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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Hochschild homology: change of ground ring

Theorem 9.1.7 in Weibel's homological algebra reads as follows (I will change the notation slightly): Let $f:k\to \ell$ be a morphism of commutative rings. Denote $\otimes = \otimes_k$. Let $A$ be a ...
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Getting the most general form of Mayer-Vietoris from the axioms of homology

I'd like to derive the most general form of the Mayer-Vietoris sequence from the Eilenberg-Steenrod axioms for homology (in particular: I do not want to use the definition of $H_\ast(X)$ in terms of ...
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41 views

Unnecessary assumptions in the four lemma?

https://en.wikipedia.org/wiki/Five_lemma The four lemma as described here requires that both rows are exact. But the diagram chase only seems to use exactness at two slots (C' and D in their ...
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Proving that $(Tor_n(\_\ ,N))_n$ is a universal homological $\delta$ functor

Problem: Let $N$ be a left $R$-module, for some ring $R$. Let $T_n$ denote $Tor^R_n(\_\ , N)$. Let $(S_n)$ be another homological delta-functor from $mod$-$R$ to $Ab$, with a natural transformation ...
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Is there an example of a ring which has weak global dimension 2 and not coherent?

Is there an example of a ring which has (weak) global dimension 2, and not coherent? We know that there exist coherent rings with weak global dimension 2 there are also Noethrian rings of global ...
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47 views

Prove that $\operatorname{Ext}^{d+1}(A, B)\cong \operatorname{Ext}^1(M_d,B)$

So, given a resolution, with $P_{i}$ projective modules: $$0\longrightarrow M_d\longrightarrow P_{d-1} \longrightarrow \cdots \longrightarrow P_0 \longrightarrow A\longrightarrow 0,$$ I'm trying to ...
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1answer
32 views

Kernel of a natural map is a direct summand of the covariant extension

I am reading chapter 2 of 'Homological Algebra' by Cartan and Eilenberg. 1/ Given a ring homomorphism $\varphi: \Lambda \rightarrow \Gamma$ and a right $\Gamma$-module $A$, we can treat $A$ and ...
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40 views

The definition of syzygies - free or projective?

For a module $M$ there is always a surjection $F \to M$ with $F$ free. As free modules are projective, there is always a surjection $P \to M$ with $P$ projective, and one may form the first syzygy ...
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Classes of modules of grade higher or equal than $n$

Good morning. For every module $N$ over a ring $R$, it is defined the grade of $N$ as $j_{R}(N)=\min\left\{i:Ext^{i}_{R}(M,R)\neq0\right\}$. In the book "Zariskian Filtrations" by Li Huishi and F. ...
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Isomorphic kernels imply pullback?

In Hilton/Stammbach's A Course in Homological Algebra, they are treating the Ext functor, and they give the following lemma: [][2 He implies (but doesn't say) that the same is not true if we ...
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A categorical perspective on the equivalence of sheaf cohomology and Cech cohomology?

In the nLab article on cohomology, I found the following passage. One can then understand various "cohomology theories" as nothing but tools for computing $\pi_0 \mathbf{H}(X,A)$ using the known ...
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Motivation for the nLab's definition of cohomology?

I am trying to penetrate the nLab article on cohomology. I don't know anything about higher category theory, but it seems like the real content here is topological. My question has two parts. First, ...
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Proving that Tor is a balanced functor using the derived category

At this end of this expository article on derived categories, R.P. Thomas says the following. There are two main advantages of this approach. Firstly that we have managed to make the complex ...
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1answer
28 views

How do you break up an exact sequence of any length to a “succession of short exact sequences”?

Note that if $\text{Hom}_R(D,-)$ functor takes short exact sequences to short exact sequences then it takes exact sequences of any length to exact sequences since any exact sequence can be broken ...
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36 views

Morphisms between long exakt sequences

I have a commutative diagram of modules of the form $$\require{AMScd} \begin{CD} @. @VVV @VVV @VVV @VVV @. \\ ... @>>> A_n @>>> B_n @>>> C_n @>>> A_{n-1} ...
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50 views

Coproducts and products are same in any preadditive category

Here is the proof that coproducts and products are same in any preadditive category from the Stack project. I have few questions regarding the above proof. I don't understand what do they mean by ...
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22 views

Equivalent roof diagrams - Gelfand-Manin seems to overcomplicate something. Or maybe I'm wrong.

I am reading Gelfand-Manin, and am a little confused about their proof that the equivalence relationship between roofs in the localization of a category $B$ at a localizing class of morphisms. In ...
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20 views

Divided Power Structure on Homology

Let $(R,\mathfrak{m},k)$ be a commutative local noetherian ring. Let $X$ be a Tate resolution of $k$ with filtration $X^0\subseteq X^1\subseteq \ldots $, i.e., $X^n=$the algebra obtained by adjoining ...
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36 views

System of Divided Powers on $\mathrm{Tor}^R(k,k)$

I was reading Gulliksen and Levin's (GL) text Homology of Local Rings, and I have a question about something in Chapter 2. Given a local commutative ring $(R,\mathfrak{m},k)$, they say that ...
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A an Abelian category, define the derived category as $D(A) = Ch(A)[Qiso^{-1}]$. Why do homotopic maps become equal?

I am defining $D(A)$ by the localization of Ch(A) for $A$ an abelian category at the set of quasi-isomorphisms, and $Q : Ch(A) \to D(A)$ is the localization map. (So $D(A)$ for this question means: ...
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Can $Ch(M)$, for $M$ the category of $R$-modules, be a category of $S$ modules? Can it be the category of quasi-coherent sheaves on some scheme?

Let $Ch(M)$ be category of chain complexes of modules in $M$, for $M$ the abelian category of $R$-modules (and $R$ a commutative unital ring). Can $Ch(M)$ be a category of $S$ modules for some (not ...
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43 views

Splitting lemma for many (at least 3) components

I am interested in such version of splitting lemma: So given short exact sequence $\hskip2.5in$ we have three equivalent statements: short exact sequence is right split, i.e there is map $t: ...
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21 views

Reference for homological algebra in abelian categories? [duplicate]

What would be nice references to learn about homological algebra in the context of abelian categories? I have already some background in category theory however that would be my first contact with ...
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40 views

Contractible chain complex defined as a direct sum

Let $\mathcal{A}$ be an algebra over a field $F$. Consider a chain complex $(X,\delta)$ if $\mathcal{A}$-modules. Denote $S_n(X) = X_{n-1} \oplus X_n$ and $D: F_n(X) \rightarrow F_{n-1}(X)$, defined ...
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Calculating $\operatorname{Ext}$ in special cases.

Is there a set of "methods" for calculating $\operatorname{Ext}$ in some special cases? For instance, I would be interested in calculating $\operatorname{Ext}_{\mathbb{Z}}^n (\mathbb{Z}/4\mathbb{Z}, ...
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44 views

$\operatorname{Ext}^{1}(M,R/m)=0$ implies $\operatorname{Tor}_{1}( M,R/m)=0$

Let $(R,m)$ be a commutative local Noetherian ring and $M$ a finitely generated $R$-module. I want to show that $\operatorname{Tor}_{n+1}(M,R/m)=0$ if and only if ...
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45 views

Homological Algebra - Tor

I am trying to prove the following: If A and B are abelian groups with mA = 0 = nB, where (m, n) = 1 , Then $Tor_{1}^{\mathbb{Z}}\left( A,B \right)=0$. Conclude that, in this case, exactness of $0\to ...
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Find all abelian groups that fit in a given short exact sequence.

I have to find all abelian groups that can appear in this short exact sequence. $0\rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Z}\oplus\mathbb{Z}_5 \rightarrow 0 $ First of all since ...
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52 views

Calculation of Group Cohomology of $\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{Z}$

I am trying to learn some group cohomology and I'm starting to get my head around the theory, but I find it hard to find some explicit examples of the calculation of group cohomology of some small ...
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39 views

Morphism in derived category

If f=0 in D(A), then $H^n(f)=0$ for all n. I am doing a exercise that shows the converse statement is not true. This is exercise 1 in page 163 of Gelfand's Methods of Homological algebra(2nd edition). ...
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Any module has a flat resolution of length 2?

Is it true that any module has a flat resolution of length 2? I mean if $A=B/C$, both $B$ and $C$ are flat module. I actually want to use free resolution of A, but the image of free module is ...
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Relation between homotopy and homology groups of realization of simplicial abelian group

Let $X$ be a simplicial abelian group, $U(X)$ the corresponding simplicial set and $|U(X)|$ its geometric realization (assumed to be path-connected). Then, for $k \geq 1$, how are $\pi_k(|U(X)|)$ and ...
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Finite dimensional algebras with finite global dimension.

Let $A$ be a finite dimensional $k$-algebra, $k$ is a field, with a finite global dimension. I wonder if that implies $A$ is tame or finite type? or more generally is there a relation between these ...
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Isomorphism between morphisms of derived category and homotopy category

This is exercise 5.1 of chapter3 of Gelfand's methods of homological algebra. I want to show $Hom_{K(A)} (X^*,Y^*)$ and $Hom_{D(A)} (X^*,Y^*)$ is isomorphic if $Y^*$ is in $ ObKom^+(I)$, the set of ...
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functor $F$ satisfies $kerF(f)\cong F(ker(f))$. $0\to M'\to M\to M''$ exact => $0\to F(M')\to F(M)\to F(M'')$ exact?

$R$ ring, $R-MOD$ is the category of $R-$modules anf $F:R-Mod\to R-Mod$ a functor such that the induced map$$Hom(M,N)\to Hom(F(M),F(N))$$ is a homomorphism of abelian groups. $F$ satisfies ...
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1answer
21 views

Definition of bar resolution

I am reading Weibel's book about homological algebra. In section 6.5, the bar resolution, he uses some notation I really do not understand. So given G, a group, what is $[g_1\otimes...\otimes g_n]$? ...
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Group cohomology as a right derived functor

In this Wiki page, it says that group cohomology can be defined as right derived functor of $F$, where $F(M)=M^G$. There are two different equivalent definition in the page, by explicit cochains and ...
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Unknown symbol for homology and surfaces

I am doing some homework currently and I came across a symbol I don't recognise. The question itself is about the euler characteristic of a space and various statements to prove. I don't want help ...
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1answer
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I want to show the group cohomology $H^{n>1}\left(F,\,M\right)$ vanishes whenever $F$ is free.

I want to show the group cohomology $H^{n>1}\left(F,\,M\right)$ vanishes whenever $F$ is free. I tried to show $\text{pdim}_{\mathbb{Z}\left[F\right]}\left(\mathbb{Z}\right)\le 1$, but we know ...
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49 views

Snake Lemma: “natural” or “functorial” [duplicate]

There are some really good discussions of the meaning/ambiguity of the term "natural" here and here. One thing I didn't quite get from those answers: when we say the Snake Lemma, for instance, is ...
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About a Corollary of Yoneda's Lemma

I am reading Assem-Simson-Skowronski's book "Elements of The Representation Theory of Associative Algebras". I do not understand a Corollary 6.2, (IV. 6.2, Functorial Aproach to almost split). It says ...
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Understanding the maps in the long exact sequence of $\operatorname{Ext}$

Suppose I have a short exact sequence in an abelian category (say abelian groups for simplicity) $$0 \to B \to X \to A \to 0.$$ If I apply $\operatorname{Ext}^*(C, \bullet)$ to this sequence, I get a ...
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1answer
48 views

Definition of $E^{\infty}_{pq}$ terms in a spectral sequence. Something strange seems to happen

I'm trying to prove the following assertion in Weibel's Homological Algebra page 125, 5.2.8 Given a homology spectral sequence, we see that each $E^{r+1}_{pq}$ is a subquotient of the previous ...
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Given $A \overset{f}{\to}B$, what measures how hard it is to factor through $f$?

Suppose $A \overset{f}{\to} B$ is a morphism in a category (say for ease they are abelian groups). When we consider other morphisms out of $A$ (say to a fixed object $C$), suppose we ask which maps ...
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1answer
89 views

Given undergraduate Algebra background, which introductory Homological Algebra textbook?

I have read the answer for graduate-level Algebra background and all answers in stackexchange and mathoverflow discussing Homological Algebra textbooks. But none of them directly answers my question, ...
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16 views

Question about adjoint functors

I am trying to do the Exercise 2.3.7 in Weibel's "An introduction to homological algebra". Let $A\in\mathcal{A}$ and $F\in\mathcal{A}^{I}$. Construct an map ...
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19 views

Caracterization of pure submodule by Hom functor

Let $M$ be an $R-$module and $N$ is a submdule of $M$. $N$ is said to be a pure submodule of $M$ if the sequence $$0\rightarrow A\otimes N \rightarrow A\otimes M$$ is exact for every $R-$module $A$. I ...
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43 views

Is $H_n(X)$ just a different way of writing$H_n(S_*(X))$?

While studying homology in algebraic topology, I sometimes see the notation $H_n(S_*(X))$, and sometime the notation $H_n(X)$. I think these are supposed to be the same, but I'm not sure. The first ...
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Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.

Suppose $X$ is a scheme and $Y$ and $Z$ are closed subschemes such that $Y$ is a finite union of closed reduced irreducible subschemes $C$ that satisfy ...
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Two-sided bar construction for algebras: $B_*(M,R^e,R)$ is quasi-isomorphic to $M\otimes_{R^e}B(R,R,R)$

Let $k$ be a commutative ring. Let $R$ be a $k$-algebra. Weibel defines a notion of "relative" Tor in his intro to homological algebra book. For a right $R$-module $M$ and a left $R$-module $N$, he ...