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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357 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
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Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
2
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49 views

Derived functors and coboundary operator

I understand that one can define the cohomology of an object $A$ in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ...
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72 views

Dedekind ring characterization via projective modules

I am looking for a book or course notes proving the following result: Let $R$ be an integral domain. Then $R$ is a Dedekind ring if and only if every submodule of a projective $R$-module is ...
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Uniqueness Theorems in Axiomatic Homology Theory

Milnor states in his paper 'On axiomatic homology theory' the following uniqueness theorem: If $H$ is a homology theory (in the sense of the Steenrod-Eilenberg axioms) on the category $\mathscr{W}$ ...
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Rank of homology group

$x_0$ is the unique a global minimum and let $c=f(x_0)$ in a Hilbert space, let $\theta$ be an other critical point of $f$ non minimum. the Morse type number of $x_0$ is ...
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free resolution of koszul complex.

Can anyone prove the following using spectral sequence? Let $(f_{1},f_{2},\ldots,f_{n})$ be a regular sequence. Prove that $K(f_{1},f_{2},\ldots,f_{n})$ is a free resolution of ...
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38 views

Is the category of chain complexes complete and cocomplete in small?

Does the category of chain complexes (let's say of modules over some ring) have all small limits and colimits? What I understand is that the category of chain complexes is certainly finitely ...
3
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Free DG modules

Let $A$ be a DG algebra and $f : F \to M$ a morphism of DG $A$-modules such that $F$ is free and the induced map $H^{\bullet}F \to H^{\bullet}M$ vanishes. Does it follow that $f$ is nullhomotopic? My ...
3
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1answer
117 views

Proof of the five lemma

How to do this using the snake lemma? this is an exercise in Lang's Algebra book. It should somehow be obvious, but I don't see it
3
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37 views

Global dimension of Artin algebras over a perfect field

Let $A$ be an Artin algebra over a perfect field $k$. Suppose that the global dimension of $A$ is finite. How one can prove that $$ \operatorname{gl}(A)=\min\{i \geq 0\mid ...
3
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35 views

Question on differential modules

Let $A,B$ be differential modules with differentiation homomorphism $d$ (such that $d^2=0$). Then let say that $g$ is an epimorphism from $A$ into $B$. Then is it possible for an induced homomorphism ...
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Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
3
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57 views

Relative homology

Let $E$ be a real banach space if $E=Y\oplus Z$ and if $S^{m-1}$ is the sphere on $Y$ ($\dim Y =m $) why $H_{m-1}(E \setminus Z)\simeq H_{m-1}(S^{m-1})$ ? please thank you
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2answers
66 views

Question on relative homology

i have this: where $|\tau|$ is the support of the chain $\tau$, i don't understand the first part why $[\sigma]=0$ in $H_{m-1}(\phi^{c+\varepsilon},\emptyset)$ ??? Please, thank you.
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relation of kernel in commutative diagram

Suppose $A,B,C$ are modules in above commutative diagram, $f$ is surjective. I read that $\ker h / \ker f \simeq \ker g$. I proved it after some diagram chasing. ($f$ induces a surjective map $f': ...
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101 views

Intuitive Understanding of Projective Modules

I was wondering if anyone could give me any sort of intuitive explanation of what a projective module is or a useful way to think about them. I know the definition(s) in terms of lifting, split exact ...
3
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1answer
183 views

A problem about an $R$-module that is both injective and projective.

Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M= \left \{ 0 \right \}$. This is exercise 7.52 of Rotman's Advanced ...
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1answer
52 views

Essential extension.

I'm trying to solve this question. My TA told me that it was easy and the information/assumption given is useless. Question We have the following inclusions of $R$-modules $M\subseteq N \subseteq ...
2
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1answer
57 views

A module with 300 elements

I have got this problem. Let it be $R=M_{2}(Z)$ the ring of square matrices over the integers. I need to find a $R-$module with $300$ elements and one question for this problem, can be there a ...
5
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2answers
161 views

Why is $ \hbox{Ext}_R^* (M,M) = H^*(\hbox{Hom}_R^*(P^*,P^*))$?

Let me first fix some notation and conventions. Let $ R$ be a ring and $ M$ a left $R$-module. Given chain complexes $P^*$ and $Q^*$ in $R$-mod, define $ \hbox{Hom}^*_R(P^*,Q^*)$ to be the graded ...
4
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98 views

On a commutative diagram

Let a commutative diagram: \begin{array}{ccccccccc} 0 & \longrightarrow & A & \overset{f}{\longrightarrow} & B & \overset{g}{\longrightarrow} & C & \longrightarrow & ...
2
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1answer
65 views

Definition of (co)homology of groups and Lie algebras: actions and augmentations

In the Chevalley-Eilenberg chain complex, what is $ux_i$? What does "trivial $\frak{g}$-module $k$" mean? Below I denote $R=k$ (any commutative unital ring). How is the augmentation (last map in the ...
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1answer
74 views

Exact sequence of $R$-modules

Let $0\longrightarrow N\overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}L\longrightarrow0$ be a short exact sequence of $R$-modules. Prove that this chain splits iff $f(N)$ is direct ...
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36 views

Simplicial Chain Complex and shifting.

Let's say that we have the following simplicial chain complex, $0 \xleftarrow{d_0} C_0 \xleftarrow{d_1} C_1 \xleftarrow{d_2} ... \xleftarrow{d_k} C_k \xleftarrow{d_{k+1}} ... \xleftarrow{d_n} C_n ...
0
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1answer
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If $A \cong A^*$, is every projective module also injective?

Suppose $A$ is a finite-dimensional algebra over $k$. Assume further that $A \cong A^* = \text{Hom}(A,k)$ as $A$-modules. My question is: is every finite dimensional projective module over $A$ ...
3
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1answer
231 views

Prerequisites for studying Homological Algebra

I have read the answers here and here and need to ask something more. I wish to study the book on Homological Algebra by Weibel but am not sure of the prerequisites. In particular how much ...
2
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2answers
52 views

Showing that a functor is Exact

Let $R$ and $S$ be rings, and let $F : Mod(R) \to Mod(S)$ be a functor which sends zero to zero. Given that $F$ is exact on short exact sequences (that is with zeros on two ends, and 3 nonzero terms ...
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1answer
50 views

Homology groups

I have to compute the groups $H_q(S^{3},S^1)$ (Singular Homology) I am new in the subject, i have compute some basics groups, but i dont know how to start with this one, if someone could help me, ...
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1answer
42 views

An exact homology sequence associated with a principal SO(n) bundle

Suppose $P$ is a principal $SO(n)$ bundle, X is its base space. Why is there an exact sequence in homology groups $$ 0 \to H^1(X;\mathbb{Z}_2) \to H^1(P;\mathbb{Z}_2) \to H^1(SO(n);\mathbb{Z}_2)\to ...
2
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1answer
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Finding $\operatorname{Ext}^{1}(\Bbb Q,\Bbb Z)$

I am trying to compute $\operatorname{Ext}^{1}(\Bbb Q,\Bbb Z)$ explicitely. Using $\Bbb Q/\Bbb Z$ I constructed a natural injective resolution of $\Bbb Z$, and I know that $\Bbb Q/\Bbb Z$ is ...
4
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135 views

Finding example of quasi isomorphism that has no quasi inverse

Between differential graded algebra $V,W$, a chain map $f\colon V\to W$ induces homomorphism between its homology. If this becomes an isomorphism between the homology of $V,W$, call this quasi ...
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“Most important absolute property in mathematics” according to Osborne

The other day I read M. Scott Osborne's book on homological algebra. On page 33 he states the following theorem. $E$ is injective if and only if $E$ is an absolute direct summand, that is, $E$ is ...
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28 views

Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that ...
2
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2answers
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Explicit computation $\operatorname{Tor}(M,N)$

Let $R=\mathbb{C}[t]/t^2$ the ring of dual numbers. Using the homomorphism $\phi:R \to \mathbb{C}=R/(t)$ we have that $\mathbb{C}$ is a $R$-module, infact we have $$\psi: \mathbb{C} \times ...
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1answer
88 views

Direct limit and products

Any of your comments (or if you know a resource which could be handy) regarding this problem would be appreciated: Show that finite products commute with filtered direct limits. Got no idea how to ...
3
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1answer
88 views

Induced isomorphism of homology implies isomorphism with coefficients in any group?

If $\alpha\colon C \to C'$ is a map of chain complexes (of free abelian groups) that induces an isomorphism on homology $a_{*} \colon H_n(C) \simeq H_n(C')$, then I know that $\alpha$ induces an ...
0
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1answer
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Chain homotopy and compositions of morphisms.

Show that if $\alpha_1 \sim \beta_1$ and $\alpha_2 \sim \beta_2$ , then (whenever composition makes sense) $\alpha_1 \circ \alpha_2 \sim \beta_1 \circ \beta_2$. I have two questions. So are these ...
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33 views

Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
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The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
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1answer
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Example on relative homology

I am trying to prove that $$H_p(B_{n+1},S_n;\mathbb{A}) \cong \left\{\begin{array}{ll} H_{p-1}(S_n,\mathbb{A}) & \text{if } p\geq2\\\ 0&\text{if } p=1, n\geq 1\\ \mathbb{A} &\text{if } ...
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1answer
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Question on relative homology [duplicate]

i have that $H_p(X,Y)$ is isomorphic to $Z_p(X,Y)/(B_p(X)+C_p(Y))$, where $Z_p(X,Y)=\lbrace \sigma\in C_p(X), \partial\sigma\in C_{p-1}(Y)\rbrace$ and i want to deduce that $H_0(X,Y)$ is the free ...
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Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
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Can we define the homology of the homology chain complex

Let $(C,\partial)$ be a chain complex where each $C_i$ is an $R$-module (R being a given ring). We know that the quotients $H_i(C,\partial)=\ker(\partial_i)/Im(\partial_{i+1}$ are also $R$-modules. I ...
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111 views

Relative homology and path connected space

I want to prove that if $X$ is a path connected space and if $Y$ is nonempty then $$H_0(X,Y)\simeq 0$$ it is sayed that we have this chain: $H_0(Y)\rightarrow H_0(X)\rightarrow H_0(X,Y)\rightarrow 0$ ...
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Constructing a cochain complex out of a chain complex

Let $(C,\partial)$ be a chain complex where $C_i$ is an $R$-module ($R$ is a given ring) , we can always construct a cochain complex out of the chain complex $(C,\partial)$ in the following way: We ...
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Homology and cohomology are basically the same

Is my following understanding correct: A chain complex $(C,\partial)$ is a family $\{C_i\}_{i\geq 0}$ of $R$-modules ($R$ is a given ring) together with a family of $R$-module homomorphisms ...
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203 views

A functor which preserves short exact sequence also preserves long exact sequence?

Let $F: C \to D$ be a functor between abelian categories (e.g. modules over the same ring), and it preserves short exact sequence, then is it also preserves long exact sequence?
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30 views

What is Kadison's process about cocycles?

My teacher told me the Kadison's process(may be not this ward, it is just my translation ) can make a 2-cocycle turn to be a cocycle(i.e.,derivation). But I can not find it in the internet. Thanks a ...
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What kind of object are the tensor products?

Now I'm reading Sze-Tsen Hu's Introduction to Homological Algebra, and it says the restricted direct sum of the natural projections $$\{p_\mu\otimes q_\nu|\mu \in M,\nu\in N\}$$ defines the ...