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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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\!: ...
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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}\{ ...
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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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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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61 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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Are Projective resolutions of commutative rings always commutative?

Suppose $X$ is a commutative $R$-algebra with $R$ a commutative ring. Is it the case that there will always be a Projective resolution of $X$ as an $R$-module that is a graded commutative DGA? Tate ...
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122 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 ...
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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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229 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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364 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 ...
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385 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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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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200 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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242 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 ...
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354 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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138 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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680 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 ...
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149 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 ...
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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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407 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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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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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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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 ...
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384 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
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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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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 ...
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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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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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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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209 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 ...
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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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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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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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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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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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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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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 ...
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277 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 ...
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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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320 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 ...
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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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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 ...
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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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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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Help to compute Tor

Consider $\mathbb{Z}_{2}$ as a $\mathbb{Z}_{4}$ module. How to compute: $Tor_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?
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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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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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244 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 ...
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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 ...