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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Complex of banach spaces is exact if and only if its dual is exact

Let's consider two complexes of Banach spaces: $ X \rightarrow Y \rightarrow Z$, with the maps $S: X \rightarrow Y$, $T: Y \rightarrow Z$. The dual complex looks like $Z^{*} \rightarrow Y^{* } \...
5
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65 views

A map of complexes which is zero on cohomology but not zero in $D(\mathcal{A})$

Yesterday I asked a very similar question about an exercise of Gelfand's book "Methods of Homological Algebra". In the comments it was pointed out that there was an easier version of that exercise but ...
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Exercise from “Methods of Homological Algebra” Gelfand

I have to show that a map of complexes $f: A^{\bullet} \to B^{\bullet}$ in $Ab$ with $H^{n}(f)=0$ is not necessarily 0 in the derived category $D(\mathcal{A})$. To find this counterexample I'm given ...
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1answer
17 views

Filtered Colimit of associative $k$-algebras that are domains

Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains. Then the colimit of the obvious diagram is an integral domain. ...
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92 views

The cohomology of the Dirac operator $d+d^{*}$

Let $(M,g))$ be a Riemannian manifold with the Hodge dual operator $d^{*}$. Is there a name (and some computation in some reference) for the cohomology of the complex of Harmonic forms with ...
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1answer
58 views

Chain complex and free resolution

If $I \subset R = k[x_1,\dots,x_n]$ is an ideal. Then why: $0 \to C_i \to \dots\to C_0 \to R \to R/I \to0$ is a free resolution of $R/I $ if and only if $0 \to C_i \to \dots\to C_0 \to I \to 0$ is a ...
2
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1answer
42 views

Regular sequence in degree 1

$R$ is a graded algebra generated by $R_1$(the degree 1 piece) over $R_0=k$ where $k$ is a infinite field and R has no negative degree. Given irrelevant ideal has depth d, then is it possible to find ...
3
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1answer
78 views

Motivation for the mapping cone complexes

I was reading some topics in Homological Algebra when I came across the concepts of cone of a map of complexes and cylinder. My knowledge of Algebraic Topology is pretty basic so I only used these ...
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Eilenberg–Zilber as abstract nonsense - why is it important?

The Eilenberg–Zilber theorem in singular homology, relating the monoidal structure of the category of chain complexes with the chain complex of the cartesian product of the underlying spaces, is used ...
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24 views

How to show that $(\Lambda^2(g))^g = H^2(g)$?

Let $g$ be a semisimple Lie algebra and $\Lambda^2(g) = g \wedge g \subset g \otimes g$ the exterior square of $g$. Consider the adjoint action of g on $g \wedge g$ and let $$(\Lambda^2(g))^g = \{x \...
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Terminology question: “boundary map” in homology theory

In homology theory the same name and symbol is used to describe two different maps: 1: the boundary maps $\partial_n: C_n(X) \to C_{n-1}(X)$ appearing in a chain complex of (say) singular n-simplices....
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38 views

What does formality of a chain complex mean topologically?

I've been told that every topological abelian group is a product of Eilenberg-Mac Lane spaces, but I don't have a reference for this fact. This confuses me because via the Dold-Kan correspondence, ...
2
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1answer
50 views

Is homology of a chain complex a universal delta-functor?

Let $\mathcal{A}$ be an abelian category and let $Ch(\mathcal{A})$ be the category of homologicaly, non-negatively graded chain complexes in $\mathcal{A}$. The sequence of homology functors $H_n:Ch(\...
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47 views

Is the functor category $[\mathsf C,\mathsf{Ab}]$ algebraic?

In this MO question, the functor category $[\mathsf C,\mathsf{Ab}]$ for $\mathsf C$ a small abelian category is examined. Reading about the acyclic models theorem, I'm wondering about the same ...
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71 views

Splitting of Short Exact sequence

Suppose $d_1d_2=n$ and let $0 \to d_1\mathbb Z_n \overset {i} \to \mathbb Z_n \stackrel {d_2\cdot} \to d_2\mathbb Z_n\to 0$ be a short exact sequence. Show that sequence splits iff $\gcd(d_1,d_2)=1$. ...
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35 views

$E_{p, 0}^2$ and $E_{0, 1}^2$ terms in sequence, in terms of homology of $K(G, 1)$, homology of $K(R, 1)$, and action of $G$ on $R$ by conjugation?

This is a followup to my previous question, reproduced here. Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R,...
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Need Counterexample to show Koszul complex is not minimal free resolution?

Recall that Koszul Complex $K.(f,g)$ of polynomials $f,g \in k[x_1,\ldots,x_n]=:R$ is defined as:$$0 \to R \overset{\phi_1} \to R^2 \overset{\phi_2} \to R \to 0$$ where $\phi_1$ and $\phi_2$ are ...
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61 views

Tensoring a connective chain complex with a simplicial set

Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ...
2
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1answer
76 views

Short exact sequence of groups, is it possible to construct an associated fibration of spaces?

Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R, 1) \to K(F, 1) \to K(G, 1)?$$
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1answer
67 views

Universal Property of free objects

I am working on free objects, I am restricting myself primarely to groups, rings and modules (with maybe algebras) so in a sense in the concrete category (if I am not mistaken. This is a thesis work I ...
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2answers
80 views

Show that the $i$th local cohomology functor is zero for $i > 0$

Let $I$ be an ideal of a Noetherian ring $R$, and let $M$ be a module over $R$. Let $\Gamma_I(M)$ be the set of all elements $m$ of $M$ for which $I^n m = 0$ for some $n \geq 1$. Then $\Gamma_I(-)$ ...
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1answer
48 views

Enough Projectives in Category of Groups

Working on homology and completion a question has arisen in my head. I know that $R$-mod as a category has enough projectives in it, and as such the category of abelian groups has it as they are in $\...
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1answer
41 views

A lemma from Hilton & Stammbach's book A Course in Homological Algebra

In orde to prove the set of equivalence classes of extensions of $A$ by $B$ is a contravariant functor of the first component and covariant functor of the second. The authors give us three lemma. I'm ...
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1answer
21 views

Why is $\ker(id\otimes \cdot b:R/(a)\otimes_R R \to R/(a)\otimes_R R)=R/(d)$?

Let $R$ be a commutative ring with unit $1_R$, $M$ a $R$-module. Let $a,b\in R\setminus \{0\}$ and $\gcd(a,b)=d$. I want to prove: $$\operatorname{Tor}_1^R(R/(a),R/(b))=R/(d).$$By definition, it is $\...
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2answers
119 views

Errata in Prof. Rotman AIHA book about projectives in the chain complex category (section 10.5)

EDIT After thinking carefully with the help of the clear answer of ZhenLin, I think I will reformulate my question the following way. The text of my original question is kept below. The claim of Prof....
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0answers
28 views

Global dimension of matrix algebra

I want to calculate the global dimension of this algebra. $$ \quad A = \begin{pmatrix} k & 0 & 0& 0 \\ k & k & 0&0\\ k&0&k&0\\ k&k&k&k \end{pmatrix} \...
2
votes
1answer
39 views

Lifting the projective property through the affine centre

Let $\mathbb{k}$ be an algebraically closed field. There are many interesting examples of $\mathbb{k}$-algebras $R$ which admit a large central subalgebra $Z_0$ such that $R$ is a free $Z_0$-module ...
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1answer
49 views

Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has Tor-...
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1answer
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Connecting homomorphism in Galois homology using the standard resolution

Let $G$ be a finite group, although this may not be necessary for almost everything that follows. One of the ways of defining Galois homology groups is using the standard resolution for the $\mathbb{Z}...
3
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1answer
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$f_*$ induces an isomorphism in homology iff the mapping cone of $f_*$ is contractible.

Let $f_*:C_*\to D_*$ be a chain map. I'm stuck in the proof of the following statement: $f_*$ induces an isomorphism in homology iff the mapping cone of $f_*$, cone($f_*$), is contractible. (For ...
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1answer
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How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [closed]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ H^4(M;\mathbb{Z})=\...
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universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
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1answer
64 views

Simple modules and homological algebra

Let $A$ be a $k$-algebra, and $M$ an $A$-module. If $\mathrm{Ext}_A^{1}(M,S)=0$ for every simple $A$-module $S$, then $M$ is projective. I know that this is true if $A$ is finite-dimensional, but if $...
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1answer
97 views

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: C_{...
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1answer
48 views

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 ...
3
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1answer
68 views

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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2answers
435 views

Positivity of the alternating sum associated to at most five subspaces

Let $V_1 , V_2 , \dots , V_n $ be vector subspaces of $ \mathbb{C}^m$ and let $$\alpha = \sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } \dim(V_{i_1} \cap \cdots \cap V_{i_r})$$ ...
0
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0answers
57 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 ...
0
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1answer
42 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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1answer
35 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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1answer
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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 $\text{Ext}_{R[G]}^s(R,T)=...
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1answer
46 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 ...
3
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1answer
65 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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0answers
29 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 ($R$ ...
4
votes
1answer
62 views

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 ...
2
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1answer
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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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40 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: $$ \...
3
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1answer
93 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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1answer
54 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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43 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 ...