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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Exercise 2.27 Atiyah-Macdonald, absolute flatness

A commutative ring $R$ is absolutely flat if every $R$-module is flat. Prove that the following are equivalent: 1) $R$ is absolutely flat 2) Every principal ideal of $R$ is idempotent 3) Every ...
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Limit of inverse system where morphisms are non-zero products

I have a rather complicated inverse limit which I am trying to compute. I will try to distil the question to its barest form, but if more details are necessary I can supply them. Let $R$ be an ...
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55 views

Tor for graded modules over a graded ring

I am confused about how this Tor is defined. Suppose $R$ is a graded ring, $M,N$ graded modules over $R$. What is $\operatorname{Tor}_{st}^R(M,N)$? I am confused about the subscripts. I realize ...
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Truncation of inverse systems

I'm trying to ascertain when (if ever) it is acceptable to ``truncate'' terms of an inverse system and arrive at an isomorphic limit. To simplify matters, assume that our directed system is $\mathbb ...
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Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
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Has this variation of Hochschild cohomology been studied?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Let $M$ be an abeliean group, and assume that it an $n$-$A$-module. That is: it has $n$ different $A$-module structures, and they are ...
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Exercise from Atiyah about flatness

This is an exercise from Atiyah. Let $N$ be a flat $B$-module, and $B$ a flat $A$-algebra where $A$ is a commutative ring with unit. Then $N$ is flat as $A$-module Any hint ?
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65 views

Derived functor vs. spectral sequence

I heard many times that because of introducing derived category, we can avoid cumbersome spectral sequence. However, I don't quite understand its meaning. Here is a precise example people talking ...
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Cohomology-Homology bilinear form of Seifert surfaces

Let $C_\ast$ be any chain complex of $R$-modules. Then for any $k\in\mathbb{Z}$ we obtain a $R$-bilinear map $$\langle-,-\rangle:H^k\!C_\ast\times H_kC_\ast\longrightarrow R, ...
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How do biadditive bifunctors extend to complexes?

$\newcommand{\Mod}{\operatorname{Mod}}$ Let $A, B$ be two rings, and let $F:\Mod A\times \Mod A \to \Mod B$ be a biadditive bifunctor. I want to extend $F$ naturally to a bifunctor from complexes ...
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Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ...
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Additive functor is exact $\iff$ quasi-ismorphisms preserved?

While reading Weibel's Homological Algebra, on pg. 391 he considers an additive functor $F:\mathcal{A}\to\mathcal{B}$ between abelian categories, and writes "If $F$ is not exact, then the induced ...
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37 views

If a left ideal I of R is an injective R-module then I is a projective R-module. Is the converse true?

I was trying to prove the first statement using some characterizations of injective and projective modules but it did not work out. Could someone drop some hints?
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If a left ideal of R is a direct summand of R, is I an injective R-module?

This was an exercise question I encountered in the book on Homological algebra by Vermani. I am unable to prove it or find a counter example.
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Space with prescribed local homology

Lets be $G_n$ sequence of abelian groups and $G_0 = \mathbb{Z}$. Is there topological space $X$ that local homology groups at every point are those $G_n$ ? ie. $$ \forall x\in X \; \forall ...
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Some projective property of projective resolution

In the proof of lemma 3.2 in this PDF, it said: Let $0 → A → A′ → A′′ → 0$ be a short exact sequence. $P′′_{*}→ A′′$ be a projective resolution. (i.e. $ ··· P_{1}′′→ P_{0}′′ → A′′ → 0$ is exact ...
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Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
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139 views

Universal coefficient theorem with ring coefficients

The universal coefficient theorem for cohomology reads: $$0 \to Ext(H_{n-1}(C), R) \to H^n(C;R) \to Hom(H_n(C), R) \to 0,$$ where $C$ is a chain complex of free abelian groups and $R$ is a ring. It ...
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free resolutions of $\mathbb Z$ in Mod(G)

Lang Algebra, XX.2,3 I'm asked to show that $E_\bullet \cong F_\bullet$ are isomorphic free resolutions of $\mathbb Z$, in Mod(G), where $E_\bullet$ is the standard complex: $E_i$ is the free ...
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64 views

Naturality condition for connecting homomorphisms?

I've been reading about the Mayer-Vietoris sequence, but I don't follow a certain naturality condition. Suppose two spaces can be written as $X=X_1^\circ\cup X_2^\circ$ and $Y=Y_1^\circ\cup ...
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Ring structure on $Ext$ and $Tor$

Wikipedia says that in certain situations, $Ext^\ast_A(R,R)$ becomes a ring, such as when $A$ is an augmented $R$-algebra, but the outline is too sketchy for me to understand. I can't find this in ...
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Conditions to ensure the chain homotopy category $K(\mathcal{A})$ is abelian?

It is known that the chain homotopy category $K(\mathcal{A})$ for an abelian category $\mathcal{A}$ need not be abelian. For example, $K(\mathrm{Ab})$ is not even abelian. Are there any known ...
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Examples of functors

Can anyone please give me examples of: 1.- An exact functor other than taking the Galois group from the category of fields. 2.- A half exact functor. 3.- A contravariant right exact functor. I ...
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Exact sequence of linear spaces

While reading Nigel Higson's book Analytic K-homology i found the result (which was known to me earlier, but I never saw the proof) that the index of the product of two Fredholms operators is the sum ...
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91 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is $1$. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that ...
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60 views

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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51 views

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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62 views

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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76 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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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 $\operatorname{Tor}(\mathbb Z_{p},\mathbb Z_{q})$?

I am self-studying Vick's Homology Theory, and now it is on the topic of free resolutions. Since I am not familiar with it, I have little ideas about how to compute $$\operatorname{Tor}(\mathbb ...
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28 views

$[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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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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69 views

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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37 views

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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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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136 views

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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37 views

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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Global Dimension 1, right annihilator

Let $R$ be a ring with right global dimension 1. Then I am trying to show that for any $a\in R$ if we define the right annihlator $r(a)=\{x\in R|ax=0\}$ then we have that $\exists e\in R$ such that ...