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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2
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1answer
502 views

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of ...
2
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0answers
43 views

Solve commutator relation $[Q,d]=-[P,d]$ for $Q$ on chain complexes with scalar product

Suppose we are given chain sequences $\dots \rightarrow C_k \rightarrow C_{k+1} \rightarrow \dots$ and $\dots \rightarrow D_k \rightarrow D_{k+1} \rightarrow \dots$ of finite-dimensional vector ...
16
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1answer
932 views

Intuition behind homology with general coefficients

We just went over homology with general coefficients in topology and did some of the usual examples ($\mathbb{Z}_2$ for projective space and manifolds being the big examples) which led me to wonder ...
8
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1answer
146 views

Generators of a certain ideal

Crossposted on MathOverflow. The MathOverflow version of the question has been rewritten. For the sake of completeness, I pasted it here in a condensed form. I also deleted the old version. Let $K$ ...
3
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0answers
174 views

Left-derived functors

Let $F:\mathcal{A}\to\mathcal{B}$ be a covariant right-exact functor between two abelian categories. Suppose $\mathcal{A}$ has enough projectives. Then we define the left derived functors of $F$ by ...
2
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2answers
161 views

Explicitly finding a cocycle in $H^3(S_3,\mathbb{Z}_3)$

I know that $H^3(S_3,\mathbb{Z}_3)\cong \mathbb{Z}_3$ (S_3 is the symmetric group for three elements). So this group is generated by any nontrivial cocycle. But I don't know how to explicitly find ...
3
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0answers
319 views

How to understand the diagonal approximation?

In the Brown's book “Cohomology of groups”, chapter 5.1, there is a concept diagonal approximation, maybe that is not a standard definition, I feel something hard to understand it. The book says that ...
2
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2answers
223 views

$\operatorname{Func}(J,Ab)$ has enough injectives.

I am trying to show that the functor category $\operatorname{Func}(J,Ab)$ has enough injectives (meaning that for each $F\in \operatorname{Func}(J,Ab)$ there is an injective object $I\in ...
2
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1answer
156 views

Why is this complex acyclic?

Let $X$ denote a set. Let $C_{n}(X)$ denote the free abelian group generated by $(n+1)$-tuples of elements of $X$. Define $$\partial_n (x_0, x_1, \ldots, x_n) = \sum_{k=0}^n (-1)^k (x_0,x_1, \ldots, ...
7
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2answers
279 views

English translation or summary of “Relevements modulo $p^2$ et decomposition du complexe de de Rham. ”

I'm looking for either an English translation or summary of the article "Relevements modulo $p^2$ et decomposition du complexe de de Rham." by Deligne. I'm attempting to read this for background ...
0
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1answer
94 views

Right derived functor of diagonal morphism equals direct image on line bundles?

Let $X$ be a smooth projective variety. The map $i:X\to X\times_k X$ induced by the identity is a closed immersion. Denote its image by $\bigtriangleup$. We have ...
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0answers
86 views

Tensor product and evaluation map in $k$-linear triangulated categories

In $k$-linear triangulated categories, there is an evaluation map $$\oplus_i \text{Hom}(E,A[i])\otimes_k E[-i]\to A .$$ I've learned that in the derived categorie of coherent sheafs on a scheme ...
9
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3answers
796 views

“The Yoneda embedding reflects exactness” is a direct consequence of Yoneda?

Let $A,B,C$ be objects of a category of modules over a ring. It is not hard to see that the Yoneda embedding "reflects exactness" (as Weibel puts it, on p. 28), i.e. if ...
4
votes
2answers
277 views

A result of flat modules that *needs* to deal with the *construction* of the tensor product?

There is a nice result concerning flat modules over a ring: If every finitely generated submodule of a module $M$ is flat, then $M$ is flat. However, the proof I've read in Rotman's Homological ...
3
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1answer
595 views

$\mathbb{Q}$ is an injective $\mathbb{Z}$-module

I just learned what an injective module is and I want to consider some basic examples. Apparently, $\mathbb{Q}$ is an injective module over $\mathbb{Z}$, but I can't find an elementary proof of this ...
6
votes
2answers
1k views

Relative homology groups of the torus

I have the following question to problem 2.1.17 in Allen Hatcher's "Algebraic Topology". So far I came up with the following exact sequences (for A and B): $$ \begin{aligned} 0&\rightarrow ...
3
votes
1answer
90 views

Map induced between Pontryagin duals

Let $f\colon A\to B$ be a group homomorphism between finite abelian groups. For abelian group $G$, let $G^\wedge=\operatorname{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ be its Pontryagin dual. Since ...
2
votes
2answers
382 views

Derived functors are Kan extensions

In this short paper by G. Maltsiniotis derived functors are presented as Kan extensions along the localization functor. I began studying derived categories only a couple of months ago, so I'm not at ...
2
votes
2answers
425 views

Two trivial questions on projective/injective modules from Hilton-Stammbach

Consider two exact sequences $0\rightarrow N\rightarrow P\rightarrow A\rightarrow 0$ and $0\rightarrow M\rightarrow Q\rightarrow A\rightarrow 0$, where $P,Q$ are projective modules. The exercise(pg ...
4
votes
1answer
329 views

Is quasi-isomorphism an equivalence relation?

Let $E^\bullet$ and $F^\bullet$ be complexes on an abelian category; what does it mean to say that $E^\bullet$ and $F^\bullet$ are quasi-isomorphic? Does it only mean that there is a map of complexes ...
3
votes
1answer
437 views

Exact sequences and functor Hom

For an abelian group $G$ we denote by $G^*$ the $\mathbb{Z}$-module $\text{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ -- group of all $\mathbb{Z}$-module homomorphisms from $G$ to ...
4
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0answers
93 views

Why does applying $-\Box_{A//B} A$ to a free coresolution preserve exactness?

Let $A$ be a Hopf algebra over a field $k$, and let $B$ be a normal subHopf algebra of $A$. Suppose we have an $A$-free coresolution of $k$ over the form $F_n=K_n \otimes_k A$. Kochman claims that ...
5
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1answer
109 views

Constructing a coresolution

I am working through computing the homotopy of Thom spectra from Kochman's book. Let $A$ be a coalgebra over a field $k$, and let $M$ be a right $A$-comodule. Kochman constructs a coresolution $F$ ...
3
votes
2answers
152 views

Vanishing of $ H^1(\mathcal{M})$ implies vanishing of $H^1(U\otimes\mathcal{M}) $ on a curve.

Let $C$ be a smooth projective curve of genus $g\geq 1$ over an algebraically closed field. Let $\mathcal{M}$ be a line bundle with $deg \mathcal{M}\geq 2g -1$. Let $T$ be torsion and denote by $U$ ...
4
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1answer
564 views

Question on the infinite direct product of projective modules

We knew that the direct sum of a family of projective modules is a projective module, and the direct product of a family of injective modules is also injective. My question is, is the infinite direct ...
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5answers
760 views

Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
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vote
1answer
110 views

Is the presheaf of continuous functions on a topological space a “complete presheaf”?

Is the presheaf of continuous functions $f:A\rightarrow B$ from a topological space $A$ to another topological space $B$ a "complete presheaf"? Can't find this, anyone have a reference?
4
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1answer
103 views

Quiver describing perverse sheaves on $\mathbb C$

I have two sources, which claim that the category of perverse sheaves on $\mathbb C$ constructible with respect to the stratification $0$ and $\mathbb C^*$ is equivalent to the category of certain ...
2
votes
1answer
705 views

When is the pullback of a linear injection a surjection on dual space?

Due to the contravariance of the dual space functor on vector spaces, one might expect the pullback of an injection to be a surjection, and the pullback of a surjection to be an injection. Indeed, for ...
20
votes
2answers
3k views

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
41
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7answers
2k views

Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra

I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra. Now, this sort of ...
3
votes
1answer
72 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 ...
1
vote
1answer
185 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
votes
1answer
194 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
votes
1answer
1k 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
votes
1answer
152 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
291 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
votes
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 ...
22
votes
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' ...
3
votes
2answers
316 views

Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?
3
votes
1answer
313 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 $H$.
0
votes
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
249 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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vote
1answer
204 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
285 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
votes
0answers
184 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
votes
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
votes
1answer
65 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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vote
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
573 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 ...