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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Research ideas in Homological algebra

I am planning to focus my research on Homological Algebra and related fields. I am on my first year, first semester and currently pursuing courses on Homological Algebra, Algebraic Geometry (first ...
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Computation of Maps Between Sheaves

Problem: Let $X$ be a locally compact topological space, $i:Z \hookrightarrow X$ inclusion of a closed subspace, and $j:U \hookrightarrow X$ inclusion of the complement. I want to compute: ...
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Homomorphisms of modules over a corner ring

Let $R$ be a Noetherian ring and suppose that we can write $1 = e_1 + e_2 + \dots + e_n$ where the $e_i$ are pairwise orthogonal idempotents. Let $S = e_1 S e_1$, and consider the right $S$-modules ...
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Where am I making a mistake with $Ext^1(A,C)$?

I am learning about $Ext^1(A,C)$ and how it forms a group under '$+$', the Baer sum and I am clearly missing the point somewhere. So, let us suppose for simplicity that $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
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For $x\in\operatorname{Ext}_R^1(C,A)$, how to construct an extension $0\to A\to B\to C\to 0$ such that $\partial (id_A)=x$? [duplicate]

Let $R$ be a ring, $C, A$ two $R$-modules. For all $x\in\operatorname{Ext}_R^1(C,A)$ I have to construct a short exact sequence $$0\to A\to B\to C\to 0$$ of $R$-modules such that $\partial(id_A)=x$, ...
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Tensor products of simple modules over algebras

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. We can regard $M$ and $N$ as $A\oplus B$-modules in natural way, namely, $AN=0$ and ...
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Bullet notation

I'm just trying to make acquaintance with homological algebra. I see there the notation $(A_\bullet,b_\bullet)$ as a short notation for $(\dots,A_{-1},A_0,A_1,\dots,\dots,b_{-1},b_0,b_1,\dots)$. ...
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Isomorphic Homology implies Isomorphic Cohomology

If two complexes have isomorphic integral homology, do the dual complexes have isomorphic integral cohomology? I can also assume that the homology, cohomology are finitely-generated abelian groups. ...
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The boundary formula and cohomology of finite groups

I've a very basic notational question on group cohomology. Let $G$ be a finite group and $M$ a $G$-module. For $i\geq 0$, let $P_i=\mathbb Z[G^{i+1}]$ be the free $\mathbb Z$-module on $G^{i+1}$, made ...
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Exact sequence associated to a acyclic complex

I am reading chapter V of 'Homological Algebra' by Cartan and Eilenberg. Regarding a module $A$ as a complex with $A^0=A$, $A^n=0$ for $n\neq 0$ and zero differentiation. The augmentation ...
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The Ext-functor and indepenence of resolution

Recall that $\text{Ext}_A^1(M, N)$ is in one-to-one correspondence with equivalence classes of extensions $$0 \to N \to - \to M \to 0.$$ (Ignore Baer sums for now, use them in your answer if strictly ...
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How to compute the homomorphism module?

I want to compute the homomorphism module $\textrm{Hom}_{\mathbb{Z}}(\mathbb{Z} /{p^{n}}, \mathbb{Z} /{p^{m}})$ for $m\leq n$. Can someone please help me!
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Exercise 2.7.3 of Prof. Weibel H-Book is wrong. Suggestion for an errata.

In this exercise, we have to prove that there is an isomorphism $$\text{Hom}(\text{Tot}^{\oplus}(P\otimes Q),I)\cong \text{Hom}(P,\text{Tot}^{\prod}(\text{Hom}(Q,I))$$ of double complexes. But if I ...
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Exercise 2.7.1.3) in Weibel's H-book

In exercise 2.7.1.3), Prof. Weibel asks to show that $\text{Tot}^{\oplus}(D)$ is not acyclic if we follow his own errata sheet for his book An Introduction to Homological Algebra 1995 edition ($D$ is ...
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projective resolution of finitely generated modules

I am in the condition where I have a noetherian ring $R$ of finite global dimension. Consider the category of finitely generated (right) modules over $R$. Then I want to show that every module admits ...
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Apply the functor $Hom(-,B)$ to the following exact sequence

Source: Weibel, Page 94. Given an ideal $I$ in a ring $R$, we have the exact sequence: $$0\rightarrow I \rightarrow R \rightarrow R/I \rightarrow 0,$$ so if we apply the contravariant left exact ...
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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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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 is not coherent?

Is there an example of a ring which has (weak) global dimension 2, and is not coherent? We know that there exist coherent rings with weak global dimension 2 there are also Noetherian rings of ...
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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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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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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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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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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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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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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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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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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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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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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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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 $\operatorname{Ext}_{\mathbb{Z}}^n (\mathbb{Z}/4\mathbb{Z}, ...
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$\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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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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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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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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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 ...