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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12
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1answer
519 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 ...
10
votes
2answers
183 views

What does $Tor^{R}_n(M,N)$ represent?

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes ...
5
votes
1answer
179 views

Mackey functor question regarding Greenlees and May

Let $G$ be a finite group. In their paper Some Remarks On the Structure of Mackey functors , Greenlees and May define a functor: $R: GMod \rightarrow M[G]$ where $GMod$ is the category of finite ...
4
votes
0answers
158 views

Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions

I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$ $0 ...
10
votes
3answers
1k views

Cokernels - how to explain or get a good intuition of what they are or might be

When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over ...
9
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1answer
505 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: ...
20
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1answer
1k 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 ...
4
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0answers
191 views

Homology and Homotopy group

$E, F, B$ are topological space, $B$ path connected. If we have given a long exact sequence.. $$\cdots\to \pi_1(F)\to \pi_1(E)\to \pi_1(B)\to\cdots$$ what will the relationship of $H_1(F,\mathbb R)$, ...
1
vote
1answer
179 views

Is there a quasi-isomorphism between a complex of sheaves and its Godement resolutions?

I have a doubt, I read somewhere that the Godement resolution of a sheaf $\mathcal{F}$ is a quasi-isomorphism $\mathcal{F} \rightarrow C^\bullet(\mathcal{F})$. Just right off the bat when I read ...
5
votes
2answers
180 views

Localization and Extension of modules

Let $R$ be a commutative ring and $S$ be an $R$-algebra. Assume that $S$ is finitely generated as an $R$-module. Let $M$ and $N$ be finitely generated $S$-modules and $\mathfrak{m}$ a maximal ideal ...
3
votes
1answer
395 views

Computing the homology of a torus by relative homology of a cylinder

I was trying to compute the homology of a torus with the long exact sequence for relative homology formed quotient, inclusion, and boundary $ \dots\to\tilde{H}_n(A)\overset{i_*}{\to} ...
1
vote
1answer
167 views

Relation between group zero-cohomology and the dual of group zero-homology

Let $\Gamma$ be a group and $A$ be an abelian group and let's take group zero-homology and zero-cohomology, $H_0(\Gamma,A)$, $H^0(\Gamma,A)$. Is there any relation between $H^0(\Gamma,A)$ and ...
3
votes
0answers
189 views

Adapted classes of objects and left (right) exact functors

I had a question about adapted classes of objects, I was confused by the definition and how it relates to left exact functors. Let $\mathcal{A}$ be an abelian category with enough injectives, let $F: ...
1
vote
0answers
41 views

Inequality of numerical invariants of complex algebraic surfaces?

Let $S, T$ be algebraic surfaces over $k=\mathbb{C}$, and $\phi: S \longrightarrow T$ a surjective morphism. Furthermore we have the numerical invariants: \begin{align*} q(S) &:= \dim H^1(S, ...
2
votes
1answer
86 views

Exact sequence of four sheaves in Beauville: associated l.e.s.?

This question is about an exact sequence of four sheaves on a smooth projective surface $S$ over $k=\mathbb{C}$, to be found in Beauville: complex algebraic surfaces, theorem I.4, page 3 (second ...
2
votes
0answers
272 views

Dual sequence and its exactness

I am reading Lang's Algebra and trying to fill in the gaps in my mathematical background while I train for the quals. So I came across the following exercise (chapter 20, ex. 26 in the third edition). ...
33
votes
3answers
1k views

Intuition behind Snake Lemma

I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
7
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0answers
308 views

Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: ...
1
vote
1answer
84 views

Is it true, that $H^1(X,\mathcal{K}_{x_1,x_2})=0$? - The cohomology of the complex curve with a coefficient of the shaeaf of meromorphic functions…

Let X be complex curve (complex manifold and $\dim X=1$). For $x_1,x_2\in X$ we define the sheaf $\mathcal{K}_{x_1,x_2}$(in complex topology) of meromorphic functions vanish at the points $x_1$ and ...
1
vote
0answers
81 views

Universal coefficient formula

Let $X$ be a compact manifold (so that all (co)homologyies have finite rank). The universal coefficient formula ($\mathbb{Z}$-coefficient for simplicity) says that we have the following short exact ...
4
votes
0answers
122 views

Ext in Dedekind domains

I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. ...
1
vote
0answers
69 views

Pontrjagin's Lemma and an application

I would appreciate any kind of help on the following issue: On page 114 of Rotman's "Homological Algebra", exercise 3.4 reads: 1) (Pontrjagin) If an abelian group $A$ is countable, torsion-free ...
1
vote
0answers
135 views

some question of calculating betti number

For a graded finitely generated $k[x_1, \cdots, x_n]$ module $V$, I know that $$ b_{i,p}(V)=\operatorname{dim}_k H_i(K\otimes V)_p$$ where $K$ be the Koszul complex of $k$. I also know that $K$ is ...
2
votes
1answer
148 views

commutativity of torsion functor

For a ring $R$ and finitely generated $R$ modules $U,W$, $${\rm Tor}_i(U,W)={\rm Tor}_i(W,U)$$ for all $i$. I saw proof in Hatcher's book, but I can understand that proof. may be I see another proof? ...
2
votes
1answer
146 views

Follow up on example computation of $\mathrm{Tor}_n$

I have a follow up question on this question of mine: I can't reconstruct how I got $\operatorname{Im}{d_1^\ast} = 0$ from the following chain: $$0 \to \mathbb Z \otimes_{\mathbb Z} (\mathbb Z / 2 ...
1
vote
1answer
103 views

Explicit 3-Cocycles of $Z_2\times Z_2$ over $U(1)$

I know that $H^3(Z_2\times Z_2, U(1))=Z_2^3$, I'd like to know all the cocycles explicitly. Is there a systematical way to find the cocycles (I guess one can always try to solve the cocycle conditions ...
5
votes
0answers
244 views

Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
2
votes
1answer
200 views

Question about the $\mathrm{Tor}$ functor

Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring. We take a projective resolution of $M$: $$ \dots \to P_1 \to P_0 \to M \to 0$$ Now ...
0
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1answer
128 views

Request for a $\mathrm{Hom}$ functor example

Let $$ P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \xrightarrow{d_0} M \to 0$$ be an exact sequence of $R$-modules. Consider $$ (*) \hspace{1 cm} P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 ...
2
votes
3answers
275 views

Quasi-isomorphism of Complexes

Let's $(K^{\bullet}, d^{\bullet})$ is the complex over field $A$ (i.e. all $K^{i}$ are vector spaces over this field) and $(L^{\bullet}, {\delta}^{\bullet})$ such that $$L^{i}=H^{i}(K)~\text{and ...
4
votes
1answer
147 views

Infinite Sum Axioms in Tohoku

In his Tohoku paper, section 1.5, Grothendieck states the following axioms that an abelian category might satisfy: AB4)Infinite sums exist, and the direct sum of monomorphisms is a monomorphism. ...
4
votes
1answer
468 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
6
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0answers
462 views

Why didn't Cartan-Eilenberg develop homological algebra on sheaf theory?

Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at ...
1
vote
1answer
246 views

Direct limit and exact sequences of abelian groups

Suppose having a set of direct systems of abelian groups $\ldots\{G_{\alpha}\}_{\alpha\in A}$, $\{G_{\beta}\}_{\beta\in B}$, $\{G_{\gamma}\}_{\gamma\in \Gamma}\ldots$ If there is a (long) exact ...
11
votes
1answer
407 views

A spectral sequence for Tor

Suppose $R \to T$ is a ring map such that $T$ is flat as an $R$-module. Then for $A$ an $R$-module, $C$ a $T$-module there is an isomorphism $$\text{Tor}^R_n(A,C) \simeq \text{Tor}^T_n(A \otimes_R ...
46
votes
2answers
1k views

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
5
votes
1answer
599 views

cohomology vs homology

I have learned the basic things about cohomology and homology. It seems that homology and cohomology both deal with the same objects, the complexes, but with a different choice of the indexes (for ...
4
votes
1answer
344 views

short exact sequences and direct product

Let $$0\longrightarrow L^{(i)}\longrightarrow M^{(i)}\longrightarrow N^{(i)}\longrightarrow 0$$ be a short exact sequence of abelian groups for every index $i$. Clearly if I take finite direct ...
6
votes
2answers
495 views

Motivation for Koszul complex

Koszul complex is important for homological theory of commutative rings. However, it's hard to guess where it came from. What was the motivation for Koszul complex?
1
vote
1answer
128 views

A particular isomorphism between Hom and first Ext.

Let $R$ commutative ring and $I$ an ideal of $R$. How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ? This question is an exercise of the course ...
9
votes
1answer
194 views

Lifting isomorphisms between derived categories

Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. I will denote by $D(A)$ and $D(B)$ the derived categories of unbounded complexes over $A$ and $B$. Suppose ...
4
votes
1answer
449 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 ...
5
votes
1answer
364 views

When $\mathbb{Z}/pq\mathbb{Z}$ is not semisimple?

Prove that for any primes $p$, $q$, $p\neq q$, the ring $\mathbb{Z}_{pq}$ (the ring of integers modulo pq) is semisimple, and for $p=q$ the same ring is not semisimple. I was told that the easiest ...
3
votes
0answers
175 views

How are injective model structures cofibrantly generated?

I have a question about the injective model structure on functor categories. As background : If $\mathcal{M}$ is a combinatorial model category and $\mathcal{C}$ is a small category, then there are ...
7
votes
1answer
307 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 ...
1
vote
1answer
82 views

Does the tensor product of two complexes with acyclic augmentation have acyclic augmentation?

More specifically, let $(K,\partial^K,\varepsilon^K)$ and $(L,\partial^L,\varepsilon^L)$ be augmented aclycic complexes of free abelian groups with augmentation module $\mathbb{Z}$; that is, ...
2
votes
1answer
185 views

Flabby sheaves and comparison of topologies

Let $A^p$ be a group of sheaves on a topological space $X$, let $F$ be the global sections functor $F(A^p) = A^p(X)$. I have to compute the cohomology of the complex $0\rightarrow A^1(X) \rightarrow ...
3
votes
1answer
149 views

The relation between betti numbers and Tor functor?

Let $M$ be finitely generated Module over the Polynomial ring $R=k[x_{1},..,x_{n}]$, then there is a free resolution of M of the form $$0\rightarrow F_{n}\rightarrow ...\rightarrow F_{0}\rightarrow ...
9
votes
1answer
452 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} ...
4
votes
1answer
228 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...