simplifies things), and that $M$ is a (left) $R$-module. Then $M$ has a projective resolution of length $n$ if and only if $\operatorname{Ext}_R^m(M,-)$ ...
5
votes
0answers
332 views

The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
4
votes
1answer
213 views

Koszul algebra of a ring

I'm studying on Cohen-Macaulay Rings of Bruns-Herzog. Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and $H_{\bullet}(R)$ its Koszul algebra. I found on the book (page 75) that "since ...
3
votes
2answers
219 views

Auslander-Buchsbaum and Ferrand-Vasconcelos

I'm studying on "Cohen-Macaulay rings" of Bruns-Herzog, here a link: http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false At page 65 there is ...
6
votes
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 ...
2
votes
1answer
227 views

Help on a proof of a Theorem of Rees

I'm studying on this book http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false on page 10 there is a Rees Theorem. I'd like to know why the ...
4
votes
3answers
468 views

Calculation of Ext

Let $A$ be an abelian group. I know that $Ext_\mathbb{Z}^1(\mathbb{Z}/p,A)=A/pA$. Are there any similar formula about $Ext_\mathbb{Z}^1(A,\mathbb{Z}/p)$? I know that $Ext_R^n(A,B)\neq Ext_R^n(B,A)$ ...
1
vote
2answers
90 views

Short exact sequence of modules generated by a set

Let $0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0$ be a short exact sequence of $R$-modules. Suppose that $A = \langle X \rangle$ and $C = \langle Y \rangle$ For each $y \in C$, ...
8
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1answer
803 views

Direct sum commuting with homology functor

I'm trying to understand a fact about commutation between homology functors and direct sums. In particular, let $G$ be a group of type $FP$ (i.e. there exists a projective resolution of finite length ...
2
votes
1answer
205 views

application of the five lemma

suppose we are given a short exact sequence of $\mathbb{Z}G$-modules $$0\to K\to F\to A\to 0$$ where $F$ is free. and we form a diagram with that first row and with a second row $0\to L\to M\to N\to ...
3
votes
1answer
322 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 ...
9
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1answer
643 views

Signs in the tensor product and internal hom of chain complexes

Let $R$ be a commutative ring and $\text{Ch}(R)$ the category of chain complexes of $R$-modules. $\text{Ch}(R)$ is first of all an abelian category, but it can also be equipped with the structure of a ...