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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9
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625 views

Exactness of the Tensor Functor

This might turn out to be a very stupid question, so I apologize in advance, but it is confusing me a little bit. I know in general that if $$M'\rightarrow M\rightarrow M''\rightarrow 0$$ is an exact ...
1
vote
1answer
54 views

Quick question about chain homotopies.

In the definition of a chain homotopy (say $h$) between two chain maps (say $f$ and $g$), are the maps $h_i$ comprising the chain homotopy required to commute with all other maps involved (the $f_i$s, ...
0
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1answer
116 views

When do we have $m\otimes n = 0$ [duplicate]

Let $M$ and $N$ be $R$-modules ($R$ a commutative ring with identity). Let $m \in M$ and $n \in N$. Is there any necessary and sufficient condition to have $m\otimes n = 0$ (as an equation in ...
1
vote
2answers
84 views

is the pullback of the cohomology of a group to the cohomology of a subgroup surjective?

If $H$ is a subgroup of $G$, is $i^*(H^*(G)$) surjective onto the cohomology of $H$? $i$ is the inclusion of $H$ in $G$.
6
votes
1answer
269 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
6
votes
1answer
146 views

Ext of an $\mathfrak{m}$-primary ideal

Let $(A,\mathfrak m,k)$ be a Noetherian local ring, $M$ a finitely generated $A$-module, and $I$ an $\mathfrak{m}$-primary ideal. If $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m},M)=0$ then ...
3
votes
0answers
76 views

Depth for intersection of prime ideals

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over field $K$. How can one compute $\operatorname{depth}(R/\bigcap_{i=1}^r p_j)$, where each $p_j$ is generated by some variables $x_i$ and have a ...
16
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1answer
714 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 ...
9
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0answers
276 views

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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0answers
107 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 ...
1
vote
1answer
106 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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0answers
53 views

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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1answer
124 views

free resolution of koszul complex. [closed]

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 ...
0
votes
0answers
78 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
votes
1answer
57 views

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 ...
1
vote
1answer
225 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
5
votes
1answer
67 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)=\max\{i \geq 0\mid ...
3
votes
1answer
40 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 ...
7
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0answers
168 views

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
votes
1answer
86 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
0
votes
2answers
100 views

Question on relative homology

I have this: $\ \ $ $3)$ Assume now that each critical point of $\varphi$ in $K_c$ is isolated in $X$. Let $\epsilon>0$ be such that $c-\epsilon >b$ and $c$ is the only critical value of ...
2
votes
0answers
74 views

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': ...
3
votes
2answers
291 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 ...
5
votes
1answer
709 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
68 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
votes
1answer
70 views

A module with 300 elements

I have got this problem. Let it be $R=M_{2}(\mathbf{Z})$ 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 ...
5
votes
2answers
193 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 ...
2
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1answer
78 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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vote
1answer
122 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 ...
0
votes
1answer
66 views

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
votes
1answer
988 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
votes
2answers
63 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
79 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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vote
1answer
90 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
votes
1answer
66 views

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
votes
1answer
475 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 ...
3
votes
0answers
89 views

“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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0answers
39 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
votes
2answers
171 views

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 ...
1
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1answer
109 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
votes
1answer
239 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
votes
1answer
153 views

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 ...
2
votes
0answers
43 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).
10
votes
1answer
387 views

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, ...
0
votes
1answer
283 views

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 } ...
1
vote
1answer
48 views

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 ...
10
votes
1answer
230 views

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 ...
2
votes
0answers
53 views

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 ...
0
votes
1answer
327 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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1answer
120 views

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 ...