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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Isomorphisms between endomorphism algebras

Assume that $R$ and $S$ are associative $\mathbb{C}$-algebras with unit $1_R$ and $1_S$, respectively. In addition, assume that $_RM$ is a simple left $R$-module and $_SN$ is a simple left $S$-module. ...
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1answer
18 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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+100

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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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 ...
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605 views

Homology commutes with direct sum and product?

I'm looking at exercise 1.2.1 from Weibel's An Introduction to Homological Algebra. (I need to show that homology commutes with direct sum and direct product.) Is it possible to show that ...
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68 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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80 views

Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
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22 views

Split Lie algebra extensions?

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras. A Lie algebra extension is a short exact sequence $$0\longrightarrow \mathfrak{h}\stackrel{\jmath}{\longrightarrow} ...
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1answer
26 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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15 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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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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41 views

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
70 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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37 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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22 views

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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198 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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253 views

Homology out of Smith normal form: simultaneous or independent diagonalization?

Let $R$ be a PID (such as $\mathbb{Z}$) and $R^m\overset{A}{\longrightarrow} R^n\overset{B}{\longrightarrow} R^o$ matrices with $BA=0$ and Smith normal forms ...
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206 views

Computing (the ring structure of) $\mathrm{Ext}^\bullet_R(k,k)$ for $R=k[x]/(x^2)$

Let $k$ be some field (say of characteristic zero, if it matters) and define $$R=k[x]/(x^2).$$ I want to compute $$\mathrm{Ext}^\bullet_R(k,k)$$ and, in particular, the ring structure on it (though I ...
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66 views

Why is the cohomological $\delta$-functor for $H^\ast(K(\pi,1),–)$ effaceable?

Let $\pi$ be an abelian group and $\mathbf Z[\pi]$ its group ring. One obtains a cohomological $\delta$-functor from the short exact sequence of $\mathbf Z[\pi]$-modules $$0\rightarrow A\rightarrow ...
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Prove that: $0\rightarrow B' \overset{i}{\rightarrow} B \overset{p}{\rightarrow} B'' \rightarrow 0$ is a split short exact sequence [closed]

Suppose map $i^{*}$ such that $Hom_{\mathbb{R}}(B,M) \overset{i^{*}}\rightarrow Hom_{\mathbb{R}}(B',M)$ is surjective for every $M$. Prove that: $0\rightarrow B' \overset{i}{\rightarrow} B ...
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1answer
51 views

Why $\bar{d_0}$ is injective in a minimal free resolution? [closed]

If $(R,\mathfrak{m},k)$ is a Noetherian local ring, $A$ a finite $R$-module. Let $L.:\cdots\rightarrow L_1\xrightarrow{d_1} L_0\xrightarrow{d_0} A\rightarrow 0$ be a minimal free resolution. ...
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19 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 ...
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1answer
113 views

If two maps induce the same homomorphism, then they are homotopic

If two chain maps $f,g:\mathcal{X} \rightarrow \mathcal{Y}$, where $\mathcal{X},\mathcal{Y}$ are chain complexes with free modules $X_p$ and $Y_p$ over a PID, $R$, induce the same homomorphism in the ...
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28 views

if $F_{\bullet}$ is a complex and $r$ an integer, what is $F_{r-\bullet}$?

While reading the paper Some results and questions on the Castelnuovo-Mumford regularity, by Marc Chardin, I encountered in the proof of Theorem 5.1 the notation $F^N_{r-\bullet}$. To provide some ...
2
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1answer
49 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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32 views

Extension Operator.

I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet $X\subseteq C(X)$ ...
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223 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ...
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Calculate $\operatorname{RHom}$ in a the derived category of graded $\mathbb{C}[x]$-Modules

I was trying to do the following exercise. Consider the category of graded $\mathbb{C}[x]$-Modules, it is clear that we can regard $\mathbb{C}[x]$ as a graded module setting $\operatorname{deg}(x)=1$. ...
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1answer
57 views

Definition of $Hom(A,B)$

I have lots of confusion about definition of $Hom(A,B)$. I would like to ask several questions with my thoughts. Hopefully I could solve my problem. -Firstly, my book write that if $A$ and $B$ is ...
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30 views

Vanishing of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $I$ be an ideal of $R$ and $N$ be an $R$-module. Let $M$ be a projective $R$-module. Is it true that $H_{I}^i(M‎, ‎N)=0$ for all ...
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How injective $\overline{f_p}$ maps $\mathfrak{m}M_{p}/\mathfrak{m}^2M_{p}$ to $\mathfrak{m}L_{p}/\mathfrak{m}^2L_p$?

If $(R,\mathfrak{m},k)$ is a local ring, $A$ a finite $R$-module. Let $L_{\bullet}:\cdots\rightarrow L_1\xrightarrow{d_1} L_0\xrightarrow{d_0} A\rightarrow 0$ be a minimal free resolution. $\bar{d_i}$ ...
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1answer
72 views

Determine $H(\mathbb{R, Q})$ and $H(\mathbb{R, Z})$

I need to determine the relative (singular) homology groups of $\mathbb{R} \text{ mod } \mathbb{Q}$ and $\mathbb{R} \text{ mod } \mathbb{Z}$. Any hints on what I need to know for this question? ...
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1answer
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Construct a free chain complex K

Let $(A_{n})_{n \in \mathbb{Z}}$ be a set of finitely presented abelian groups. Construct a chain complex $\mathbf{K}$, with each $K_{n}$ a free abelian group, such that for each $n \in \mathbb{Z}$, ...
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Why are we interested in cohomology?

I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, ...
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21 views

projective resolution for an $I$-torsion $R$-module

Let $R$ be a commutative Noetherian ring with non-zero identity, $I$ be an ideal of $R$ and $M$ be an $I$-torsion $R$-module. We know that there exists an injective resolution of $M$ in which each ...
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1answer
82 views

Variation on localization of Tor

It is known that Let $R$ be a commutative ring with unit and $S \subset R$ a multiplicative sistem. If $M$ and $N$ are $R$-modules there is a isomorphism of $S^{-1}R$-modules: ...
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1answer
53 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.
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1answer
69 views

Compute Ext with Macaulay2

I want to compute Ext with Macaulay2. I see in the website they write how to do but I can not do. Can anyone help me with an example? For example, let $S=k[x,y,z,t]$. How compute ...
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Is this functor representable?

Fix a group $G_0$ and $R$ a subset of $G_0$. Consider the functor $F$ from $\textbf{Grps}$ to $\textbf{Sets}$, sending every object $G$ in $\textbf{Grps}$ to $F(G)$, the subset of $\varphi \in ...
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250 views

Coproducts and products of modules

I've just looked at the book A Course in Homological Algebra (by Hilton and Stammbach). They show the universal property of the direct sum (coproducts) using injections and the universal property of ...
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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, ...
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Global dimension of Artin algebras over a perfect field

Let $A$ be an Artin algebra over a perfect field $k$. Suppose that the global dimension of $A$ is finite. How one can prove that $$ \operatorname{gl}(A)=\max\{i \geq 0\mid ...
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Homomorphisms of Chain Complexes

Let $(K, d^{K})$ and $(L, d^{L})$ be chain complexes. For $n \in \mathbb{Z}$ define $$ \mathrm{Hom}(K, L)_{n} := \prod_{j \in \mathbb{Z}} \mathrm{Hom}(K_{j}, L_{j+n})$$ and $$ d_{n}^{K,L} \ \colon ...
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A short exact sequence that cannot be made into an exact triangle. (Weibel 10.1.2)

The following exercise is in Weibel Chapter 10. Regard the groups $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ as cochain complexes in degree 0. Show that the short exact sequence $$ 0 ...
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1answer
31 views

Projective Dimension and Schanuel's Lemma

Let $R$ be a ring and $M$ a (say, left) $R$-module of projective dimension $n$. According to Noncommutative Noetherian Rings, any projective resolution of $M$ can be terminated at length $n$, and this ...
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Proof of Birger Iversen “Cohomology of Sheaves” Theorem 6.8

I am having troubles completing the proof of theorem 6.8 (page 44) from Birger Iversen, Cohomology of Sheaves. (pdf here) Previously we had constructed a functor $\rho$ from $K^+(A)$ (the homotopy ...
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51 views

Some question in relative homology

When we consider the pair (X,A) in relative homology, do we assume A is a sub complex of X? And why don't we just consider X/A instead of (X,A)? Is there an better advantage to consider (X,A) ...
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Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker ...
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60 views

Does trivial cohomology imply trivial homology? Does $\operatorname{Hom}(A,\mathbb Z) = \operatorname{Ext}^1(A, \mathbb Z) = 0$ imply $A = 0$?

Is there a topological space $X$ such that $H^i(X; \mathbb{Z}) = 0$ for all $i > 0$, but $H_n(X; \mathbb{Z}) \neq 0$ for some $n > 0$? In his answer to the question Is homology determined ...