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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135 views

A simple second homology question

What is $H_2(\mathbb{Q},\mathbb{Z})$ where the action is trivial. Thanks in advance
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687 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 M_{\...
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1answer
581 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\!: i\!\...
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1answer
719 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}\{ \mathrm{...
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0answers
217 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: ...
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1answer
210 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 $\...
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1answer
90 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 $$\mbox{...
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1answer
155 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 R^{i+1}F(M)...
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2answers
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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
304 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 $I=(a_1,\ldots,a_n)\subset A$. ...
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1answer
868 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 please?...
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1answer
595 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 ...
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1answer
260 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
290 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 ...
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1answer
348 views

Is $\mathbf{R}F(Z^\bullet)$ equal to $\mathbf{K}F(Z^\bullet)$ when $Z^\bullet$ consists of $F$-acyclic objects?

I'm not sure how I can show the following: If $F \colon \mathcal{A} \to \mathcal{B}$ is a left exact functor from an abelian category $\mathcal{A}$ to an abelian category $\mathcal{B}$, whose ...
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1answer
1k 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 ...
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1answer
234 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 $M$? (...
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2answers
1k 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
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1answer
271 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 version"...
4
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1answer
326 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 ...
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1answer
1k 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
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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 ...
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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 ...
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0answers
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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 (Vakil'...
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1answer
863 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
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1answer
605 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 ...
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1answer
8k 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 I'm ...
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2answers
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$\mathbb{Z}/2\mathbb{Z}$ 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
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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 $\operatorname{Ext}_R^m(M,-)$ ...
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0answers
530 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 ...
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1answer
317 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 $H_{\...
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2answers
287 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 ...
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2answers
5k 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 ...
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1answer
399 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 ...
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3answers
588 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 $Ext_R^n(A,B)\neq Ext_R^n(B,A)$ ...
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2answers
99 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 $y \in C$, ...
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1answer
2k 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 $...
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1answer
374 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 $0\to L\to M\to N\to 0$...
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1answer
546 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 $R$-...
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1answer
1k 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 ...
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6answers
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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 ...
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0answers
588 views

Are homology and cohomology really dual to each other?

I don't remember if I've already seen this question even here or in MO or in my mind. This is partly related to questions arose about differences between homology and cohomology; I'm wondering if some ...
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1answer
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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 ...
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4answers
2k views

Short exact sequence of exact chain complexes

If $0 \rightarrow A_{\bullet} \rightarrow B_{\bullet} \rightarrow C_{\bullet} \rightarrow 0$ is a short exact sequence of chain complexes (of R-modules), then, whenever two of the three complexes $A_{\...
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2answers
746 views

Tensor product of abelian group and a free abelian group

I am trying to show that if $F,H$ are abelian groups with $F$ free abelian, and if $a \in F$ and $h \in H$ are non-zero, then $a \otimes h \ne 0$ in $F \otimes H$. This is specifically in a section ...
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2answers
345 views

Free module implies trivial Tor

Let $A$ be a commutative ring. If $M$ or $N$ aree free $A$-module then $Tor_{n}^{A}(M,N)=0$. Since $Tor_{n}^{A}(M,N)=Tor_{n}^{A}(N,M)$ it suffices to deal with the case say when $N$ is flat right? ...
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1answer
386 views

Help to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$? [closed]

Consider $\mathbb{Z}_{2}$ as a $\mathbb{Z}_{4}$-module. How to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?
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0answers
591 views

Chain map inducing isomorphism in homology

If $X$ is a CW complex, show that there is a chain map $W_*(X) \to S_*(X)$ inducing isomorphisms in homology. Here $W_p(X) = H_p(X^p,X^{p-1})$ Let $E$ be the CW decomposition of $X$ and let $M$ ...
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1answer
594 views

Exact sequence and torsion

I've come across another exact sequence, where (I guess) I need to deduce the result using some properties of torsion. I am calculating the homology of the Klein bottle using attaching maps. I start ...
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1answer
511 views

Torsion and torsion-free abelian groups

I am missing some knowledge about torsion and torsion-free groups that I need to understand an example (let's say I have not seen these expression before). We have the exact sequence of abelian groups:...