1
vote
0answers
30 views

The Betti numbers

can someone explain me this definition from :http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the ...
2
votes
1answer
35 views

If R and S are artinian and finite dimensional algebras respectively, then the tensor product of them is artinian.

Let $R$ be an artinian algebra and $S$ be a finite dimensional algebra over the field $k$. How can i show that $R\otimes_kS$ is artinian? I know that $S$ is also artinian since it is finite ...
0
votes
0answers
47 views

Describe the kernel and the fibers of $\phi$ geometrically (as subsets of the plane).

Define $\phi : \mathbb{C}^{\times} \mapsto \mathbb{R}^{\times}$ by $\phi(a+bi) = a^2 + b^2$. Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of ...
4
votes
1answer
248 views

Mayer-Vietoris sequence for local cohomology

Update 7:35pm UTC 3/23/14: I've reposted this quesion on MathOverflow here. As an assignment in my commutative algebra class, I need to prove the Mayer-Vietoris sequence for local cohomology: Let ...
0
votes
1answer
33 views

Homotopy between two homomorphisms and homology

If I have two chain complexes $C$ and $D$ and I suppose that there is a homotopy between $\phi, \psi:C \rightarrow D$ (i.e there is a sequence of homomorphisms $(K_n: C_n\rightarrow D_{n+1})$ such ...
0
votes
2answers
58 views

short exact sequence

Let $0 \rightarrow L \stackrel{\alpha}\rightarrow M\stackrel{\beta}\rightarrow N \rightarrow 0$ be an exact sequence, and $M_1$, $M_2$ be two submodules of $M$; then whether the follwing implications ...
-1
votes
1answer
69 views

Relative homology of ball and sphere

What is the result of $H_k(B^n,S^{n-1}; \mathbb{A })$ and in any book can i found the proof ? And what about $H_n(S^{n};\mathbb{A})$ (sigular homology of the sphere )?? Please help me. Thank you
1
vote
1answer
58 views

Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
4
votes
1answer
91 views

Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
1
vote
1answer
31 views

Small question on relative holology

if $Y\subset X$ , what is $\ker \delta$ such that $\delta: H_k(X,Y)\rightarrow H_k(Y)$ ? is it $\ker \delta = H_k(X,Y)$ ? $\delta$ is the usual connecting homomorphism from the long exact sequence ...
0
votes
1answer
66 views

Homology and Exact Sequence

I have this exact sequence: $$0\stackrel{f}{\rightarrow} H_k(X,C)\stackrel{g}{\rightarrow} H_k(X,A)\stackrel{h}{\rightarrow} 0$$ Can I say that $H_k(X,A)=H_k(X,C)$ and why? Please; Thank you.
1
vote
0answers
42 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 ...
3
votes
1answer
67 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
69 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.
1
vote
1answer
56 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, ...
0
votes
1answer
113 views

decomposition of cokernel

Lets work over $\mathbb Z$ and represent maps $f:\mathbb Z^l \longrightarrow \mathbb Z^n$ as matrices. For the following matrices write an equivalent diagonal matrix. I want to write a decomposition ...
0
votes
1answer
66 views

Question on “Homotopy invariance”

i have this from Hatcher's book "Algebric topology" And i don't understand why $\displaystyle \partial P(\sigma)=\sum_{j\leq i}(-1)^i(-1)^j F\circ (\sigma\times ...
1
vote
0answers
59 views

Question on Singular homology

i have this example : The homology of the space $X=\lbrace x \rbrace$ . for all $p\geq 0$, there is a unique singular p-simplex $\sigma_p:\Delta_p\rightarrow X$, and for $p>0$ we have ...
0
votes
1answer
89 views

Prove that a module is projective or not

Let $R=\left(\begin{array}{cc}\mathbb{Q}&\mathbb{Q}\\0&\mathbb{Q}\end{array}\right), J=\left(\begin{array}{cc}0&\mathbb{Q}\\0&0\end{array}\right)$. Prove that $R/J$ is not a ...
1
vote
0answers
61 views

Question on Snake lemma

we have Short exact sequence of chain complexe $0\rightarrow C\xrightarrow[]{f}D\xrightarrow[]{g}E\rightarrow 0$ i want to prove that there existe a longue exact sequence of modules $$...\rightarrow ...
2
votes
1answer
30 views

A functor on commutative diagrams

As suggested by Daniel Rust I'll pose this as a separate question. Let $C$ be a category. Denote by $Ar(C)$ the following category: an object in $Ar(C)$ is a morphism $X_1 \rightarrow X_2$ in $C$. ...
2
votes
1answer
29 views

Morphisms as Objects, Comm. Diagrams etc.

Let $C$ be a category. Denote by $Ar(C)$ the following category: an object in $Ar(C)$ is a morphism $X_1 \rightarrow X_2$ in $C$. A morphism in $Ar(C)$ from $X_1 \rightarrow X_2$ to $Y_1 ...
0
votes
3answers
18 views

Choose a set that makes a sequence exact

What choice of $X \in \mathrm{Ab}$ and maps between the groups would make the following sequence exact? $$0 \rightarrow \mathbb{Z}/3 \rightarrow X \rightarrow\mathbb{Z}/2 \rightarrow 0$$ I'm thinking ...
0
votes
1answer
50 views

Diagram chasing, and more

1) Assume that $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \rightarrow 0$ and $0 \rightarrow C_1 \rightarrow C_2 \rightarrow D \rightarrow 0$ are exact, $i=1,2$. Show, using a diagram chase, ...
0
votes
1answer
57 views

Show that $C$ is a chain complex

I suppose it's a common exam question to show that a certain sequence actually is a chain complex. What is it that has be shown, minimally? A chain complex is a sequence of modules and module maps, ...
2
votes
1answer
67 views

Map between two direct limits

Let $\{ M_i, ϕ_j^i\}_{i\in I}$ be a direct system of $R$-modules over a direct index set $I$. Show that there exists a direct system $\{P_i,\psi_j^i\}_{i\in I}$ of projective $R$-modules and a ...
3
votes
2answers
94 views

Show that $[l_1 \cdot l_2 \cdot l_3 ] = [l_1 + l_2 + l_3] \in H_1(X)$ The first Homology group of X

Let $l_1$ , $l_2$ and $l_3$ be three paths in X with $l_1 (0) = l_3 (1)$, $l_1 (1) = l_2 (0)$ and $l_2 (1) = l_3 (0)$. Define the loop $l = l_1 \cdot l_2 \cdot l_3 $ (based at $l_1 (0)$). Show that ...
2
votes
1answer
81 views

How to prove the global dimension of the polynomial ring $F[x_1,…,x_n]$ is $n$?

I am trying to prove that the global dimension of the polynomial ring $F[x_1,\dots,x_n]$, where $F$ is a field , is exactly $n$. By Koszul complex, I know its global dimension is greater than or ...
4
votes
1answer
63 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
1
vote
1answer
60 views

Change of base rings for exterior algebra

This may be not a good question. But I really get tough. I am studying basic knowledge about homological algebras and I am dealing with Koszul's Complex and Hilbert's Syzygy Theorem. At the very ...
5
votes
2answers
234 views

$A$ PID, $M$ flat (i.e., torsion-free). Then $\operatorname{Ext}_A^1(M,N)$ is injective, for all $N$.

Let $A$ be a PID and $M$ a flat (i.e., torsion-free) $A$-module. Then, for every $A$-module $N$, $\text{Ext}_A^1(M, N)$ is injective in $A\text{-}\mathbf{Mod}$. It is easy when $M$ is finitely ...
9
votes
0answers
209 views

cohomological proof of Maschke's theorem

I have been working on the following problem.. I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
3
votes
1answer
126 views

Group extension: $H^n(\mathbb{Z},M)$

Let $M$ be any $\mathbb{Z}$-module. Find $H^n(\mathbb{Z}, M)$ for all $n$. Use different approaches for the case $n=2$. First approach: because $\mathbb{Z}$ is a free $\mathbb{Z}$-module, it is ...
4
votes
1answer
138 views

Projective dimenson of tensor product

I've been struggeling for some time with the following problem Let $k$ be a field and $A$ and $B$ two $k$-algebras. We can then view the tensor product $A\otimes_k B$ as a $k$-algebra by $(a_1\otimes ...
5
votes
1answer
187 views

Bicartesian squares of abelian groups

A commuting square is called bicartesian if it is both a pullback and a pushout. I would like to show that given any diagram of abelian groups $A \stackrel{f}{\twoheadrightarrow} ...
2
votes
3answers
202 views

Quasi-isomorphism of Complexes

Let's $(K^{\bullet}, d^{\bullet})$ is the complex over field $A$ (i.e. all $K^{i}$ are vector spaces over this field) and $(L^{\bullet}, {\delta}^{\bullet})$ such that $$L^{i}=H^{i}(K)~\text{and ...
4
votes
1answer
247 views

Koszul Complex Homology

I'm attempting to understand Eisenbud's proof that: If $x_1,x_2,\ldots,x_i$ is an $M$-sequence, then $H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M$. Here ...
2
votes
2answers
184 views

$\operatorname{Func}(J,Ab)$ has enough injectives.

I am trying to show that the functor category $\operatorname{Func}(J,Ab)$ has enough injectives (meaning that for each $F\in \operatorname{Func}(J,Ab)$ there is an injective object $I\in ...
17
votes
2answers
2k views

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
4
votes
0answers
299 views

The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...