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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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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40 views

Long exact sequence to short exact sequence [duplicate]

$$\dots\xrightarrow{p_*}\pi_{n+1}(B)\xrightarrow{\partial}\pi_n(F)\xrightarrow{\text{inc}_*}\pi_n(E)\xrightarrow{p_*}\pi_n(B)\xrightarrow{\partial}\pi_{n-1}(F)\xrightarrow{\text{inc}_*}\dots$$ From ...
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35 views

$\mathcal{T}^i(X/Y,\mathcal{F})$ forms a sheaf

In Hartshorne's Deformation Theory, given an $A$-algebra $B$ and a $B$-module $M$, he defines these functors $T^i$ for $i=0,1,2$ that outputs $B$-modules $T^i(B/A,M)$. In Exercise 3.5, he asks the ...
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I'm searching for $X$ and $V$ such that the evaluation map $f:H^n(X,A;V)\to \hom_\mathbb{Z}(H_n(X,A;\mathbb{Z}),V)$ is not injective

Let $V$ be a $\mathbb{Z}$-module $(X,A)$ a pair of topological spaces. We define the pairing-map in singular co- homology$$H^n(X,A;V)\times H_n(X,A;\mathbb{Z})\to V$$ $$([\xi ],[\alpha])\mapsto \xi ...
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38 views

Contractibility of an exact chain complex

How can one prove that an exact (acyclic) chain complex of projective modules that is trivial in negative degrees is contractible? I would appreciate some nudges in the right direction more than ...
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48 views

A chain complex is split if and only if it splits as a direct sum.

This is the first part of Exercise 1.4.2 in An Introduction to Homological Algebra by Weibel. The first part is showing that a chain complex, $C$, with boundaries $B_n$ and cycles $Z_n$ in $C_n$ is ...
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30 views

Counterexample to, if $f$ is acyclic then $kerf$ and $cokerf$ acyclic.

This is the second half of exercise 1.3.5 in An Introduction to Homological Algebra by Weibel, it simply asks if this statement is true of false and I believe it is false but cannot construct a ...
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65 views

Why is Hom$_A(M,A)$ a right $\Gamma$ comodule?

I'm reading through appendix I (Hopf algebroids) of Ravenel's green book, and I came across a line I can't understand in a proof. The part of the lemma I'm interested in states: $\mathbf{Lemma ...
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1answer
38 views

Baer Sum notation requires clearence.

I am working on Baer sum and I have my book by Rotman, Introduction to Homology, and also MacLanes book Homology and they use notation I am puzzled on. I have understood baer sum of extensions ...
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24 views

Compute directly that the mapping cone of a homotopy equivalence is contractible

Let's consider the category $Ch_R$ of cochain complexes of modules over a commutative ring $R$. I'm trying to prove that if the chain map $\phi:M\rightarrow N$ is a homotopy equivalence then its ...
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23 views

Automorphism of semisimple Lie algebra corresponding to a simple reflection

Let $\mathfrak{g}$ be a complex, finite-dimensional Lie algebra. Let $\mathfrak{h}\subset \mathfrak{g}$, $W$ and $\Pi$ be a Cartan subalgebra, its Weyl group and the set of all simple roots, ...
3
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1answer
42 views

Baer sum of $\mathbb{Z}_9$ and $\mathbb{Z}_9$

I am working on trying to figure out the third extension of $\mathbb{Z}_3$ by $\mathbb{Z}_3$, I know one is $\mathbb{Z}_9$ and the neutral element (with respect to baer sum) $\mathbb{Z}_3\oplus ...
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1answer
41 views

Getting a double complex that computes Ext

Suppose $C$ is an abelian category and I am trying to compute $Ext^i(M,N)$ for some objects $M,N$. Suppose there is an exact sequence $0 \rightarrow A_1 \rightarrow A_2 ... \rightarrow A_n ...
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51 views

$T^i$ functors in Hartshorne's Deformation Theory

In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a ...
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55 views

An exact sequence of inverse systems of $R$-modules

Let $$0\longrightarrow \big\{A_n,f_{mn}\big\}_{m \leq n} \overset{\Phi}\longrightarrow \big\{B_n,g_{mn}\big\}_{m \leq n} \overset{\Psi}\longrightarrow \big\{C_n,h_{mn}\big\}_{m \leq n} ...
3
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41 views

An identity map which is not null-homotopic

I have some problems in understanding how the definition of a null-homotopic cochain map actually works. Maybe I lack concrete examples. Let $f^{.}:X^{.}\longrightarrow Y^{.}$ a cochain map of ...
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30 views

Analogue of the trivial extension for higher Ext.

I've been doing some homological algebra and some work on showing some extensions are equivalent, and a thought just came to me, which is that I didn't know how to write down what the analogue of the ...
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27 views

Hochschild homology of dgas with nontrivial differential

In this question, we see how to compute the Hochschild homology of a dga with zero differential: it's just the same as computing its Hochschild homology as a graded algebra. I want to know about ...
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1answer
49 views

Is $M_R\otimes _R {_R}N\cong M_{\mathbb Z}\otimes_{\mathbb Z} {_{\mathbb Z}}N$?

Suppose $M$ is a right $R-$module and $N$ is a left $R-$ module. Also $M$ and $N$ are naturally $Z-$ module, both in left and right side. So we will denote $M_R$, $M_{\mathbb Z}$, and $_RN, _{\mathbb ...
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71 views

Exact Sequences in algebraic geometry [closed]

A very basic question. I am going to take my first course in Algebraic Geometry next semester and I am now repeating some commutative algebra to be prepared. I just came up to the part of Homological ...
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113 views

Mayer Vietoris for locally finite singular homology

Usually one defines the traditionnal singular homology (let's say in $\mathbb{Z}$ and on a topological space $X$) by using singular $p$-chains. A singular $p$-chain is a finite formal sum $\sum ...
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23 views

Bockstein homomorphism and the universal coefficient theorem

The following statement is given in the third comment of kernel of the mod $2$ Bockstein on the first cohomology group: Statement: Let $X$ be a path-connected finite $CW$-complex. Suppose $$ ...
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$\text{Hom}(M \otimes_A N, L) \approx \mathscr{L}(M,N; L)$ The $A$-linear homs from the tensor product into $L$ are isomorphic with bilinear maps.

Let $M,N, L$ be two $A$-modules over a commutative ring $A$. Let $\mathscr{L}(M,N;L)$ be the $A$-module of bilinear maps $M \times N \to L$. Then $\text{Hom}_A(M \otimes_A N, L) \approx ...
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27 views

Symbol for exactness in diagram

When making commutative diagrams there are many symbols one can use and even omit in favours of others. For example in the following short exact sequence $$0\to A\to B\to C \to 0$$ we can rewrite it ...
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64 views

In Kunneth Formula for Cohomology, the finitely generated condition is necessary.

Kunneth formula for cohomology: The cross product $H^*(X;\mathbb Z)\otimes H^*(Y;\mathbb Z)\to H^*(X\times Y;\mathbb Z)$ is an isomoprhism of rings if $X$ and $Y$ are CW complexes and $H^k(Y,R)$ is ...
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37 views

Generator of $\tilde{H}_n(S^n;G)$

I'm struggling with the following exercise: Let $\sigma_n: \Delta^n \to\Delta^n/ \partial \Delta^n$ be the quotient map. Show that $[\sigma_n]$ generates $\tilde{H}_n(S^n;G)$ with $G$ any abelian ...
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1answer
52 views

Are those two ways to relate Extensions to Ext equivalent?

Given an extension $\xi$ of $R$-modules $0\to B\to X\to A \to 0$, one usually associates $x\in\operatorname{Ext}^1(A,B)$ by taking the long exact sequence $$\ldots\to \operatorname{Hom}(A,X) \to ...
3
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1answer
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Where can I found an explanation of group cohomology from the point of view of invariants?

I heard once that we can view group cohomology as the right derived functor quantifying precisely (i.e. by the usual long exact sequence) how much the functor of "taking the invariants" is not right ...
2
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35 views

chain map induces an isomorphism in homology, but as a cochain map, $f^*$ does not induce an isomorphism in cohomology

Let $f:(X,A)\to (Y,B)$ be a continuous map of pairs $(X,A), (Y,B)$ of topological spaces. $f$ induces a chain map on singular chain complexes $$f_*:C_*(X,A;R)\to C_*(Y,B;R),\; ...
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1answer
39 views

Finding an example of an extension of length 2

I am working on extensions in general but for sake of simplicity we can assume it's a module here. I am interested in an extension of the form $$0\to B \to E_2\to E_1 \to A \to 0$$ which is an ...
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3answers
128 views

Bijection between Extensions and Ext (Weibel Theorem 3.4.3)

I was wondering about one step in the proof of surjectivity of $\Theta$ constructed for Theorem 3.4.3 in Weibel's "An introduction to homological Algebra". For an extension $\xi:0\to B\to X\to A\to0$ ...
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1answer
31 views

definitition of projective resolution of $R$-modules (with homology)

Let $R$ be a commutative ring with unit $1_R$, $M$ be a $R-$module. I have a small question about different definitions of projective resolutions of $M$ (and I'm confused with the degrees of the ...
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41 views

Group Homology with Coefficients in Chain Complex of trivial G-modules

I am reading through Kenneth Brown's Cohomology of Groups, and right now I am learning about spectral sequences. He seems use to some kind of "intuitive" idea of spectral sequences that I would very ...
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1answer
51 views

What is a direct limit of exact sequences?

(Hatcher Section 3.3, page 243) First, recalling the definition of a directed system of groups: Suppose one has abelian groups $G_\alpha$ indexed by some partial ordered index set $I$ having the ...
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1answer
29 views

Splitting Lemma where $C=\mathbb{Z}.$

Given a short exact sequence $$ 0 \xrightarrow{\theta_3} A \xrightarrow{\theta_2} B \xrightarrow{\theta_1} \mathbb{Z} \xrightarrow{\theta_0} 0 $$ show that $B \cong A \oplus \mathbb{Z}.$ So far I ...
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1answer
38 views

Estimating regularity of sheaves with rank of certain modules and zeroth cohomology

I'm studying Eisenbud's book "Geometry of syzygies", in particular the Gruson-Lazarsfeld-Peskine theorem for Castelnuovo-Mumford regularity. I'm concerned about an intermediate step in the proof. Let ...
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1answer
36 views

Why does the antisymmetrization map factor through $n$-forms?

Consider a $k$-algebra $A$ and a bimodule $M$. One can construct two complexes, the Hochschild complex $C_n(A,M)$ and the Chevalley-Eilenberg complex $C'_n(A,M)=M\otimes \Lambda^n(A)$. Given an ...
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1answer
80 views

A tensor identity - $\text{Hom}_{R}(A,B \otimes_S C) \cong \text{Hom}_{R}(A,B)\otimes _SC$

Let $R,S$ be associative algebras over $\mathbb{C}$. Let $A$, $B$ and $C$ be, a left $R$-module, a $(R,S)$-bimodule, a left $S$-module, respectively. Assume that $B\otimes_S C$ is finite-dimensional. ...
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1answer
50 views

$C_\infty$ analog of the correspondence between $A_\infty$-alg. structures on $A$ and dg coalg. strucures on $(\bar T(sA),\Delta)$

There is a 1-1-correspondence between $A_\infty$-algebra structures on a graded vector space $A$ and dg. coalgebra structures on the bar construction $(\bar T(sA),\Delta)$. My question: Is there any ...
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75 views

Relation between ranks of free sheaves and cohomology

Suppose that $\mathbb{P}^r=\mathbb{P}^r_K$ is the projective space over a field $K$. Let $\mathcal{O}_{\mathbb{P}^r}(-1)^n\longrightarrow \mathcal{O}_{\mathbb{P}^r}^m$ be a morphism of vector ...
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1answer
63 views

Show that a sequence is a free resolution

Let $I \subset R = k[x_1,\dots,x_n]$ be an ideal and $f \in R$ such that $I = \left < f \right >$ ($k$ is a field, so R is commutative ring). How do I show that (1) $I$ has a free resolution ...
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1answer
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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^{* } ...
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1answer
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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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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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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
57 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 ...
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1answer
39 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 ...
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1answer
50 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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1answer
1k views

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 ...