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.

3,134 questions
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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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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)... 2answers 1k 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 ... 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. ... 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?... 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 ... 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 ... 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 ... 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 ... 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 ... 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? (... 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 ... 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"... 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 ... 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 ... 6answers 2k 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 ... 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 ... 0answers 360 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 (Vakil'... 1answer 863 views what is a faithfully exact functor? Could any of you give me a definition of faithfully exact functor, please? 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 ... 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 ... 2answers 2k 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? 0answers 1k 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,-) ... 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 ... 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_{\... 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 ... 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 ... 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 ... 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) ... 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, ... 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 ... 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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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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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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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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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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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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