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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$\mathbf{Ch}(\mathcal{A})$ has enough projectives

This is the Exercise 2.2.2 from Weibel's book. Suppose an Abelian category $\mathcal{A}$ has enough projectives, then so does the category $\mathbf{Ch}(\mathcal{A})$ of chain complexes over ...
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2answers
31 views

Auslander-Buchsbaum formula without minimal/finite resolutions

Does anybody know a proof of Auslander-Buchsbaum's formula that uses only projective/injective/flat resolutions and homological functors Ext and Hom without using minimal/finite resolutions?
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Homological group $Ext^R$ for P.I.D or noetherian rings

I'm still very new to homological algebra. I would like to know what are the groups cohomology derived from the functor $Hom_R(\_, D)$ of the $R$-module $A$ (i.e. compute $Ext^n(A,D)$ ), -in the ...
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1answer
20 views

Showing a class of objects is $F$-acyclic

There's a lemma from homological algebra that I'm using in sheaf cohomology, and I can't remember where else I've seen it. Where are some other key places it is applied? Lemma: Let $F:\mathcal{C} ...
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1answer
33 views

Chain map with one-sided inverse between isomorphic chain complexes quasi

Suppose that $C_\bullet$ and $D_\bullet$ are finitely generated isomorphic chain complexes. Let $f:C_\bullet\rightarrow D_\bullet$ a chain map with a one-sided inverse. Is it true that f is a ...
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A spectrum $I$ is $E$-injective iff the map $i:I\rightarrow I\wedge E$ is an inclusion of a retract.

I was reading some notes on stable homotopy theory and I came across the statement in the title of this question. "Suppose $E$ is a ring spectrum, then $I$ is $E$-injective if and only if the ...
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20 views

Exact sequences of pairs

Take a triple of topological spaces $(X, A, B)$ consists of a topological space $X$ and two subspaces $A,B$ with $B \subseteq A \subseteq X$. Why is the following sequence of pairs exact? $$ 0 ...
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63 views

I don't get it, does “augmented chain complex” actually mean anything?

If I understand correctly, chain complexes make sense in any category enriched in the world of pointed sets. In practice, there's also a notion of an augmented chain complex, where we have an extra ...
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Weibel Exercise 4.5.6

Let $R$ be a regular local ring with residue field $k$. Show that $Tor_{p}^{R}(k,k) \cong Ext_{R}^{p}(k,k) \cong \Lambda^{p}k^n \cong k^{{n}\choose{p}}$. Because of Koszul Resolution and we know ...
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54 views

What does $(0:x)$ mean?

The following excerpt is from Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry" on pg. 424 We can decide whether an element $x\in R$ is a nonzerodivisor from the homology of ...
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45 views

Several problems in homological algebra

I met so many problems when I study homological algebra by myself. Thus, I really would like to see the answers. Hopefully, everyone can help me (my big thanks). 1) When we create torsion functor ...
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Where can I learn about differential graded algebras?

I want to learn more about differential graded algebras so that I can construct explicit examples of derived schemes over characteristic 0, compute smooth resolutions of morphisms of schemes, and ...
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49 views

adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
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67 views

Barycentric subdivision proof

In the proof of barycentric subdivision in singular homology, we take the subdivision operator $b: C_q(X) \to C_q(X)$, and do some algebra to define an operator $b^\infty$, which applies $b$ "as many ...
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1answer
22 views

Calculating $\mathrm{Ext}_R^i(\mathbb{Z},\mathbb{Z})$ for $R=\mathbb{Z}[x,y]/(xy)$

First of all, sorry for the picture – it was something that would have taken me quite a while to LaTeX up. Below is my workings on the problem stated in the title: calculating ...
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1answer
21 views

Demonstrating Isomorphism in Commutative Rectangle with Short Exact Rows

I am working on the first exercise in Hilton & Stammbach's "A Course in Homological Algebra", wherein I am given the following commutative diagram: I am asked to show that $\alpha'$ is an ...
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1answer
135 views

Detail in the proof that sheaf cohomology = singular cohomology

Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of ...
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1answer
20 views

Trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$

I am trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$ I know that $RP^2$#$T^2$~$RP^2$#$K^2$ and that $X(M$#$N)$=$X(M)+X(N) -2$ where X is the ...
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89 views

Splitness of a short exact sequence on a curve

Let $C$ be a curve with genus $g > 1$. Consider the product $C \times C$, with natural projections $p_1$ and $p_2$ (from the first and second factor, respectively) to $C$. Consider the following ...
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1answer
76 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow ...
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Exercise 1.5.7 in Weibel's book about mapping cone and mapping cylinder

Given a short exact sequence of chain complexes $$0\rightarrow B \xrightarrow{\ f\ }C \xrightarrow{\ g\ }D\rightarrow 0$$ The problem asks to show that there is a quasi-isomorphism $B[-1]\rightarrow ...
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21 views

f is null homotopic if and only if (-s,f):cone(C)->D

Actually this question is from Weibel, exercise 1.5.2. Let $f:C\to D$ be a map of complexes. Show that $f$ is null homotopic if and only if $f$ extends to a map $(-s,f):$cone($C$)$\to D$. ...
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33 views

Degree of maps $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$

In the book I am reading right now, it is defined that for a map $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$ the degree is the degree of the direct image cycle $\mu_{*}[\mathbb{P}^1]$. We are ...
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62 views

About alternative ways of computing $H^1(X,\mathcal{O}_{\mathbb{P}^n}(m))$.

This is a follow up to my question : Applications of $Ext^n$ in algebraic geometry In the case of $\mathcal{O}_X$-Modules it is clear that $Ext^i(\mathcal{O}_X, \mathcal{F}) \cong H^i(X,\mathcal{F})$ ...
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25 views

Left exact functors and long exact sequences

I wonder whether in any Abelian category $\mathcal{C}$ when we have a long exact sequence $0\to M_1\to M_2\cdots\to M_n\to 0$ and a (covariant) left exact functor $F$ we have $0\to FM_1\to FM_2\to ...
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3answers
41 views

Is $Z/mZ\otimes Z \cong Z/mZ$?

I'm reading a Homological Algebra book that states this in some point without proving. I was trying to prove it and it seems to me that the first module is infinite and the second is not.
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1answer
30 views

Isomorphism on top cohomology implies isomorphism on homology

Let $F$ be a finite field (for example I could take $\mathbb{Z}_2$) and $f:X\longrightarrow Y$ a continuous map between compact, orientable and connected manifolds of dimension $n$. Suppose I have an ...
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1answer
24 views

Understanding proof of Universal coefficient theorem for cohomology

I am working through Cohomology chapter on Hatcher's book and I am having trouble with the proof of Universal Coefficient theorem for Cohomology. To be concrete I don't understand the last part of the ...
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How does the universal coefficient theorem give a map $H_k(M;\mathbb{Q})\to H_k(M;\mathbb{C})$?

On the wikipedia page for Hodge cycles, it is stated that the universal coefficient theorem gives us a map $$H_k(M;\mathbb{Q})\to H_k(M;\mathbb{C})$$ But I don't see how. From what I know we would ...
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Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
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1answer
74 views

Splitness of quotient sequence

Let $A, B, C$ be holomorphic vector bundles over some complex manifold $X$. Let $A', B', C'$ be sub bundles, respectively. Suppose that we have short exact sequences: $$0 \rightarrow A \rightarrow B ...
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about chain nullhomotopy and projective objects

Let $\mathcal{A}$ be any abelian category and let $Ch(\mathcal{A})$ denote the category of chain complexes on $\mathcal{A}$. Let $P\in Ch(\mathcal{A})$ be a projective object. How can I show that any ...
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1answer
38 views

Isomorphism on Cohomology implies isomorphism on homology

Say I am given a chain map $f:C \to D$ of complexes of (free if necessary) abelian groups. Assume that this map induces isomorphisms of cohomology with all coefficient rings. How do you prove that ...
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1answer
37 views

Bounded derived category and hereditary categories

Let $\mathcal{A}$ be an abelian category with enough projectives (injectives). I tried to prove that if every element $M$ of $\mathcal{D}^{b}(\mathcal{A})$ satisfies $$ M \cong \bigoplus_{i} ...
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28 views

Is the BGG category $\mathcal{O}$ a Serre subcategory of $\mathfrak{g}$-mod? [duplicate]

Let $\mathcal{O}$ be the BGG category for a be a finite-dimensional, semi-simple complex Lie algebra $\mathfrak{g}$. Let $\mathfrak{g}$-mod be the category of all $\mathfrak{g}$-modules. Is the BGG ...
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64 views

Homology of the $n$-torus using the Künneth Formula

I'm trying to apply the Künneth Formula $$H_{n}(X \times Y) \simeq \displaystyle \bigoplus_{r+s=n} H_{r}(X) \otimes H_{s}(Y)$$ to compute the homology groups of the $n$-torus. For the double torus, ...
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Weibel IHA exercise 1.2.6 : Example of total complex

1.2.6 is below; Give examples of (1) a second quadrant double complex C with exact columns such that $Tot^{\prod}(C) $ is acyclic but $Tot^{\oplus}(C)$ is not; (2) a second quadrant double complex ...
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1answer
46 views

Ext functors and extensions.

Given two short exact sequences, $$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 $$ $$ 0 \to B \xrightarrow{\alpha} Y \xrightarrow{\beta} X\to 0$$ I want to show that if the short exact ...
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1answer
22 views

The existence of a non-split composition series in a indecomposable module

Assume that $R$ is a ring with unit and $M$ is a indecomposable left $R$-module with finite length. That is, $M$ has a composition series. Is it true that there is a composition series ...
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1answer
98 views

Computation of $\mathrm{Ext}^2_{\mathbb{C}[x,y]}(\mathbb{C}[x,y]/(x^2,xy,y^2), \mathbb{C}[x,y]/(x,y))$

I need to evaluate left derived functors of $\mathrm{Ext}^2_{\mathbb{C}[x,y]}(\mathbb{C}[x,y]/(x^2,xy,y^2), -)$ on $\mathbb{C}[x,y]/(x,y)$ but i have no idea how to evaluate zeroth functor.. I wrote ...
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72 views

Showing epimorphism without using the Freyd-Mitchell Embedding Theorem

In an Abelian category $\mathscr{C}$ consider a commutative diagram as follows: $$\require{AMScd}\begin{CD} 0@>>>\ker f@>\theta>>W @>{f}>> Y\\ @. @. @V{\phi}VV @|{id} \\ @. ...
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1answer
29 views

$\operatorname{Ext}^{n}$ as the class of Yoneda extensions of degree $n$.

Given an abelian category $\mathcal{A}$, we can define $\operatorname{Ext}^{n}(A,B)$ as the class of extensions of degree $n$ of $A$ by $B$. How can one prove that $\operatorname{Ext}^{n}(A,B)$, is ...
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1answer
18 views

Extending Semiring $\mathbb{N}$ to $\mathbb{Z}$ through exact sequence

I am working on extensions in the form of $$A\hookrightarrow B\twoheadrightarrow C$$ in my thesis and I am just wanting to add as an extra note, IF POSSIBLE, this. We have that that $\mathbb{Z}$ is ...
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1answer
18 views

Module induced from projective is projective

Let $A,B$ be rings such that $A$ is $B$-module, $P$ be projective $B$-module. I want to prove that $A\otimes_B P$ is projective. I have that $\mathrm{Hom}_A(A\otimes_B P,M) \simeq ...
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The bigger picture the Five Lemma fits into

The Five Lemma is a statement in category theory about certain conditions under which certain maps in exact sequences are isomorphisms. It has a few relatives like the 4 lemmas and maybe the Nine ...
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1answer
38 views

Translation functor in a triangulated category under certain hypotheses

Let $\mathcal{T}$ be a $\Bbbk$-linear triangulated category which is Hom-finite and Krull-Schmidt, with translation functor $\Sigma$ satisfying $\Sigma^2 = \text{id}$. Suppose that $\mathcal{T}$ has ...
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Yoneda extensions and $\operatorname{Ext}$ functor.

I am reading this this entry http://stacks.math.columbia.edu/tag/06XU of the Stacks Project. I'm having problems in understanding how $\left(L^{-i+1}\oplus A\right) / L^{-i}$ is constructed. I mean, ...
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79 views

What comes after diagram chasing?

An early edition of Lang's algebra textbook gives the famous exercise to Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book. Here ...
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1answer
20 views

Show that the maps are chain homotopic

Let $\Delta _{2}$ be a 2-simplex, $I=\left [ 0,1 \right ]$. Given are two maps $i_{0}:\Delta _{2}\rightarrow \Delta _{2}\times I$, defined by $x \mapsto (x,0)$ and $i_{1}:\Delta _{2}\rightarrow ...
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A direct summand of a sequence, Rotman, Homological Algebra, ex. 10.15 [duplicate]

If $0 \rightarrow A' \xrightarrow{\delta} A \rightarrow A'' \rightarrow 0$ is a split short exact sequence in an abelian category $\mathcal{A}$ (if you like, let $\mathcal{A}$ be the category of ...