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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64 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 ...
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1answer
45 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'...
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1answer
32 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 ...
2
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1answer
32 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 $...
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0answers
38 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 ...
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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
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0answers
44 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):=\...
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4answers
150 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 ...
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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 ...
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0answers
36 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 "...
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33 views
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23 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 $b:C^n(A,M) \to C^{n+1}(A,M)$...
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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{...
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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 ...
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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 $A$-...
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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
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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 ...
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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
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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 ...
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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: \...
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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;\...
4
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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 ...
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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\...
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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{...
3
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1answer
56 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
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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
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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 ...
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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 ...
3
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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
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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 ...
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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 ...
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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 $M$ ...
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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 ...
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0answers
27 views

Minus signs in internal hom

Consider the internal hom of chain complexes $A^\bullet$ and $B^\bullet$ $$\operatorname{Hom}^\bullet(A^\bullet, B^\bullet):=\{\text{degree $n$ maps}\} \qquad df:= d^B\circ f - (-1)^{|f|}f \circ d^A$$...
2
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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 ...
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1answer
27 views

Choosing an injective resolution of a short exact sequence of complexes

Lemma: Given a short exact sequence of cochain complexes in an abelian category $\mathcal{C}$ with enough injectives, $$0 \to P^\bullet \xrightarrow{f} Q^\bullet \xrightarrow{g} R^\bullet \to 0,$$ ...
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1answer
25 views

Finite product exists implies finite coproduce exist.

Let $C$ be a category such that the law composition of morphisms is bilinear, and there exists a zero object $0$, and the products exists for arbitrary finite sets of objects of $C$. Then the ...
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0answers
82 views

Chain Homotopy in abelian category

When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
3
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3answers
106 views

Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
2
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1answer
55 views

Meaning of functorial

It's known that for a short exact sequence of complexes, $0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$, it associates a homology sequences $...\rightarrow H(E')\rightarrow H(E)\rightarrow ...
0
votes
1answer
24 views

Seifert matrix, linking numbers, generators

I have been asked to compute the seifert form of a knot, the twist knot. I know how to compute the seifert surface, and then the seifert matrix seems to be defined accordingly (according to all the ...
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1answer
37 views

Why solving linear equations is taking a quotient by some subspace?

Linear equation can be represented by a linear form, and its solution space is the same thing as kernel of this form. The same is true for system of linear equations. But this lecture notes suggest ...
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0answers
34 views

The relation between Weyl character formula and Frobenius characteristic map

Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence ...
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2answers
36 views

If $\text{Ext}_R^1(A,I) = 0$ for all $A\in ob(_R\text{Mod})$, then $I\in ob(_R\text{Mod})$ is injective. [duplicate]

Let $\text{Ext}^1(A,I)=0$ for all $A\in ob(_R\text{Mod})$, then $I\in ob(_R\text{Mod})$ is injective. I got stuck by this problem. Any ideas?
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1answer
52 views

A left exact functor preserves quasi-isomorphisms between acyclic complexes

A homological algebra theorem states Theorem: Let $T: \mathscr{A} \to \mathscr{B}$ be a left exact functor between abelian categories, and let $X^\bullet \xrightarrow{f} Y^\bullet$ be a quasi-...
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1answer
50 views

How do you find the free resolution of the module $M$ and of $F/M$ where $F=(K[x,y])^3$?

$M$ is a module generated by $$f_1=(xy,y,x), f_2=(x^2+x,y+x^2,y), f_3=(-y,x,y),f_4=(x^2,x,y).$$ We're to use the lex ordering with $x<y$ and $e_1>e_2>e_3$, where terms are given preference ...
2
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0answers
37 views

Ext group of bundles on moduli space of curves

Let $\mathcal{M}_{g}$ be the moduli space of curves of genus $g$. Let's suppose $g \geq 2$. Let $T$ be the tangent bundle of $\mathcal{M}_{g}$. Is the Ext group $\text{Ext}^1(\bigwedge^2T, T)$ trivial?...
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71 views

Homology Groups of Tangent 2-Spheres

I have been trying to compute the Homology Groups $ H_n $ of two tangent 2-Spheres (we will call this space, X). By previous results, I am able to easily determine that $ H_0(X) $ is isomorphic to the ...
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0answers
18 views

Mapping cylinder of chain complexes via $-\otimes \Delta$

An instructor gave me a homework set where the mapping cylinder of a chain map $C_\bullet \xrightarrow{f} D_\bullet$ is defined as $(\Delta^1_\bullet \otimes C_\bullet) \oplus_{C_\bullet} D_\bullet$, ...
4
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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 ...