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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Cellular homology of the real projective space $\mathbb R P^n$

I've been able to calculate the cellular homology of $\mathbb R P^2$ but I'm struggling to do the same for higher dimensions. My problem is that I don't exactly see how one get to the result $d_i: ...
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Confusion regarding definition of adjoint functor - Hilton and Stammbach

While defining Adjoint functors in their book A Course in Homological Algebra, Hilton and Stammbach said the follwing: Let $F:\mathfrak{S}\rightarrow \mathfrak{M}_{\Lambda}$ be the free functor ...
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counterexample for $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence

I want to understand a counterexample for: Let $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence, because the statement ...
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47 views

Sheaf cohomology of long exact sequence of sheaves

Let $X$ be an algebraic variety and $F_1,...,F_n$ be a collection of coherent sheaves on $X$. Suppose we have a long exact sequence $$0\to F_n\to F_{n-1}\to...\to F_1\to0.$$ Knowing all sheaf ...
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40 views

Translate this theorem from Endliche Gruppen

In a paper I was doing a reference is given from "Endliche gruppen" by Huppert. I do not understand german and google translator was also not much helpful. Can some translate this theorem or much ...
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30 views

show that $H^1(Q,Z(P))=0$

Suppose $P$ is a normal sylow $2$-subgroup of $G$. Now let $\varphi$ is an automorphism of $G$ and $Q=G/P$. Now suppose we have this commutative diagram If $(|P|,|Q|)=1$ then I want to show that ...
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24 views

The Ext-functor and inverible modules

I need some help regarding an argument from the proof of Proposition 4.2.1 in John Rognes article Galois Extensions of Structured Ring Spectra. We are supposed to prove that ...
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27 views

Cohomology induces a functor

Let $\mathcal{A}$ be an abelian category and $(X^{\cdot}, d_{x}^{\cdot})$ a cochain complex in $\mathcal{A}$. Let $Ch(\mathcal{A})$ be the category of cochain complexes in $\mathcal{A}$. I define the ...
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30 views

References on completion and Tor/Ext

I am currently working on a thesis that relates to the Functors $\text{Tor}$ and $\text{Ext}$. I have found some work on localization with respect to them when it comes to information in my books but ...
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54 views

Is the cohomology ring (coefficients in a field) functor right adjoint to something? Or, why does it commute with products?

Take coefficients in a field, so as to not have the correction from Tor. I am thinking about the functor sending a topological space $X$ to its cohomology ring $H^*(X)$. So specifically, I am ...
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23 views

How to find a counter-example that the centralizer of differential graded algebras does not preserve quasi-isomorphism?

Let $A^{\bullet}$, $B^{\bullet}$ be two differential graded algebras (dga) and $f: A^{\bullet}\to B^{\bullet}$ be a differential graded homomorphism between them. Now let $R$ be another algebra ($R$ ...
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Homology with local coefficients as a functor from pointed, path-connected spaces and $\pi_1$-modules.

A local system of coefficients on a space $X$ is a functor $F\colon \Pi(X)\rightarrow Ab$ from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups ...
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37 views

Higher self-extension $\text{Ext}^i_{\mathcal{O}}(L(\lambda), L(\lambda))$ between two irreducible modules in BGG category $\mathcal{O}$

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $\mathcal{O}$ be the BGG category for $\mathfrak{g}$. It is well-known that the set of irreducible ...
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21 views

Calculate cohomology of the special double complex

Let $B,B'$ be finitely generated $R$-modules. So we have two short exact sequences: $$0\to A\to R^n\to B \to 0$$ $$0\to A'\to R^m\to B' \to 0$$ Using these triples i obtain double complex: $$ ...
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85 views

Degeneracies of simplex $y$ which appears as any face of some simplex $x$

Let $K$ be simplicial set and $d_i:K_n\rightarrow K_{n-1}$, $s_i: K_n\rightarrow K_{n+1}$ ($i = 0,...,n$) face and degeneracy maps respectively. Suppose we have some $x\in K_n$ with $d_0x = ... = ...
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47 views

The category of (completable) rings has enough projectives in it

I am working on functors and projective resolutions and of course the issue of "Enough projectives" comes up. I know $R$-modules have enough but I am curious about the category of rings in general? ...
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25 views

Relation between compactly supported cohomology and locally finite homology

Building up on a previous question, I am currently investigating in the properties of locally finite homology. Suppose that $X$ is a reasonably well-behaved space. I want to find out whether there is ...
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$E_{p,q}^r$ spectral sequence. Find a l.e.s. $\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} E_{p-1,1}^2\to A_p\to E_{p,0}^2\to\cdots$

Let $E_{p,q}^r$ be a spectral sequence which converges to $A_n$. Let $E_{p,q}^r=0$ for $q\ge 2$. How to construct a long exact sequence $$\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} ...
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A general definition for a character of a (not necessarily associative) algebra

Let $A$ be a algebra over a algebraically closed field $k$. Is there certain definition of a "character" $f: A \rightarrow k$? That is, what is the common and useful condition for a linear map $f: A ...
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relations between homology and cohomology

Let $p$ be a prime number and $X$ a topological space. Are the following equivalent? (1) In the homology module $H_*(X;\mathbb{Z})$ there does not exist any element of order $p$. (2) In the ...
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Let $G$ be a group. Why is $ \operatorname{Ext}_{\mathbb{Z}G}^1(\mathbb{Z},\mathbb{Z})\cong \operatorname{Hom}_{Grp}(G,\mathbb{Z})$?

Let $G$ be a Group and $\mathbb{Z}G$ is the ring of the formal sums $$\sum_{g\in G}n_gg$$ with multiplication $$(\sum_{g\in G}n_gg)(\sum_{h\in G}m_hh)=\sum_{g\in G}(\sum_{h\in ...
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38 views

Homology of Hom and Hom of homology

In 'Homological Algebra' by Cartan & Eilenberg: (page 203) For complexes $X$ and $Y$, consider the map $\alpha':H^{p+q}(\text{Hom}(X,Y))\rightarrow \text{Hom}(H_p(X),H^q(Y))$. Let $h_1\in ...
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34 views

Group cohomology of Z/2Z

Let $G=Z/2Z=\{1, g\}$. Consider the ring of integer $Z$ with the alternating action of G, i.e., $g\cdot n=-n$ for $n\in Z$ and an abelian group $M$ on which $G$ acts. It is well-known that the group ...
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36 views

Intuition behind $Ext^1(A,\,C)$

So I recently asked a question concering $Ext^1(A,\,C)$ regarding the connection between isomorphism and the congruence '$\equiv$' (Where am I making a mistake with $Ext^1(A,C)$?). Suppose, for ...
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68 views

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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17 views

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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40 views

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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28 views

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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1answer
60 views

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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When do we have $\operatorname{Hom}_{R}(L,M\otimes N)\cong \operatorname{Hom}_R(L,M)\otimes N$? [duplicate]

Let $R$ be a commutative ring and $L,M,N$ be $R$ modules. I would like to know that when is the natural map $$ \operatorname{Hom}_R(L,M)\otimes_R N\to \operatorname{Hom}_R(L,M\otimes_R N) $$ is an ...
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1answer
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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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9 views

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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24 views

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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1answer
34 views

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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1answer
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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24 views

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
31 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 ...
3
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1answer
39 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 ...
2
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46 views

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. ...