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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7
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294 views

Condition for a ring on projective and free modules problem

Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
3
votes
2answers
541 views

Split exact sequences

Let $0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0$ be a split-exact short exact sequence of $R$-modules, where $R$ is any ring. Let $T$ be an additive functor from $R$-modules ...
9
votes
1answer
267 views

Hochschild homology of Weyl algebra

Could someone explain to me how one can compute Hochschild homology of Weyl algebra $A_n$ (i.e. algebra of differential operators with polynomial coefficients in $n$ variables)?
2
votes
0answers
79 views

A question on the standard chain complex

Suppose that $G$ is a group, and $\mathbb{Z}[G]$ the group ring. Then $\mathbb Z$ can be considered as a $\mathbb{Z}[G]$-module if every $g (\in G)$ acts trivally, and every $n (\in \mathbb Z)$ acts ...
10
votes
2answers
352 views

Computing intersection multiplicity using Tor - explicit example

When trying to compute the (Serre-generalized) intersection number of two varieties at a closed point, I came to a need to compute the following $\operatorname{Tor}$: Let $k$ be an algebrically ...
4
votes
1answer
171 views

Flabby sheaves and exact sequences of sheaves - Question about proof

I was going through this proof from Rotman's 'Introduction to homological algebra' (Pages 381-382) and I just can't seem to make sense of it, am not super well-versed in this so I don't know if it's ...
3
votes
1answer
189 views

$\mathrm{Tor}$ functor not left exact

Is there an example which shows that the functor $B\otimes_R(-)$ is not left-exact, given a ring $R$ and a right $R$-module $B$?
3
votes
1answer
413 views

Is the dual of a projective module always projective?

I'm able to prove it for finitely generated modules, by appealing to the characterization of a projective module as a summand of a free module, and the fact that finite-rank free modules are ...
7
votes
1answer
226 views

Injective Maximal Cohen-Macaulay modules

Let $R$ be a Gorenstein (not necessarily commutative) ring and let $I$ be an injective finitely generated module over $R$. Is it true that if $\operatorname{Ext}_R^i(I, R)=0$ for $i > 0$, then $I$ ...
19
votes
4answers
974 views

Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module

Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module? The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
3
votes
1answer
194 views

Question about proof that Flabby sheaves are acyclic

Can anybody help me understand this proof? In Rotman's 'An introduction to homological algebra' in Proposition 6.75 (iii). Flabby sheaves $\mathcal{L}$ are acyclic (Page 381), in the proof it says ...
11
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2answers
463 views

Computation of $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$

Can you please give some examples of computation of the derived functors $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$ for some simple cases, say $R=\mathbb{Z}$ or $R=\mathbb{Z}[G]$ for some ...
2
votes
1answer
52 views

when is $R\!\times\!0$ not a free $R\!\times\!R'$-module

If $R,R'$ are two rings, then $R\!\times\!0$ is a projective $R\!\times\!R'$-module, since it is a direct summand of a free module: ...
3
votes
2answers
184 views

The subcomplex of degenerate simplices has trivial homology

Let $A_\bullet$ be a (non-augmented) simplicial object in an abelian category, with face maps $d_i : A_n \to A_{n-1}$ and degeneracy maps $s_i : A_n \to A_{n+1}$, $0 \le i \le n$, for each $n \ge 0$. ...
5
votes
1answer
256 views

Some questions about the Tor functor as a two-variable functor related to the arbitrary character of the choice of projective resolutions

Given a ring $R$, we can consider the following functors: any $A\in Mod-R$ and choice of projective resolutions $P_\bullet(B)$ for every $B\in R-Mod$ defines a functor $Tor_n^R(A,-):R-Mod\to Ab$, ...
1
vote
1answer
421 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
votes
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 ...
13
votes
1answer
828 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
votes
1answer
143 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
votes
0answers
170 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
votes
2answers
151 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 ...
4
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0answers
277 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
votes
2answers
198 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
151 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
votes
2answers
252 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
votes
1answer
93 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 ...
1
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0answers
85 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 ...
7
votes
3answers
679 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
251 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
votes
1answer
493 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
764 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
88 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
343 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
379 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
284 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
387 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
votes
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
votes
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
151 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
votes
1answer
514 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 ...
9
votes
5answers
701 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 ...
1
vote
1answer
106 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
93 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
636 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 ...
18
votes
2answers
2k 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 ...
39
votes
7answers
1k 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 ...
2
votes
1answer
65 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
165 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
185 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 ...
5
votes
1answer
852 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 ...