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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The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
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Show that Q, as a Z module, is a direct summand in a direct product of copies of Q/Z.

Prove:Q, as a Z module, is a direct summand in a direct product of copies of Q/Z. This is a problem from P.J.Hilton&Stammbach's Homological Algebra. If this is true, then there exists a ...
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Why we can consider both modules as modules over $R_{(p)}$? (Bruns and Herzog, Theorem 1.5.9)

I'm reading Bruns-Herzog's book Cohen Macaulay rings and have a probably elementary question. Why we may consider both modules as modules over $R_{(p)}$ in this theorem? ... i know that ...
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Injective dimension and depth

Here is Bruns and Herzog's book Cohen-Macaulay Rings, Theorem 3.1.17: Let $R$ be a Noetherian local ring, and $M$ a finite $R$-module of finite injective dimension. Then $\operatorname{inj\ ...
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Tor sheaves on schemes

I was trying to understand the definition of "Tor sheaves", but since it is defined in the derived category of sheaves of $\mathcal{O}_X$-modules and since I am not acquainted with derived categories ...
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Why is $\mathsf{HTAG}$ (Hausdorff, Topological, Abelian Groups) preabelian?

The category of Hausdorff topological abelian groups are commonly cited as an example of a category which is preabelian, but not abelian. I think one reason that is is not abelian comes from the ...
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S-modules and Schur functors

I am reading the book "Algebraic Operads" by Loday and Vallette. (I will refer to their version 0.999 here : http://math.unice.fr/~brunov/Operads.pdf) In Chapter 5, they define an $\mathbb{S}$-module ...
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Direct proof for the independence of $\operatorname{Tor}$

It is known that $\operatorname{Tor}$ is independent of the choice of the resolution. More specifically, I am trying to do the exercise 1 (c) of Vick's homology theory. The author gives the ...
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60 views

History of five lemma

I am interested in the history of five lemma. Who was first to prove it and What was the purpose of proving it ? http://en.wikipedia.org/wiki/Five_lemma
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37 views

A Sort of Exact Sequence

I have not given a lot of thought to this question: It may be very easy or very hard or somewhere in between. Suppose we have a sequence of modules and morphisms which looks like $ \ldots \to A_1 ...
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How can I compute $Tor\left(Z_{p},Z_{q}\right)$?

I am self-studying Vick's Homology theory, and now it is on the topic of free resolution. Since I am not familiar with it, I have little ideas about how to compute $$Tor\left(Z_{p},Z_{q}\right)$$ ...
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$[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets

how to show: if $F \to E \to B $ is a fibration then for any space $X$ the sequence $[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets. any hints, thanx.
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31 views

Example of excision in Hochschild homology

The excision theorem for Hochschild homology introduced by Wodzicki seems like a very powerful tool (as scision was hyper-useful in topology). However, I cannot actually seem to think of a result ...
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Homology of the fixed points of the singular complex of a G-space

Suppose $X$ is a topological space and $G$ a finite group acting on it. We can form the singular complex $C_\bullet(X),$ and then taking homology gives singular homology: $H_*(X) = h_* C_\bullet(X).$ ...
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The space $\Delta^n$ with all faces of the same dimension.

If the space $A$ is obtained from $\Delta^n$ by identifying all faces of the same dimension; What is a $\Delta$-complex structure on the space $A$? And how can you compute the Simplicial Homology ...
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Is any right exact sequence of modules induced by free modules?

Let $R$ be a ring and let $M \to N \to K \to 0$ be an exact sequence of $R$-modules. Is there an exact sequence of free modules $A \to B \to C \to 0$ and a commutative diagram $$\begin{array}{c} M ...
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Tor of submodule

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between: $Tor_{A^e}$ and $Tor_{B^e}$?
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Coproducts and Hochschild

I $\{X_i\}$ is a small family of associative $\mathbb{C}$-algebras and $X$ is their free product. Then I have two questions: 1) Why is $X$ their coproduct? 2) Is the Hochschild homology of X ...
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Natural map of extension groups

Let $\Lambda$ be a cocommutative Hopf algebra over a commutative ring $R$. For two left $\Lambda$-modules $M$ and $N$, interpret $\mathrm{Ext}_{\Lambda}^n(M,N)$ as the set of equivalence classes of ...
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36 views

Relative singular chains basis

If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
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Book for advanced homological algebra

I already read the books: 1.- An introduction to homological algebra - Rotman (the two versions of it) 2.- An introduction to homological algebra - Weibel 3.- A course on homological algebra - ...
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Why is Hochschild cohomology just a group and mot a module?

This is probably a very basic question in Hochschild theory. Let $k$ be a field, and let $A$ be a $k$-algebra (which is not commutative). If $M$ is an $A$-bimodule, then the $n$-th Hochschild ...
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Definition/existence/uniqueness of a minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm trying to understand the following discussion on page $32$ in which he introduces the notion of a minimal projective ...
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Hom($P$, $R$) $\neq 0 $ if $P$ is a nonzero projective left $R$-module (Rotman)

I've found this exercise, number $3.11$ from Introduction to homological algebra. Prove that $\operatorname{Hom}(P, R) \neq 0 $ if $P$ is a nonzero projective left $R$-module. Any hint?
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$\operatorname{inj.dim}_R N= \operatorname{inj.dim}_R \widehat{N}$?

$(R,m)$ is a local ring. For an $R$-module $N$, we know that $\operatorname{inj.dim}_R N= \operatorname{inj.dim}_\widehat{R} \widehat{N}$. Is it true that $\operatorname{inj.dim}_R N= ...
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Question about the definition of homology

i have this paragraphe: Can someone explaine me what it means ? if i understand $H_n$ measure the numbers of holes with dimension $n$ but what about $H_0$ what is the relation between the holes of ...
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40 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
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Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals?

Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals ? So by dual, I mean the linear maps on $H_{k}$. I need this to understand the Poincare duality i.e. $H_{k}\cong ...
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Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...
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Local Cohomology - Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings

This question arises in the context of Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings. Let $(R,m)$ be a local complete Cohen-Macaulay ring of dimension $d$. Denote by $H_m^d(-),\omega_R$ ...
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Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
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Question about relative singular homology groups

I know that the sphere $S^{\infty}$ is contractible, but why if $H$ is a Hilbert space then we have $$H_q(H,S^{\infty})=0, q\in \mathbb{N}?$$ Please help me Thank you
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Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
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Is it possible to compute homology groups of a space given the Pontryagin ring?

Or similarly, given the cohomology ring of a space, is it possible to compute its cohomology groups? I'm mainly interested in integer and mod 2 homology and cohomology.
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Bounds dimension, scheme and projective dimension

Is the dimension of a (commutative unital associative) algebra always bounded above by its protective (injective) dimension? If not is it always bounded above by its global dimension?
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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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Exercise from Assem-Simson-Skowronski

I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory. The exercise in question is from chapter IV. So, 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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If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also

EDIT: The original title was: If a morphism of diagrams of complexes is composed of quasi-isomorphisms, is the induced arrow a quasi-isomorphism? Let $J$ be a small category and $C$ be the category ...
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existence of a cofiber sequence

can anyone help me with this problem. thanx. Show that there are cofiber sequence $S^{n+3} \to S^{n+2} \to \sum^{n}\mathbb{C}P^2$ for each $n \in \mathbb{Z}^+$. Conclude that a space of the form ...
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Does $\operatorname{id} M =\dim R$ hold for finite modules of finite injective dimension?

When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for ...
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Doubt about the tensor product

Suppose $M$ is an abelian group and $F(X)$ is the free abelian group over $X$. Is it true that any element of $M\otimes F(X)$ can be written as a finite sum $$m_{1}\otimes x_{1}+ \cdots+m_{n}\otimes ...
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Having trouble understanding the Tor functor

I am having trouble understanding the Tor functor as presented in Dummit and Foote. Given $\dotsb\to P_n\to P_{n-1}\to\dotsb\to P_0\to B\to 0$ as a projective resolution with homomorphisms ...
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Problem about tensor products and modules over a group algebra

Can anyone help me with hints for these problems? I would appreciate it a lot. 1) Supose $G$ is a group and $k$ is a field. If $U,V$ and $W$ are $kG$-modules then $\operatorname{Hom}_{kG}(U\otimes ...
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Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}_n^R(A,B)$ is also torsion.

Given an integral domain $R$, and a left torsion $R$-module $A$ (i.e. $\forall{a}\in A,\exists{r}\in R$ such that $ra=0$) how would you show that $\mathrm{Tor}_n^R(A,B)$ is also a torsion $R$-module?
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Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
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An easy infinite free resolution

I'm doing exercise 1.23 on Eisenbud's Commutative algebra, and I have the following situation: let $k$ be a field and $R = k[x]/(x^n)$. They ask for a free resolution of $R/(x^m)$, for some $m \leq ...
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Acyclic resolution but not projective

Suppose $\mathfrak{C}$ is an abelian category which does not have enough projectives and we're interested in computing the right derived functors of some covariant functor $F$. If however, every ...
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Does additivity of (equivariant) cohomology hold at the algebra level?

The additivity property of many (co)homology theories is that if $X = \bigsqcup_{i \in I} X_i$ then $H^*(X) = \bigoplus_{i\in I} H^*(X_i)$. This is usually either an axiom of the theory, can be proven ...