# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
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### 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 ...
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### 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 ...
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### 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.
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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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### 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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### 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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### 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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### 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 ...
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i have this from Hatcher's book "Algebric topology" And i don't understand why $\displaystyle \partial P(\sigma)=\sum_{j\leq i}(-1)^i(-1)^j F\circ (\sigma\times ... 0answers 59 views ### Question on Singular homology i have this example : The homology of the space$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$we have ... 1answer 89 views ### Prove that a module is projective or not Let$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 ... 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 $$...\rightarrow ... 1answer 30 views ### A functor on commutative diagrams As suggested by Daniel Rust I'll pose this as a separate question. Let C be a category. Denote by Ar(C) the following category: an object in Ar(C) is a morphism X_1 \rightarrow X_2 in C. ... 1answer 29 views ### Morphisms as Objects, Comm. Diagrams etc. Let C be a category. Denote by Ar(C) the following category: an object in Ar(C) is a morphism X_1 \rightarrow X_2 in C. A morphism in Ar(C) from X_1 \rightarrow X_2 to Y_1 ... 3answers 18 views ### Choose a set that makes a sequence exact What choice of X \in \mathrm{Ab} and maps between the groups would make the following sequence exact?$$0 \rightarrow \mathbb{Z}/3 \rightarrow X \rightarrow\mathbb{Z}/2 \rightarrow 0$$I'm thinking ... 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, ... 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, ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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. ... 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 ... 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 ... 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} ... 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 ... 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 ... 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 ...
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 ...