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.

learn more… | top users | synonyms

0
votes
1answer
44 views

Why are spherical objects so named?

Let $S$ be an object in an abelian category. Then we say S is spherical if $Ext^p(S,S)$ is 0 unless $p = 3$. I know that the cohomology of the three sphere bears some formal resemblence, but it doesn'...
1
vote
1answer
707 views

Euler-Poincaré characteristic and homology

$\DeclareMathOperator{rk}{\text{rk}}$ $\DeclareMathOperator{im}{\text{im}}$ The problem Let $$C = ( C_n \overset{\partial_n}\to C_{n-1} \overset{\partial_{n-1}}\to \dots \overset{\partial_2}\to C_1 ...
0
votes
0answers
23 views

torsion in reduced K-theory

If $f\colon C\to C'$ is a chain equivalence of finite chain complexes over a ring $R$, then there is a well-defined (reduced) torsion $\tau(f)\in\widetilde{K}_1(R)$. My question involves the reduced $...
2
votes
2answers
61 views

What does “is natural in $A$” mean in this context?

While reading Bredon's Topology and Geometry, I've come across the following claim: Naturality in $A$ of the sequence defining $\text{Ext}(A,G)$ shows that $\text{Ext}(A,G)$ is a contravariant ...
1
vote
1answer
31 views

Let $\mathcal{A}$ be an Abelian category (as defined in Stacks), then all monomorphisms are kernels.

I'm struggling to prove this statement, using the definitions below (I'm assuming the proof for the statement about epimorphisms is analogous). I know that a morphism $f:x\to y$ is monic if and only ...
4
votes
0answers
35 views

Grading of Cech-de Rham cohomology

I am currently self-studying Bott and Tu. In chapter 2 the Cech-de Rham cohomology is introduced and I thought I had understood it well enough. However when I got to chapter 3 on spectral sequences I ...
1
vote
4answers
149 views

Survey articles in Commutative/Homological algebra

I am a graduate student interested in Commutative algebra/Homological algebra. I am comfortable with first eight chapters of Atiyah. I am familiar with some algebraic geometry, first two chapters of ...
2
votes
2answers
118 views

Minimal injective resolution of a module

Let $R$ be a commutative Noetherian ring and $M$ an $R$-module. Let $0\rightarrow M\rightarrow E^{\bullet}$ be a minimal injective resolution of $M$ and $0\rightarrow M\rightarrow I^{\bullet}$ be an ...
0
votes
0answers
43 views

Koszul complex: isomorphism between $K(a_1,\ldots, a_n;A) \simeq K(a_1;A) \otimes \cdots \otimes K(a_n;A)$

Given $a_1,\dots,a_n\in A$, with $A$ a suitable ring, my algebra teacher defined the Koszul complex associated to $a_1,\dots,a_n$ with coefficients in $A$ in this way: $$K(a_1,\dots,a_n;A):=\...
0
votes
0answers
29 views

Minimal injective resolutions isomorphism [closed]

How can I prove that given an $A$-module $M$ two injective resolutions of $M$ are isomorphic as complexes? Thank you, have a nice day Asdrubale
0
votes
1answer
49 views

Are “noncommutative resolutions” a thing?

I am interested in looking at "resolutions" of modules in noncommutative rings, making some obvious necessary modifications to the definition of a resolution. However, whenever I try to search the ...
14
votes
2answers
579 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
6
votes
1answer
233 views

Tensor product of two chain homotopic maps are again chain homotopic

Let $C$, $C'$, $D$, $D'$ be chain complexes, $f$, $f'\colon C\to C'$ and $g$, $g'\colon D \to D'$ two pairs of homotopic chain maps. How to show $f \otimes g$ and $f' \otimes g' \colon C\otimes D\to C'...
0
votes
0answers
35 views

Derived tensor product

Let $A$ be a commutative ring, $M$ and $N$ $A$-modules. Is the derived tensor product $M[0]\otimes^L N[0]$ isomorphic to $M\otimes_A N$? I know that the derived tensor product is supposed to be a "...
3
votes
1answer
104 views

$\operatorname{Ann}_RA+\operatorname{Ann}_RB\subseteq\operatorname{Ann}_R\operatorname{Ext}^n_R(A,B)$?

Let $R$ be a commutative unital ring and $r\in R$. Let $A$ and $B$ be $R$-modules. Does $rA=0$ or $rB=0$ imply $r\operatorname{Ext}^n_R(A,B)=0$ for all $n\in\mathbb{N}$? For $n=0$ it holds, but I'm ...
1
vote
0answers
22 views

Maps inducing identity in Hochschild and cyclic theories

Let $A$ be a unital algebra over $\mathbb{C}$, $M$ be an $A$ bimodule, $C^n(A,M)$ be a space off all $n$-linear maps $f:A^{n} \to M$ (to be called $n$ cochains) and define $b:C^n(A,M) \to C^{n+1}(A,M)$...
0
votes
0answers
25 views

If $x\in \mathrm{Ann}(N)$ then $x$ annihilates $\mathrm{Ext}_i(N,M)$ for all $i$, why? [duplicate]

Matsumura in his Commutative Ring Theory, for the proof of theorem 16.6, uses a fact as follows: Let $A$ be a unital commutative ring, $N$ a (finitely generated?) $A$-module, and $M$ any $A$-...
0
votes
1answer
48 views

Prove that $A \otimes_R B \cong (A \otimes_{\mathbb{Z}} B)/ H$

I am working on the following problem: Let $A \in Mod-R$ and $B \in R-Mod$. Prove that $A \otimes_R B \cong (A \otimes_{\mathbb{Z}} B)/ H$ where $H=\langle ar\otimes_{\mathbb{Z}}b - a\otimes_{\mathbb{...
0
votes
2answers
60 views

Ideals and Tensor Products

I'm reading Osbourne's Basic Homological Algebra, and on page 18 he has this situation where we've got a ring $R$ and a right-ideal $I$, and some left $R$-module $B$. He says $I\otimes B$ is not a ...
0
votes
1answer
29 views

Generalization of Universal Coefficient Theorem

Suppose we are in an abelian category $\mathscr{A}$. Given a fixed monomorphism $A \overset{i}{\hookrightarrow} B$, and an object $C$, I would like to express concisely the notion of the group of maps ...
2
votes
1answer
54 views

Why does this homological lemma hold?

Let $A$ be a noetherian ring; let $C^{\boldsymbol\cdot}$ be a bounded above complex of flat $A$-modules in positive degrees, let $L^{\boldsymbol\cdot}$ be a bounded above complex of free $A$-modules ...
0
votes
1answer
31 views

Right exact functor applied to epimorphism of cohomology is still epimorphism?

Let $\mathcal A,\mathcal B$ be abelian categories and $F$ an additive, right exact functor $\mathcal A\rightarrow\mathcal B$. Suppose I have a morphism of chain complexes (in positive degrees) $C^{\,\...
2
votes
1answer
198 views

Tor for graded modules over a graded ring

I am confused about how this Tor is defined. Suppose $R$ is a graded ring, $M,N$ graded modules over $R$. What is $\operatorname{Tor}_{st}^R(M,N)$? I am confused about the subscripts. I realize ...
2
votes
1answer
76 views

Is R/m a flat R-module?

Let $(R,\frak m)$ be a commutative Noetherian local ring. Is $R/\frak m$ a flat $R$-module? Thanks.
2
votes
1answer
47 views

Why should a DG-module homomorphism also be a chain map?

According to every definition in the literature, (1) a homomorphism of DG-modules must be a chain map [Stacks Project]. This is perfectly reasonable at first glance, especially if you want to use the ...
1
vote
0answers
66 views

When does the Grothendieck spectral sequence converge?

I am trying to understand spectral sequences in algebraic geometry. One has the Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$, and $\mathcal G: \...
1
vote
2answers
67 views

What is the kernel of the map $H_i(X;\mathbb Z)\to H_i(X;\mathbb Z_2)$?

Let $X$ be a topological space. By Universal Coefficient Theorem for Homology we have the exact sequence $$0\to H_i(X;\mathbb Z)\otimes\mathbb Z_2\to H_i(X;\mathbb Z_2)\to \text{Tor}_1(H_{i-1}(X;\...
10
votes
3answers
148 views

An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$

Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are self-...
4
votes
1answer
80 views

What's wrong with my understanding of the Freyd-Mitchell Embedding Theorem?

It's truly bizarre that there exists no full modern exposition of this theorem, as noted elsewhere. Anyway, I thought I'd poke through and see if I could get the gist of how it works as somebody who ...
0
votes
1answer
32 views

$(\mathbb{Z}/n\mathbb{Z})$-homology isomorphism is also a $(\mathbb{Z}/n^k\mathbb{Z})$-homology isomorphism

I'm trying to prove that if a map $f \colon X \to Y$ induces isomorphisms on singular homology with coefficients in $\mathbb{Z}/n\mathbb{Z}$, then the same is true for coefficients in $\mathbb{Z}/n^k\...
3
votes
1answer
167 views

Which book would you recommended as help (assistance) for reading the so-called “Tohoku Paper”?

Recently I thought that maybe is a good time to try, read Grothendieck's "Tohoku paper" as a sort of inspiration for the future and to read some of the ideas of this great mathematician, which (among ...
3
votes
1answer
55 views

Stronger version of Acyclic Models Theorem

Let $\mathscr{C}$ be an abelian category. If $P_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C})$ is a bounded below complex of projectives, and $C_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C}...
2
votes
1answer
25 views

Functoriality of internal hom of chain complexes

The internal hom of chain complexes $[-,-]$ is supposed to form a bifunctor $$\operatorname{Ch}_\bullet(\mathsf{Mod}_R)^\mathrm{op} \times \operatorname{Ch}_\bullet(\mathsf{Mod}_R) \to \operatorname{...
4
votes
1answer
115 views

Homotopy Colimit of Truncations

Let $\mathcal{A}$ be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in ...
1
vote
1answer
76 views

Homological dimension of categories of modules

Let $A$ be a Noetherian ring. We have two categories: (a) category of $A$-modules (b) category of finite type $A$-modules. Do their homological dimensions agree? The homological dimension of an ...
1
vote
0answers
246 views

chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$ C^n(Y_i) \oplus C^{n-1}(X_i) $$ with boundary operator: $$ (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
2
votes
1answer
59 views

Ext and Tor over noncommutative rings

This might be a stupid question, but I could not find a good reference that thoroughly explains the matter. I will start with some lengthy introduction. If $\mathcal{A}$ is any abelian category, then ...
0
votes
1answer
33 views

Homology group versus group homology

If we have a simplical complex $K$, then we are able to define $C_i(K)$ as the free abelian group over $\mathbb Z_2$ with the basis of all $i$-dimensional simplices. By using the boundary map we are ...
3
votes
1answer
95 views

$0\to C'\to C\to C''\to0$ splits if $C\cong C'\oplus C''$ as a chain complex?

Question Given a unitary ring $A$ and an exact sequence $$0\to C'\xrightarrow iC\xrightarrow pC''\to0$$ in the Abelian category of chain complexes over $A$, where $C,C',C''$ are chain complexes of ...
52
votes
1answer
2k views

Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
4
votes
3answers
342 views

How to do diagram chasing effectively?

I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3. I have spent a lot of time ...
4
votes
1answer
78 views

Is a kernel in a full additive subcategory also a kernel in the ambient abelian category?

Setting: Let $\mathscr{C}\subset \mathscr{A}$ be a full additive subcategory of an abelian category. Let $C,C'$ be objects of $\mathscr{C}$ and let $f\in \operatorname{Hom}_\mathscr{C}(C,C')=\...
0
votes
0answers
24 views

Definition of contractible chain complex

A relatively simple question. A book I'm reading states "a complex homotopic to the zero complex is called contractible"... but I don't understand the statement. I know what it means for chain maps ...
1
vote
1answer
43 views

Faithfully flat descent of projectivity and freeness

I am reading this paper. It is proven there that if $f:A\rightarrow B$ is a faithfully flat morphism of rings and $M$ an $A$-module such that the $B$-module $M\otimes_A B$ is projective, then $M$ ...
1
vote
0answers
30 views

Interaction of a functor with internal hom

An additive functor between abelian categories $F: \mathscr{C} \to \mathscr{D}$ induces a functor on categories of chain complexes $F: \mathscr{C}^\bullet \to \mathscr{D}^\bullet$. The internal hom ...
2
votes
1answer
38 views

On selfinjectivity of Hopf algebras

Any group algebra $kG$ of a finite group is selfinjective. More generally Gentile proves that for a group ring $RG$ with $R$ commutative and torsion free as a $\Bbb Z$-module, $RG$ is selfinjective ...
4
votes
1answer
609 views

Projective resolution of tensor product

Let $M,N$ are $R$ modules and $P^\bullet, Q^\bullet$ are their projective resolutions. Can we obtain projective resolution $M\otimes N$ using $P^\bullet, Q^\bullet$. If i understand correctly homology ...
0
votes
2answers
60 views

Cohomology of free group acting trivially

I read here the formula for cohomology of a free group with trivial action: $$H^q(G, M) = \begin{cases} M &\text{for } q = 0\\ M^n &\text{for } q = 1\\ 0 &\text{for } q \geq 2\\ \end{cases}...
0
votes
0answers
27 views

Minus signs in internal hom

Consider the internal hom of chain complexes $A^\bullet$ and $B^\bullet$ $$\operatorname{Hom}^\bullet(A^\bullet, B^\bullet):=\{\text{degree $n$ maps}\} \qquad df:= d^B\circ f - (-1)^{|f|}f \circ d^A$$...