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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Minimal injective resolutions isomorphism [on hold]

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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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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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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46 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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33 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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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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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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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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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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28 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 ...
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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^{\,\...
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1answer
46 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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60 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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66 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;\...
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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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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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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{...
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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}...
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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 ...
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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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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 ...
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165 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 ...
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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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23 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
42 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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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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27 views

Minus signs in internal hom

Consider the internal hom of chain complexes $A^\bullet$ and $B^\bullet$ $$\operatorname{Hom}^\bullet(A^\bullet, B^\bullet):=\{\text{degree $n$ maps}\} \qquad df:= d^B\circ f - (-1)^{|f|}f \circ d^A$$...
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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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26 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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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 ...
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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 ...
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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 ...
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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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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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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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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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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 ...
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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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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 $(\Delta^1_\bullet \otimes C_\bullet) \oplus_{C_\bullet} D_\bullet$, ...
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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 ...
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72 views

Splitting a short exact sequence of complexes of vector spaces

It's well-known that any complex of vector spaces is isomorphic to a direct sum of two types of indecomposable complexes (a one-dimensional space concentrated in one degree, or two one dimensional ...
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28 views

Exercise about Ext functor and 'tri-module'

While trying to understand the Hochschild-Kostant-Rosenberg theorem, I learned that $Ext_{R \otimes R}^1(R, R) = Der_K(R)$, where $R$ is an regular affine (commutative) $\mathbb{C}$-algebra. I am ...
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103 views

The endomorphisms of an $A_\infty$-algebra as a (bi)module over itself

Let $A$ be a unital associative algebra. A well-known exercise states that the ring of $A$-bimodule endomorphisms of $A$ are isomorphic to the center of $A$. That is, $\text{End}_{A-\text{mod}}(A) \...
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32 views

Derived functors in abelian categories and homotopy theory

For two Abelian categories $\mathcal A,\mathcal B$ and a right exact additive function $F\colon\mathcal A\to\mathcal B$, there is a left derived functor $LF$ acts on chain complexes $K_+(\mathcal A)$ ...
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A embedding of tensor product over semisimple algebras

Let $R$ be a semisimple Artinian algebra over the complex number field $\mathbb{C}$, that is, $R$ is isomorphic a finite direct product of matrix rings over $\mathbb{C}$. Let $S$ be a ideal of $R$, ...