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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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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30 views

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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31 views

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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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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65 views

All local cohomology modules being zero

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$? The converse of ...
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How does Local Cohomology detect UFD?

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a ...
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217 views

Vanishing of a local cohomology module

I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ It is well known $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, therefore $$\operatorname{Supp} ...
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33 views

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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Endomorphism rings, Book [on hold]

anyone know a book that discusses the concept of endomorphism ring (End(M)) where M in a module. a detailed study of endomorphism ring. Thanks!!
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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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46 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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430 views

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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35 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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31 views

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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1answer
66 views

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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366 views

Motivation for Koszul complex

Koszul complex is important for homological theory of commutative rings. However, it's hard to guess where it came from. What was the motivation for Koszul complex?
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37 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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54 views

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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49 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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63 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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How to calculate $Ext(M,N)$?

I am confused about the calculation of $\text{Ext}(M,N)$. If $N$ is a fixed module and if we consider the projective resolution $$\cdots \to C_1 \to C_0 \to M \to 0,$$ then $\text{Ext}_n(M,N)$ is the ...
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232 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 ...
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29 views

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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25 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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32 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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80 views

For $R$-modules $M,N$, what are sufficient conditions for $\operatorname{Supp}(M\otimes_R N)\subseteq \operatorname{Supp}(\operatorname{Hom}_R(M,N))$?

Let $R$ be a commutative ring, $M$ and $N$ be finitely generated $R$-modules. What additional conditions will ensure $\operatorname{Supp}(M\otimes_R N)\subseteq ...
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1answer
49 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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1answer
64 views

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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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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35 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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380 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
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1answer
31 views

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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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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38 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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114 views

$\operatorname{inj.dim}_R N= \operatorname{inj.dim}_R \widehat{N}$?

$(R,m)$ 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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Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
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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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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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55 views

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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1answer
213 views

Projective resolution of tensor product

Let $M,N$ are $R$ modules and $P^\bullet, Q^\bullet$ are their projective resolutions. Can we obtain projective resolution $M\otimes N$ using $P^\bullet, Q^\bullet$. If i understand correctly homology ...
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Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...
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54 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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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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Under what conditions are the resolutions of two modules subcomplexes of the resolution of the tensor product?

I have that $S=k[x_1, \dots, x_n]$, $I$ is a lattice ideal, and $J$ is a monomial ideal. I am interested in the resolution of $S/(I+J)\cong S/I\otimes S/J$. In particular, I am interested in knowing ...
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2answers
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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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2answers
42 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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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 ...